a tunable and narrow–linewidth diﬀerence–frequency ... · nonlinear crystals, having...

Universita degli Studi di Firenze

Facolta di Scienze Matematiche Fisiche e Naturali

Dipartimento di Fisica

Tesi di Dottorato di Ricerca in Fisica (XII Ciclo):

A tunable and narrow–linewidthdifference–frequency spectrometer

around 4.25 µm for CO2

high–resolution spectroscopy

Candidato:

Dott. Davide Mazzotti

Relatore: Co–relatore:

Prof. Massimo Inguscio Dott. Paolo De Natale

Relatore esterno: Coordinatrice del Dottorato:

Prof. Francesco Saverio Pavone Prof.ssa Annamaria Cartacci

Dicembre 1999

Contents

1 Introduction 1

2 Theoretical background 7

2.1 Three–wave interactions in nonlinear uniaxial crystals . . . . . 7

2.2 Formalism of wave propagation . . . . . . . . . . . . . . . . . . 9

2.2.1 Optical difference–frequency generation . . . . . . . . . 12

2.2.2 Phase–matching in difference–frequency generation . . 13

2.2.3 Difference–frequency generation with focused

Gaussian beams . . . . . . . . . . . . . . . . . . . . . . 14

2.3 Quasi–phase–matching . . . . . . . . . . . . . . . . . . . . . . 17

2.4 The CO2 molecule . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4.1 Rotational energy levels . . . . . . . . . . . . . . . . . 23

2.4.2 Vibrational energy levels . . . . . . . . . . . . . . . . . 24

3 Experimental set–up 29

3.1 Laser sources . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.1.1 The master diode laser . . . . . . . . . . . . . . . . . . 30

3.1.2 The slave diode laser . . . . . . . . . . . . . . . . . . . 32

3.1.3 The injection–locking scheme . . . . . . . . . . . . . . 32

i

ii CONTENTS

3.1.4 The Nd:YAG laser . . . . . . . . . . . . . . . . . . . . 35

3.2 The nonlinear crystal . . . . . . . . . . . . . . . . . . . . . . . 37

3.2.1 LiNbO3 as nonlinear material . . . . . . . . . . . . . . 39

3.2.2 Fabrication process . . . . . . . . . . . . . . . . . . . . 41

3.2.3 Temperature stabilization . . . . . . . . . . . . . . . . 43

3.2.4 Bandwidths . . . . . . . . . . . . . . . . . . . . . . . . 46

3.3 Optical scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.4 Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.4.1 Preamplifier . . . . . . . . . . . . . . . . . . . . . . . . 53

3.4.2 Detection noise . . . . . . . . . . . . . . . . . . . . . . 54

3.4.3 Responsivity calibration . . . . . . . . . . . . . . . . . 56

3.5 Vacuum apparatus . . . . . . . . . . . . . . . . . . . . . . . . 60

3.5.1 Pumping system . . . . . . . . . . . . . . . . . . . . . 61

3.5.2 Vacuum gauges . . . . . . . . . . . . . . . . . . . . . . 62

3.5.3 Gas cell . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.5.4 Flushing box . . . . . . . . . . . . . . . . . . . . . . . 63

4 Measurements 65

4.1 Noise analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.1.1 Frequency noise . . . . . . . . . . . . . . . . . . . . . . 65

4.1.2 Intensity noise . . . . . . . . . . . . . . . . . . . . . . . 66

4.2 Efficiency analysis . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.3 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . 71

4.4 Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.4.1 Wavelength measurements . . . . . . . . . . . . . . . . 73

4.4.2 Direct–absorption spectroscopy . . . . . . . . . . . . . 73

CONTENTS iii

4.4.3 Saturated–absorption spectroscopy . . . . . . . . . . . 81

4.5 Beam–shape analysis . . . . . . . . . . . . . . . . . . . . . . . 92

5 Summary 103

A Optics of uniaxial crystals 105

A.1 Phase–matching conditions . . . . . . . . . . . . . . . . . . . . 107

A.2 Types of phase–matching . . . . . . . . . . . . . . . . . . . . . 108

A.3 Crystal symmetry and effective nonlinearity . . . . . . . . . . 109

B Direct–absorption lineshape 115

iv CONTENTS

List of Figures

2.1 Evolution of the difference–frequency phasor . . . . . . . . . . 19

2.2 Normal vibrational modes of CO2 molecule . . . . . . . . . . . 21

2.3 CO2 spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.1 Extended–cavity mounting of the master diode laser . . . . . . 31

3.2 Optical scheme for the injection locking . . . . . . . . . . . . . 34

3.3 Nd:YAG ring cavity photo . . . . . . . . . . . . . . . . . . . . 35

3.4 Intensity noise spectrum of the Nd:YAG laser . . . . . . . . . 37

3.5 Nd:YAG wavelength calibration curves . . . . . . . . . . . . . 38

3.6 Schematic drawing of the PPLN crystal . . . . . . . . . . . . . 39

3.7 Transparency range for three different nonlinear crystals . . . 40

3.8 Temperature tuning of the PPLN crystal . . . . . . . . . . . . 46

3.9 Angular tuning of the PPLN crystal . . . . . . . . . . . . . . . 47

3.10 Optical set–up for DFG . . . . . . . . . . . . . . . . . . . . . 50

3.11 Equivalent circuit for the InSb detector. . . . . . . . . . . . . 53

3.12 Preamplifier circuit . . . . . . . . . . . . . . . . . . . . . . . . 54

3.13 Calibration of the infrared detector . . . . . . . . . . . . . . . 56

3.14 Black–body signal . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.15 Vacuum apparatus . . . . . . . . . . . . . . . . . . . . . . . . 61

v

vi LIST OF FIGURES

3.16 Schematic drawing of the gas cell . . . . . . . . . . . . . . . . 63

3.17 Wooden box for nitrogen flow . . . . . . . . . . . . . . . . . . 64

4.1 Frequency fluctuations . . . . . . . . . . . . . . . . . . . . . . 66

4.2 Intensity noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.3 Intensity noise fit . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.4 Background intensity noise . . . . . . . . . . . . . . . . . . . . 70

4.5 R(16) air–broadened CO2 transition . . . . . . . . . . . . . . . 74

4.6 R(58) air–broadened CO2 transition . . . . . . . . . . . . . . . 75

4.7 R(58) self-broadened CO2 transition . . . . . . . . . . . . . . . 76

4.8 R(82) self–broadened CO2 transition . . . . . . . . . . . . . . 77

4.9 CO2 spectrum around 2296.0 cm−1 . . . . . . . . . . . . . . . 78

4.10 CO2 spectrum around 2304.7 cm−1 . . . . . . . . . . . . . . . 79

4.11 Pressure broadening for R(64) CO2 transition . . . . . . . . . 80



4.14 Fit of R(64) pressure–broadening coefficient . . . . . . . . . . 83



4.17 Experimental set–up for saturation spectroscopy . . . . . . . . 85

4.18 Saturated–absorption dip on the Doppler profile (first derivative) 86

4.19 Zoomed saturated–absorption dip (first derivative) . . . . . . . 87

4.20 First–derivative slope dependence on the modulation depth. . 88

4.21 Saturated–absorption dip (third derivative) . . . . . . . . . . . 89

4.22 Third–derivative slope dependence on the modulation depth. . 90

4.23 Third–derivative slope dependence on the gas pressure. . . . . 90

LIST OF FIGURES vii

4.24 Recorded beam intensity profiles for θi = 0 mrad . . . . . . . . 93

4.25 Focusing function . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.26 Calculated beam intensity profiles for θi = 0 mrad . . . . . . . 97

4.27 Calculated vs. recorded beam intensity profiles . . . . . . . . . 98

4.28 Calculated beam intensity profiles for θi = 5 mrad . . . . . . . 99

4.29 Calculated beam intensity profiles for θi = 10 mrad . . . . . . 100

4.30 IR power generated at different tilting angles . . . . . . . . . . 101

viii LIST OF FIGURES

List of Tables

2.1 Character table of the symmetry point group D∞h. . . . . . . 21

2.2 CO2 normal vibrations . . . . . . . . . . . . . . . . . . . . . . 26

3.1 Sellmeier’s coefficients . . . . . . . . . . . . . . . . . . . . . . 44

3.2 Numerical calculation . . . . . . . . . . . . . . . . . . . . . . . 60

4.1 CO2 detection sensitivities . . . . . . . . . . . . . . . . . . . . 72

A.1 Expression for deff in uniaxial crystals . . . . . . . . . . . . . 113

ix

Chapter 1

Introduction

Spectral coverage of regions where a continuous tunability of laser sources

was not available, has driven most of the research in nonlinear optics. As

a consequence, one of the main aims for the development of nonlinear de-

vices has been the efficient mixing and conversion of input frequencies. In

the microwave region, the availability of diode rectifiers and other strongly

nonlinear elements permit straightforward design of efficient “lumped” mix-

ers, smaller than a wavelength in size. At optical frequencies, by contrast,

nonlinear responses are quite weak, so that efficient mixing requires “dis-

tributed” devices many wavelengths long. Many of the practical problems

associated with the optical frequency mixing result from the distributed na-

ture of the mixing process. In particular, the difference in phase velocities of

the interacting waves of different frequencies in a nonlinear medium produces

a phase difference that accumulates along the length of the device and can

significantly limit the efficiency of the mixing process. Thus, special steps

have to be taken in order to “phase–match” nonlinear processes.

Almost 40 years after the first demonstration of frequency conversion from

1

2 CHAPTER 1. INTRODUCTION

the red to the blue [1], nonlinear optical devices have widespread applications

in fields as diverse as laser fusion, biomedical instrumentation, femtosecond

spectroscopy, and precision metrology. Nonlinear optical technology is enor-

mously growing, driven by improvements in solid–state and nonlinear optical

material technology.

The availability of high–power semiconductor diode lasers [2] has opened

the door to rapid advances in diode–pumped solid–state laser sources such

as Nd:YAG, Nd:YLF, Er:YAG, Tm-Ho:YAG, and Yb:YAG. These lasers

operate at various discrete near–infrared (NIR) wavelengths between 946

and 2010 nm in continuous wave (CW), Q–switched, or mode–locked oper-

ation. Highly stable single–frequency diode–pumped monolithic YAG lasers

[3] provide the frequency and amplitude stability as well as the power level

required for highly efficient nonlinear frequency conversion. Single–mode

GaAlAs diode lasers followed by high–power broad–area or tapered semicon-

ductor amplifiers [4, 5] also serve as intense pump laser sources for nonlinear

frequency conversion in the visible and NIR wavelength range. The large

gain bandwidth of Ti:Al2O3 lasers, whose output can be tuned from 700 to

1060 nm, makes it possible to generate mode–locked pulses as short as 10 fs [6]

to pump nonlinear optical devices for applications in ultrafast time–resolved

spectroscopy.

New and improved crystalline nonlinear materials, such as BBO, LBO,

KTP, KNbO3, LiNbO3, AgGaS2, and AgGaSe2, which all meet requirements

for efficient nonlinear optical frequency conversions, can be used to generate

virtually any wavelength from the ultraviolet (UV) to the far–infrared (FIR).

In latest years the availability of domain–engineered, quasi–phase–matched

3

nonlinear crystals, having increased frequency conversion efficiency, has rep-

resented a breakthrough for the realization of CW tunable sources. By using

quasi–phase–matching (QPM), virtually any nonlinear process can be phase–

matched throughout the entire transparency range of the nonlinear optical

crystal. The nonlinear frequency conversion efficiency can be significantly

enhanced by introducing the nonlinear medium into the pump laser cavity

[7, 8, 9] or a passive optical buildup cavity that resonates the interacting light

waves [10, 11, 12, 13, 14] or by using guided–wave nonlinear optical devices

[15, 16].

In the 3 ÷ 11 µm region the fundamental ro–vibrational transitions of

most molecules are found, which have linestrengths several orders of mag-

nitude larger than overtone transitions. If suitable IR sources are available,

the IR region can be the best choice to maximize the sensitivity for molec-

ular detection. The other requirement for high sensitivity is to approach a

quantum–noise limited regime for the spectrometer. Since QPM crystals can

offer an efficiency about 20 times larger than birefringent devices, difference–

frequency generation (DFG) can be done in single–pass geometry to produce

IR powers in the microwatt range, which is sufficient for high sensitivity

detection of molecular spectra.

Different strategies have been followed for what concerns pump laser

sources:

• a pair of different solid state lasers is the best choice to get the max-

imum power of generated IR radiation, but narrow tunability is the

other side of the coin [17];

• vice versa, a pair of different semiconductor diode lasers is the best


choice to get the maximum tunability of generated IR radiation, but

relatively low power [18, 19];

• combination of a solid state laser and a semiconductor diode laser can

be used to provide an intermediate situation with good power and

tunability of the generated IR radiation [20, 21, 22];

• combination of a solid state laser and an external–cavity diode–laser

(ECDL) can be used to get wider reliable tuning and narrow linewidth

operation, at the expense of a lower output power [23, 24];

• one (in combination with a solid state laser) or two semiconductor

diode lasers/ECDL power–enhanced by fiber amplifiers [25, 26, 27, 28]

or semiconductor amplifiers [29, 30] can be used for increased output

powers.

Presently, fiber amplifiers have a wavelength range limited to the 1.5 µm te-

lecommunication band (Erbium/Ytterbium–doped fibers) or the 0.9÷1.1 µm

interval (Ytterbium–doped fibers) [31].

The alternative we thought up is a pumping scheme based on a CW

Nd:YAG laser and a diode laser injection–locked by an ECDL. This lets us

achieve a narrow linewidth and a wide tunability in the IR. In this the-

sis a careful analysis of the noise for such a source is presented and the

sensitivity limits of present day DFG spectrometers are discussed. Detec-

tion sensitivities as high as a few parts per billion (ppb) result from the

very high oscillator strength typical of fundamental ro–vibrational bands in

the IR [24]. Above threshold optical parametric oscillators (OPOs) have

also started to prove useful for recording Doppler–limited absorption [32], as

5

well as photo–acoustic spectra [33]. The large molecular dipole moments for

these IR transitions also make them suitable candidates for observing narrow

saturated–absorption dips. The required saturation intensity is proportional

to the inverse squared dipole moment. Considering that pump lasers used

for frequency conversion generally emit in the visible or near IR, where many

frequency standards already exist, narrow lineshapes can help bridging the

gap to the mid IR. Hence, a grid of accurate frequency standards in a very

wide range, extending up to 4.5 µm, could be created, with implications for

metrology, molecular spectroscopy and fundamental physics.

Chapter 2

Theoretical background

2.1 Three–wave interactions in nonlinear u-

niaxial crystals

Conversion of a light–wave frequency (multiplication, division, mixing) is

possible in nonlinear optical crystals for which the refraction index n is func-

tion of the electric field strength vector E of the light wave

n(E) = n0 + n1E + n2E2 + . . . (2.1)

where n0 is the refractive index in absence of the electric field, and n1, n2,

and so on are the coefficients of the series expansion of n(E).

In nonlinear optics a vector of dielectric polarization P (dipole moment

of unit volume of the matter) is introduced. It is related to the field E by

the matter equation

P (E) = εoχ(E)E = εo

[χ(1)E + χ(2)E2 + χ(3)E3 + . . .

](2.2)

where χ(1) is the linear dielectric susceptibility, and χ(2), χ(3), and so on are

7

8 CHAPTER 2. THEORETICAL BACKGROUND

the nonlinear dielectric susceptibility coefficients.

The square nonlinearity takes place (χ(2) 6= 0) only in acentric crystals, i.e.

in crystals without symmetry center; in crystals as well as in isotropic matter

χ(2) ≡ 0. On the contrary, the cubic nonlinearity exists in all crystalline and

isotropic materials.

Propagation of two monochromatic waves with frequencies ω1 and ω2 in

crystals with square nonlinearity gives rise to new light waves with com-

bination frequencies ω3,4 = ω1 ± ω2; the sign plus corresponds to sum–

frequency, the sign minus to difference–frequency (three wave interaction).

Sum–frequency generation (SFG) is frequently used for conversion of long–

wave radiation, for instance, infrared (IR) radiation, to short–wave radiation,

namely ultraviolet (UV) or visible light. Difference–frequency generation

(DFG) is used for conversion of short–wave radiation to long–wave radia-

tion.

At ω1 = ω2 we obtain two special cases of conversion, namely, second–

harmonic generation (SHG) as a special case of SFG, ω3 = 2ω1, and optical

rectification (OR) as a special case of DFG, ω4 = 0.

The effect of optical parametric oscillation (OPO), is the inverse process

of SFG and involves the appearance of two light waves with the frequencies

ω1,2 in the field of the intense light wave with frequency ω3 = ω1 + ω2.

2.2. FORMALISM OF WAVE PROPAGATION . . . 9

2.2 Formalism of wave propagation in non-

linear media

In this section the main results that govern the propagation of electro-

magnetic waves in nonlinear media are summarized. The starting point is

Maxwell’s equations

∇×H = J +∂

∂tD (2.3)

∇×E = −µ∂

∂tH (2.4)

and

D = εoE + P (2.5)

J = σE (2.6)

where σ is the conductivity. If we separate the total polarization P into its

linear and nonlinear portions according to

P = εoχ(1)E + P NL (2.7)

P NL = 2εod ·E ·E (2.8)

the equation 2.3 becomes

∇×H = σE + ε∂

∂tE +

∂

∂tP NL (2.9)

with

ε ≡ εo

[1 + χ(1)

](2.10)

Taking the curl of both sides of equation 2.4, and the vector identity

∇× (∇×E) = ∇(∇ ·E)−∇2E (2.11)


and taking ∇ ·E = 0, we get

∇2E = µσ∂

∂tE + µε

∂2

∂t2E + µ

∂2

∂t2P NL (2.12)

Next we go over to a scalar notation and rewrite 2.12 as

∇2E = µσ∂

∂tE + µε

∂2

∂t2E + µ

∂2

∂t2PNL (2.13)

where we assumed, for simplicity, that P NL is parallel to E. Let us limit our

consideration to a field made up of three waves propagating in the z direction

with frequencies ω1, ω2, and ω3 according to

ej(z, t) ≡ Ej(z)ei(kjz−ωjt) (j = 1, 2, 3) (2.14)

where the complex notation has been adopted. Then the instantaneous field

is

e(z, t) ≡ ∑

j

ej(z, t) (2.15)

Next we substitute 2.15, using 2.14, into the wave equation 2.13 and sepa-

rate the resulting equation into three equations, each containing only terms

oscillating at one of the three frequencies. The nonlinear polarization P NL

in 2.13 contains the terms

εod2ω1E21e

2i(k1z−ω1t) εod0|E1|2

εod2ω2E22e


εod2ω3E23e


2εodω1+ω2E1E2ei[(k1+k2)z−(ω1+ω2)t] 2εodω1−ω2E1E

∗2e

i[(k1−k2)z−(ω1−ω2)t]


∗3e

i[(k1−k3)z−(ω1−ω3)t]


∗3e

i[(k2−k3)z−(ω2−ω3)t]


These oscillate at the new frequencies and, in general being non–synchronous,

will not be able to drive the oscillation at ω1, ω2, or ω3. An exception in the

last statement is, for example, the case when

ω3 = ω2 + ω1 (2.16)

In this case the term

2µεod∂2

∂t2E3E

∗2e

i[(k3−k2)z−(ω3−ω2)t] (2.17)

oscillates at ω3−ω2 = ω1 and can thus act as a source for the wave at ω1. In

physical terms, we have power flow from the fields at ω3 and ω2 into that at

ω1, or vice versa. Assuming that 2.16 holds, we return to 2.13 and, writing

it for the oscillation at ω1, obtain

∇2e1 = µ1σ1∂

∂te1 + µ1ε1

∂2

∂t2e1 + 2µ1εod

∂2

∂t2E3(z)E∗

2(z)ei[(k3−k2)z−(ω3−ω2)t]

(2.18)

Next we observe that, in view of 2.14,

∇2e1 =∂2

∂z2E1(z)ei(k1z−ω1t)

=

[−k2

1E1(z) + 2ik1d

dzE1(z)

]ei(k1z−ω1t) (2.19)

where we assumed that

∣∣∣∣∣k1d

dzE1(z)

∣∣∣∣∣ À∣∣∣∣∣d2

dz2E1(z)

∣∣∣∣∣ (2.20)

If we use 2.16 and 2.18, and take ∂/∂t = −iω1, we obtain

[−k2

1E1(z) + 2ik1d

dzE1(z)

]ei(k1z−ω1t)

= −(iω1µ1σ1 + ω2

1µ1ε1

)E1(z)ei(k1z−ω1t)

−2ω21µ1εodE3(z)E∗

2(z)ei[(k3−k2)z−ω1t] (2.21)


Recognizing that

k2 = ω2µε (2.22)

we can rewrite 2.21 after multiplying all the term by e−i(k1z−ω1t)/2ik1 as

d

dzE1 = −σ1

2

√µ1

ε1

E1 +iω1d1

cn1

E3E∗2e

i(k3−k2−k1)z (2.23)

and, similarly,

d

dzE∗

2 = −σ2

2

√µ2

ε2

E∗2 +

iω2d2

cn2

E∗3E1e

−i(k3−k2−k1)z (2.24)

d

dzE3 = −σ3

2

√µ3

ε3

E3 +iω3d3

cn3

E2E1e−i(k3−k2−k1)z (2.25)

for the fields at ω2 and ω3. These are the basic equations describing nonlinear

parametric interactions. We notice that they are coupled to each other via

the nonlinear constant d.

2.2.1 Optical difference–frequency generation

In the following treating, we assume the absorption to be negligible, so

σ1,2,3 = 0. Assuming that µ1,2,3 ≡ µo, equation 2.23 becomes

d

dzE1 =

iω1d1

cn1

E3(z)E∗2(z)ei∆k′z (2.26)

where

∆k′ ≡ k3 − k2 − k1 (2.27)

To simplify the analysis further, we may assume that the depletion of the

input waves at ω3 and ω2 due to conversion of their power to ω1 are negligible.

Under those conditions, which apply in the majority of the experimental

situations, we can take E3(z) and E2(z) constant in 2.26 and neglect their


dependence on z. Assuming no input at ω1, that is E1(0) = 0, we obtain

from 2.26 by integration the output field at the end of the crystal of length l

E1(l) =iω1d1

cn1

E3E∗2

ei∆k′l − 1

i∆k′(2.28)

If the three beams are confined to a cross section A, then the power per unit

area (intensity) is related to the field by

I ≡ P

A=

εocn

2|E|2 (2.29)

The conversion efficiency from ω3,2 to ω1 can be written as

ηpw(A, l) ≡ P1

P2P3

=8π2

εoc

d21

λ21n1n2n3

l2

Asinc2

(∆k′l

2

)(2.30)

where

sinc(x) ≡ sin x

x(2.31)

2.2.2 Phase–matching in difference–frequency genera-

tion

According to 2.30, a prerequisite for efficient difference–frequency generation

is that ∆k′ = 0 or

k3 = k2 + k1 (2.32)

If ∆k′ 6= 0, the sum–frequency power generated at some plane, say z1, hav-

ing propagated to some other plane, say z2, is not in phase with the sum–

frequency wave generated at z2. This results in the interference described by

the factor

sinc2

(∆k′l

2

)(2.33)


in 2.30. The main peak and the first zero of spatial interference pattern are

separated by the so–called “coherence length”

lc ≡ 2π

∆k′=

2π

k3 − k2 − k1

(2.34)

The coherence length lc is thus a measure of the maximum crystal length

that is useful in producing the sum–frequency power. Under ordinary cir-

cumstances it may be no larger than 100 µm. This is because the index of

refraction n(ω) normally increases with ω.

The technique that is used widely to satisfy the phase–matching require-

ment ∆k′ = 0 takes advantage of the natural birefringence of anisotropic

crystals. Using the relation

k(ω) =ω

cn(ω) (2.35)

equation 2.32 becomes

n3ω3 = n2ω2 + n1ω1 (2.36)

In normally dispersive materials the index of the ordinary wave or the ex-

traordinary wave along a given direction increases with ω. This makes it

impossible to satisfy 2.36 when all the three beams are of the same type,

that is, when are all ordinary or extraordinary. We can, however, under cer-

tain circumstances, satisfy 2.36 by making the three waves be of different

types.

2.2.3 Difference–frequency generation with focused

Gaussian beams

The analysis of difference–frequency generation leading to 2.30 is based on a

plane wave model. In practice one uses Gaussian beams that are focused so


as to reach their minimum radius (waist) inside the crystal.

The incident Gaussian beams are characterized by confocal parameter b

(which is supposed to be the same for both of them), which is the distance

between points at which the beam “area” is double that of the waist. We

recall that

b ≡ kw2 (2.37)

where w is the waist. If b À l, the beam area, hence the intensity, of the

incident waves is nearly independent of z within the crystal, and we may

apply the plane wave result. The transverse profile of electric field is taken

as

E(x, y) = Eoe−x2+y2

w2 (2.38)

as appropriate to a fundamental Gaussian beam. An electromagnetic wave

with transverse profile E(x, y) for its electric field has an integrated optical

power

P =εocn

2

∫

section|E(x, y)|2 dx dy =

εocn

2

πw2

2|Eo|2 (2.39)

Making use of the following relation which holds among the three waists

1

w21

=1

w22

+1

w23

(2.40)

we can easily obtain the efficiency

ηcb(w2, w3, l) =16π

εoc

d21

λ21n1n2n3

l2

w22 + w2

3

sinc2

(∆k′l

2

)(2.41)

We must recall that this relation was derived for Gaussian beams input with

b À l.

According to 2.41 in a crystal of length l and with given inputs P2,3, the

output power P1 can be increased by decreasing the waists w2,3. This is


indeed the case until b becomes comparable to l. Further reduction of w will

lead to a situation in which the beams begin to spread appreciably within the

crystal, thus leading to a reduced intensity and reduced difference–frequency

generation. It is thus reasonable to focus the beams until l ≈ b. At this point

w2 ≈ l/k which is referred to as confocal focusing. A more exact analysis

of difference–frequency generation with focused Gaussian beams [7] shows

that 2.41 becomes

ηfb(l) =32π2

εoc

d21

λ21n1

l

n3λ2 + n2λ3

h(µ, σ, ξ) (2.42)

where

µ ≡ k2

k3

(2.43)

σ ≡ ∆k′b2

(2.44)

ξ ≡ l

b(2.45)

and the focusing function is defined as

h(µ, σ, ξ) ≡ 1

4ξ

∫ ξ

−ξdτ

∫ ξ

−ξ

e−iσ(τ−τ ′)

1 + ττ ′ − i2

(1+µ1−µ

+ 1−µ1+µ

)(τ − τ ′)

dτ ′ (2.46)

The complex nature of this function is only apparent: the integration, in fact,

cancels the i dependence out and yields a real function. The main difference

between 2.42 and the plane wave result 2.41 is that the conversion efficiency

in this case increases as l instead of l2. This reflects the fact that a longer

crystal entails the use of a larger beam spot size w so as to keep b ≈ l, which

reduces the intensities of the fundamental beams.

2.3. QUASI–PHASE–MATCHING 17

2.3 Quasi–phase–matching

An alternative technique for achieving phase–matching in crystals is referred

to as quasi–phase–matching, a reference to a crystal fashioned in such a way

that the direction of one of its principal axes, say z, is reversed periodically.

This, in a properly chosen crystal orientation and polarization directions of

the participating optical fields, results in a periodic modulation of the non

linear coefficient tensor element dij responsible for the interaction.

The coupled wave equations 2.23, 2.24, 2.25 remain unchanged, except

that d is replaced by d(z), which, being periodic, can be expanded in a Fourier

series

d(z) = dbulk

∞∑

m=−∞ame−im 2π

Λz (2.47)

where Λ is the period of d(z). The effect on the equation 2.26, as an example,

is to transform it to

d

dzE1 =

iω1dbulk

cn1

E3E∗2

∞∑

m=−∞amei(k3−k2−k1−m 2π

Λ )z (2.48)

Phase–matching is obtained if for some integer m the condition

k3 − k2 − k1 −m2π

Λ= 0 (2.49)

is satisfied. Let us define

K ≡ m2π

Λ(2.50)

∆k ≡ ∆k′ −K (2.51)

so that the condition 2.49 becomes ∆k = 0. Ignoring non–phase–matched

terms in 2.48, (their contribution averages out to zero over distances that are


large compared to the coherence length), we rewrite 2.48 as

d

dzE1 =

iω1amdbulk

cn1

E3E∗2e

i(k3−k2−k1−m 2πΛ )z (2.52)

am =1

Λ

∫ Λ

0

d(z)

dbulk

eim 2πΛ

zdz (2.53)

One of the simplest cases of a spatially periodic d(z) is one in which d(z)

switches from dbulk to −dbulk with period Λ and duty–cycle D. In this case

am =2 sin(πmD)

πmeiπmD for m 6= 0 (2.54)

so that, choosing m = 1 and D = 1/2, the effective nonlinear constant is

deff = a1dbulk =2

πdbulk (2.55)

It is clear from 2.52 that, in principle, a quasi–phase–matched configuration

can give rise to the same conversion efficiency as in the ideal ∆k = 0, phase–

matched case, except that a longer interaction path is required to achieve it.

The length penalty factor is dbulk/deff = 1/a1.

To appreciate quasi–phase–matching on an intuitive basis, we note that

it involves reversing of the sign of the nonlinear interaction at

z = qlc2

q = 0,±1,±2,±3, . . . (2.56)

with lc defined by 2.34. These are the locations where the power flow would

reverse direction in the non–phase–matched case. This keeps the power flow-

ing in the same sense along the length of the crystal and leads to cumulative

buildup of E1(z). This can be best visualized using a phasor plot of the

interaction. Taking the specific case of difference–frequency generation as an

example, we can divide the interaction path l into sufficiently short segments,

each of length ∆z, such that ∆k′∆z ¿ π and obtain from 2.26

∆E1(z) =iω1d(z)

cn1

E3E∗2e

i∆k′z∆z (2.57)

2.3. QUASI–PHASE–MATCHING 19

where ∆E1 is the complex increment to the phasor E1(z) due to the segment

of length ∆z centered on z. By adding the increments ∆E1 vectorially, or

rather phasorially, we obtain the phasor diagram shown in Figure 2.1.

E (z)1

E (z)1

E (z)1

z=0 z=0z=0

(a) (b) (c)

Figure 2.1: The evolution of the difference–frequency phasor E1(z) in (a) a

non–phase–matched case, (b) quasi–phase–matched, and (c) bulk birefrin-

gent phase–matching (∆k′ = 0).

In the non–phase–matched case (a), the generated difference–frequency

field keeps growing, reaching a maximum, usually insignificantly small, at

z = π/∆k′. At longer distances, E1 begins to shrink, returning to zero at

z = 2π/∆k′. In the quasi–phase–matched case (b), the sign of the interaction

reverses every ∆z = π/∆k′. This is done by reversing the sign of d(z). The

resultant E1 thus keeps growing monotonically, albeit at a (spatial) rate

smaller than the case of ideal bulk phase–matched (c).

Finally, we quantitatively compare the efficiencies for birefringent phase–


matching and quasi–phase–matching. From Table A.1 it can be seen that,

for LiNbO3, which belongs to the point symmetry group C3v, the allowed

absolute values for birefringent effective nonlinear coefficients are [34]

|dooe| ≤ 3.76± 0.38 pm/V (2.58)

|deoe| = |doee| ≤ 2.10± 0.21 pm/V (2.59)

The QPM effective nonlinear coefficient is

|deee| = 2

π|d33| = 17.3± 1.7 pm/V (2.60)

Since the efficiency quadratically depends on the effective nonlinear coeffi-

cient, it turns out that the QPM gives an efficiency that is at least a fac-

tor 20 higher. QPM has further advantages with respect to birefringent

phase–matching: wider temperature and wavelength acceptance bandwidths,

collinear geometry always available (no angular restrictions and bounds). On

account of this higher efficiency, quasi–phase-matched nonlinear crystals en-

able three–wave interaction in single pass, that, even without the need for

enhancement cavities, still generates optical powers that are significant for

many applications.

2.4 The CO2 molecule

We shall first examine the vibrational structure classically, and later extend

it to the quantum mechanical description. The CO2 is a linear symmetric

molecule which belongs to the symmetry point group D∞h (see Table 2.1).

There are three normal modes of vibration, ν1, ν2, and ν3, which are associ-

ated with the species Σ+g , Πu, and Σ+

u , respectively (Figure 2.2). The species

2.4. THE CO2 MOLECULE 21

D∞h I 2Cφ∞ 2Sφ

∞ ∞C2 ∞σv σh i |li| νi symmetries

Σ+g +1 +1 +1 +1 +1 +1 +1 0 ν1 x2 + y2; z2

Σ+u +1 +1 −1 −1 +1 −1 −1 0 ν3 z

Σ−g +1 +1 +1 −1 −1 +1 +1 0 Jz

Σ−u +1 +1 −1 +1 −1 −1 −1 0

Πg +2 +2 cos φ −2 cos φ 0 0 −2 +2 +1 (Jx, Jy); (xz, yz)

Πu +2 +2 cos φ +2 cos φ 0 0 +2 −2 +1 ν2 (x, y)

∆g +2 +2 cos 2φ +2 cos 2φ 0 0 +2 +2 +2 (x2 − y2, xy)

∆u +2 +2 cos 2φ −2 cos 2φ 0 0 −2 −2 +2

. . . . . . . . . . . . . . . . . . . . . . . . . . .

Table 2.1: Character table of the symmetry point group D∞h.

C OO

n1=1388 cm

symmetric stretch

-1

n3=2349 cm

asymmetric stretch

-1

n2=667 cm

bending

-1

Figure 2.2: Normal vibrational modes of CO2 molecule with corresponding

frequencies.


Πu represents a doubly degenerate vibration, usually indicated by ν2a and

ν2b, which occurs with equal frequency both in the plane and perpendicular

to the plane of the paper.

The potential V and kinetic T energies expressed in terms of Cartesian

and normal coordinates are

V =1

2

∑

ij

k′ijq′iq′j

=1

2

[k1q

21 + k2(q

22a + q2

2b) + k3q23

](2.61)

T =1

2

∑

ij

(m′−1)ijp′ip′j

=1

2

[p2

1

m1

+p2

2a + p22b

m2

+p2

3

m3

](2.62)

where kij and (m−1)ij are elasticity and inverse mass tensors, respectively.

The frequencies of normal modes can be expressed in terms of such tensors

as

νi =1

2π

√ki

mi

(2.63)

It should be emphasized that the concept of normal vibrations rests on

the assumption that the amplitudes of oscillation are infinitesimally small.

Actually the amplitudes of the quantized oscillations, though small, are by

no means infinitesimal and therefore the oscillations are more or less an-

harmonic. In other words, in addition to the quadratic terms, higher–order

terms in the potential must be introduced into the wave equation. As a con-

sequence, the energy is no longer a sum of independent terms corresponding

to the different normal vibrations but contains cross terms corresponding to

the vibrational quantum numbers of two or more normal vibrations. It may

be worth pointing out that the anharmonic corrections to the vibrational en-

ergy levels, in most cases, increase monotonically with increasing vibrational


quantum numbers. In addition, certain IR bands, like ”overtone” bands,

which are forbidden in the harmonic approximation are partially allowed

when anharmonic constants are taken into account.

2.4.1 Rotational energy levels

The rotational Hamiltonian of a rigid linear molecule has the simple form

Hr = hBvJ2 (2.64)

where J is the dimensionless angular momentum and the subscript v is to

indicate the dependence of the rotational constant B on the particular vi-

brational state, which is of the form

Bv = Bo −∑

i

αi

(vi +

di

2

)(2.65)

where vi are the vibrational quantum numbers and di = 1 the respective

degeneracies. The energy levels are simply given by

Er(J)

h= BvJ(J + 1) + DvJ

2(J + 1)2 + HvJ3(J + 1)3 + LvJ

4(J + 1)4 (2.66)

The Dv, Hv, and Lv terms are due to the non–rigidity of the rotating molecule

(centrifugal distortion) and are increasingly related to higher and higher J

transitions to fit experimental data. A dependence on the vibrational state

similar to 2.65 holds for those rotational constants, too.

The population of the various rotational levels can be described by the

Boltzmann distribution as

n(J) ∝ g(J)e−Er(J)

kBT (2.67)

where g(J) is the statistical weight. For CO2 molecule, or molecules belong-

ing to point group D∞h, alternate levels have different statistical weights. In


the case of CO2 the spins of the two identical nuclei are zero. Therefore,

the antisymmetric rotational levels are missing entirely. That is, for Σ+g elec-

tronic states, the odd rotational levels are absent in accordance with Bose

statistics. Therefore

g(J) =

0 for J even

2J + 1 for J odd(2.68)

The Jmax at which n(J) is maximum is obtained from equations 2.67 and 2.68

and is given by

Jmax ≈√

kBT

2hBv

− 1

2(2.69)

The selection rules for rotational transitions are ∆J = ±1 for molecules

having permanent dipole moment. Since CO2 molecule has no permanent

dipole moment, transitions between rotational levels in a given vibrational

state are forbidden.

2.4.2 Vibrational energy levels

In the harmonic approximation (small displacements) the Schrodinger equa-

tion can be resolved into three uncoupled equations in terms of normal co-

ordinates. The energy values are

Ev(vi)

h=

∑

i

νi

(vi +

di

2

)(2.70)

In the harmonic approximation, the selection rules for both the IR and Ra-

man transitions in a polyatomic molecule are ∆ν1 = ±1 for each normal

vibration. The occurrence of certain fundamentals in the IR or Raman spec-

trum depends on the difference of either the dipole moment or the polar-

izability, respectively, between the involved states. As it can be seen in


Table 2.1, under symmetry operations, only the symmetric vibration ν1 of

the CO2 molecule can change its quantum number v1 by ±1 owing to the

Raman effect, whereas the bending ν2 and the asymmetric vibration ν3 can

change their quantum numbers, v2 and v3, by ±1 owing to electric dipole

interaction.

If the anharmonicity is taken into account, the energy eigenvalues 2.70

undergo small corrections given by

Ev(vi, li)

h=

∑

i

νi

(vi +

di

2

)

+∑

j

[xij

(vi +

di

2

) (vj +

dj

2

)+ gijlilj

] (2.71)

The first summation in the above equation is the dominant one; the double

sum gives the higher–order correction terms which arise from cross coupling

between different modes of vibration through the anharmonic force constants

xij; the last summation is called `–type doubling, where gij represents small

constants of the order of xij. For non–degenerate vibrations, li ≡ 0, while

for a vibrational state corresponding to a normal mode νi with degeneracy

di and quantum number vi the total number of vibrational states becomes

ni(vi) =

(vi + di − 1

vi

)=

(vi + di − 1)!

vi(di − 1)!(2.72)

and the allowed values for vibrational angular momentum are li = −vi,−vi +

2, . . . , vi − 2, vi. In the case of CO2 there is only one degenerate vibration,

ν2, with n2(v2) = v2 + 1. This mode of vibration is doubly degenerate on

account of the equivalence of the two directions of the angular momentum `

(in equation 2.71 there is no distinction between±li). This double degeneracy

can be removed by increasing rotation as a result of the Coriolis effect. Thus,


for each J a splitting into two components occurs, whose separation increases

with increasing J .

For every vibrational state there exists a set of rotational levels, but

with slightly different spacings for the different vibrational levels. It must

be realized that for degenerate vibrational levels, J must be larger than or

equal to |li|; in other words, the first |li| − 1 rotational states are missing in

the degenerate vibrational state. We summarize the vibrational features of

CO2 molecule in Table 2.2. We shall indicate the vibrational state of CO2

i 1 2 3

νi (cm−1) 1388.3 667.3 2349.3

di 1 2 1

Si Σ+g Πu Σ+

u

Table 2.2: Main features of CO2 normal vibrations: frequencies νi, degener-

acy di, symmetry species Si.

molecule with the notation∣∣∣v1, v2

l2 , v3

⟩.

Finally, we report in Figure 2.3 the spectrum of the CO2 molecule in

the wavelength region of interest. The spectrum is calculated [35] at room

temperature (298 K) and it includes the three most abundant isotopic species:

12C16O2 (98.420 %), 13C16O2 (1.106 %), and 12C16O18O (0.395 %). The two

strongest bands are taken into account for each isotope: the fundamental

asymmetric vibration ν3 (|0, 00, 0〉 → |0, 00, 1〉), and the so–called “hot band”

(|0, 11, 0〉 → |0, 11, 1〉).


2200 2250 2300 2350 240010-27

10-25

10-23

10-21

10-19

10-17

(c2)

(c1)

(a2)

(b2)

(b1)

(a1)

Line

str

engt

h (c

m)

Frequency (cm-1)

Figure 2.3: CO2 spectrum around 4.25 µm, with indication of the

linestrength. The legend for the band labels, with respect to isotopic

species and quantum numbers, is: a=12C16O2; b=13C16O2; c=12C16O18O;

1=(|0, 00, 0〉 → |0, 00, 1〉); 2=(|0, 11, 0〉 → |0, 11, 1〉).

Chapter 3

Experimental set–up

In this chapter we will describe the main components of the experimental

set–up: the laser sources, the nonlinear crystal, the optics, the detector, and

the vacuum apparatus.

3.1 Laser sources

In this section we will describe our primary laser sources used to generate

difference–frequency. We will focus our attention on features such as wave-

length tunability, power range, temperature stabilization, amplitude and fre-

quency noise. We chose to generate 4.25 µm radiation (idler beam) starting

from a master/slave couple of semiconductor lasers (pump beam) operating

in an injection–locked scheme and a diode–pumped solid–state laser (signal

beam).

29

30 CHAPTER 3. EXPERIMENTAL SET–UP

3.1.1 The master diode laser

The master diode laser is based on a commercial Fabry–Perot type GaAlAs

diode laser (SDL mod. 5401–G1), which can emit up to 50 mW optical power

around 852 nm, with a total emission bandwidth (90 % integrated power)

of about 3 nm. The frequency tuning is done by changing the operating

temperature (∼ 0.3 nm/K), the drive current (∼ 1.5 GHz/mA), or both. This

laser combines a quantum well structure and a real–refractive index–guided

single mode waveguide to provide high power, low astigmatism, relatively

narrow (∼ 10 MHz) spectral width and a single spatial mode Gaussian far

field. It can also operate in single longitudinal mode under some conditions.

Like all Fabry–Perot index guided diode lasers, spectral broadening, mode

hopping and longitudinal mode instability may occur due to small changes

in drive current, diode junction temperature or, more frequently, optical

feedback.

The emission bandwidth is increased up to about 8 nm, by using a feed-

back cavity with a diffraction grating mounted in Littrow–type configuration

[36]: the zeroth–order diffracted beam is reflected back to the laser cavity and

the first–order one is available as output laser beam (see Figure 3.1). The

grating is mounted on a PZT translator whose frequency/voltage tuning co-

efficient was measured to be 285 MHz/V. This allows a narrower linewidth

emission (∼ 500 kHz) at the cost of a smaller free spectral range for longi-

tudinal modes of the resonant cavity (∼ 7.5 GHz). One of the other effects

of extended–cavity operation is that the frequency/current tuning coefficient

for the diode laser gets about one order magnitude lower (∼ 200 MHz/mA)

than the free–running operation. The maximum power coming out of the ex-

3.1. LASER SOURCES 31

Figure 3.1: Extended–cavity mounting of the master diode laser: top view

(above) and front view (below).

tended cavity is about 25 mW, but it is still sufficient to accomplish optical

injection on the slave diode laser.


3.1.2 The slave diode laser

The slave diode laser is a commercial model (SDL mod. 5422–H1), very simi-

lar to the master laser, but operating without extended cavity. It can emit up

to 150 mW optical power around 852 nm, with a typical emission bandwidth

of about 3 nm. In free–running operation, without a feedback cavity, the

spectral quality of the emitted radiation, compared with the master diode

laser (not very stable single mode operation, larger linewidth).

3.1.3 The injection–locking scheme

One of the possible ways to combine the high spectral purity of the master

diode laser with the high emission power of the slave one is to operate them

in an injection–locking scheme. A fraction of the power coming out of the

former is optically injected into the latter. The injecting (master) laser forces

the injected one (slave) to emit radiation with its own spectral features.

The coupling between the two oscillators makes their frequency and phase

strongly correlated [37].

This kind of phenomenon was first experimentally demonstrated in 1966

by using two He–Ne lasers [38]. The injection of a weak signal with frequency

ωi into an oscillator oscillating at its own frequency ωo (in general different

from ωi) can force, under proper coupling conditions, the injected oscillator

to oscillate at frequency ωi too, and the whole available power to switch

from one frequency to the other. Defining Q as the quality factor for the

cavity of the slave laser, Po its maximum emission power, Pi the power from

the master laser that is coupled into the slave cavity, the condition for an


optimum injection locking is [39]

2∆ωlock ≡ 2|ωi − ωo| ≈ ωo2

Q

√Pi

Po

(3.1)

This equation states that, as Pi gets lower with respect to Po, the locking

interval 2∆ωlock gets narrower. It must be noted that Q appears at the

denominator, so that for a low Q oscillator, like a diode laser, the condition

for ωi and Pi is less stringent.

Let’s give a numerical estimation of the wavelength interval for efficient

injection locking in case of diode lasers, which can have Q ≈ 3, in case of

uncoated cavity surfaces

2∆λlock ≈ λ

√Pi

Po

(3.2)

The surprising result is that, even with Pi ≈ 1/100Po, the locking interval

may be several tens of nanometers. Actually this is only a theoretical, hardly

achievable value, because of optical coupling (alignment and mode–matching)

of the master beam into the slave cavity and complications due to diode lasers

intrinsic features. Actually we measured the power difference between free–

running and injected slave diode laser and we found it to be about 0.5 mW,

much less than the injecting power coming from the master diode laser (about

4 mW). A more thorough discussion about injection–locked diode lasers can

be found in [40].

The employed optical scheme make use of an optical isolator with escape

ports (see Figure 3.2). Playing with the two relative polarizations, this iso-

lator lets the injecting light from the master diode laser enter the slave one,

preventing the light of the slave diode laser from coupling with the cavity of

the master itself.


CL

slave DL

rampgenerator

master DL

PZT

l/2

l/2

Figure 3.2: Optical scheme for the injection locking. The following legend

holds: CL=cylindrical lens; λ/2=half–wave plate.

Once the injection–locking condition for the two diode lasers is achieved,

basically by adjusting their operating temperature and currents, the best

way to span the frequency over a range as wide as possible, without loosing

the lock, must be found. Feeding a voltage ramp to the PZT translator of

the grating mounted on the master diode laser, we could achieve a maximum

frequency span of about 3 GHz, measured by the Fabry–Perot spectrum

analyzer. This span is narrower than the typical full width at half maximum

(FWHM) of pressure–broadened lines of atmospheric CO2. In fact with an

air–broadening coefficient of about 22 kHz/Pa [35], the linewidth 2γair ∼4.3 GHz is obtained. Since we desired to have a full recording of such lines, we

tried to extend the continuous span. The easiest way we found to get this aim

was to feed a proportional current ramp to both diode lasers, while feeding

the voltage ramp to the PZT. After optimization of the relative weights

to give to the 2 drive currents with respect to the voltage ramp on PZT,


we succeeded in spanning the frequency of the injected diode laser over a

∼ 12 GHz interval. The continuity of the span was verified both by the

FPSA and the wave–meter, obtaining the same results.

3.1.4 The Nd:YAG laser

The signal beam (the most powerful one) is delivered by a commercial Nd:

YAG laser (Innolight mod. Mephisto 800) based on a monolithic non–planar

ring cavity, which is shown in Figure 3.3. It can deliver up to 800 mW optical

Figure 3.3: Nd:YAG ring cavity photo. The optical path of radiation inside

the crystal is visible.

power at 1064 nm wavelength. The emitted radiation has diffraction limited

beam quality (longitudinal TEM00 spatial mode) and is single frequency with

a typical spectral linewidth of 1 kHz on 100 ms time scale, while the drift

is ∼ 1 MHz/min. The beam emerges perpendicular to the plane of the


aperture with a waist wo ≈ 180 µm and a half–divergence θ = λ/πwo ≈1.9 mrad. The laser radiation represents an eigenmode of the ring cavity:

the polarization results to be elliptic, with the ratio Is/Ip ≈ 7/1. Frequency

tuning can be achieved in 2 ways: slow (bandwidth ∼ 1 Hz), by changing

crystal temperature, with a slope dν/dT ≈ 3 GHz/K and a tuning range

(discontinuous) of > 30 GHz; fast (bandwidth ∼ 100 kHz), by applying a

voltage to the piezo–mechanical transducer (PZT) which properly stresses

the crystal, with a slope dν/dV ≈ 1 MHz/V and a tuning range > 100 MHz.

The laser is equipped with an integrated noise reduction system, which

can decrease intensity fluctuations of the laser beam by several orders of

magnitude. These fluctuations are largely due to a phenomenon called re-

laxation oscillations, that is caused by periodical fluctuations between the

population of the energy levels of the gain media, and the laser cavity field.

The spectrum of these fluctuations resembles the spectral behaviour of a

white–noise–excited classical oscillator, featuring a large peak and a signifi-

cant amount of low frequency noise (see Figure 3.4). Both components can

be cut down by using the noise eater option. For this, a small fraction of the

generated Nd:YAG laser light, transmitted through a mirror, is focused onto

a photo–detector to analyze the variation of the generated output power.

This signal, appropriately filtered and amplified, is fed to the diode laser

current to stabilize the output power.

By use of a wave–meter (see 4.4.1) we performed an accurate wavelength

calibration of the radiation emitted by the laser, as a function of the drive

current and the crystal temperature. The resulting curves, which are plot-

ted in Figure 3.5, could help us to tune the spectrometer at the desired IR

3.2. THE NONLINEAR CRYSTAL 37

Figure 3.4: Intensity noise spectrum of the Nd:YAG laser with intensity–

noise eater on and off.

wavelength, resonant with some CO2 transition.

3.2 The nonlinear crystal

The heart of our experimental apparatus is a periodically poled lithium nio-

bate (PPLN) crystal, built in collaboration with the National Institute of

Standards and Technology (NIST) in Boulder, Colorado, which is shown in

Figure 3.6. The pump beams enter the crystal through one of these facets

(we will call it front facet) and leaves it through the other (the back facet),

together with the generated difference–frequency radiation. There is no anti–

reflection (AR) coating on the facets, so that part of the pump power as well

as of the IR power goes lost.

Successful quasi–phase–matching entails varying the nonlinear coefficient


20 25 30 35 401064.3

1064.4

1064.5

1064.6

Temperature (°C)

Drive current 1200 mA 1400 mA 1600 mA 1800 mA

Wav

elen

gth

(nm

)

Figure 3.5: Nd:YAG wavelength calibration curves, as a function of the crys-

tal temperature, for different values of the drive current.

of an optical material on a periodic basis. The most efficient method of con-

structing such a grating in a ferroelectric such as lithium niobate (LiNbO3)

is to alternate the direction of the spontaneous polarization, and hence the

sign of the nonlinear coefficient. The period of this grating is determined by

the coherence length of the interaction being phase–matched; each domain

of spontaneous polarization is constrained to be an odd number of coherence

lengths long.

The most successful method of forming this pattern of alternating do-

mains in ferroelectrics is the electric field poling technique, which entails pho-

tolithographically defining electrodes with the desired grating period on the

LiNbO3 wafer. An electric field of sufficient strength to shift the anisotropy

in the crystal, and hence the direction of the ferroelectric domain, is then


x

y

z

l

w

h

L optical axis

Figure 3.6: Schematic drawing of the PPLN crystal: the up/down arrows

represent the ferroelectric polarization, determining the χ(2) sign. The crystal

is a parallelepiped with dimensions l × w × h = 17.5 × 10 × 0.5 mm. The

poling period is Λ = 22.0 µm.

applied along the optical axis of the sample. The end result is a grating

pattern of alternating domains, where the domains are reversed from the

original orientation in the regions that were covered by the electrodes.

3.2.1 LiNbO3 as nonlinear material

LiNbO3 is a human–made ferroelectric material that consists of planar sheets

of oxygen separated by off–center lithium and niobium atoms. This asym-

metry in the crystal structure below the Curie temperature (∼ 1200 C)gives

rise to the spontaneous polarization that results in the optical non–linearity

of the material. LiNbO3 has a high nonlinear coefficient (d33 ≈ 27 pm/V)


that can be used for QPM interactions and is transparent over a wide range

of wavelengths (0.35 ÷ 5 µm). Figure 3.7 shows the spectral transmission

curve of three different nonlinear materials: KTiO2PO4 (KTP), LiNbO3,

and RbTiOAsO4 (RTA). It must be noted that at our generated wavelength

Figure 3.7: Transparency range for three different, 10 mm long nonlinear

crystals.

(4.25 µm) the absorption losses (apart from Fresnel reflection losses) are not

negligible at all. A rough estimation can be done just by observing the graph

above: α ≈ 0.5 cm−1.

The grating periods needed to phase–match interactions in the trans-

parency range vary from 2.5 µm to around 35 µm. These periods can be ob-

tained using standard photolithographic processing and techniques. LiNbO3

is already used in a variety of commercial applications; as such it is readily

available in large, reasonably homogeneous wafers for a relatively low cost.


Several disadvantages to the use of LiNbO3 as nonlinear material for elec-

tric field poling include a high room temperature coercive field needed for

poling (∼ 21 kV/mm) that is close to, and sometimes above, the dielectric

breakdown field of the material.

Currently, LiNbO3 is generally only available as grown from a congruent

melt (a ratio of the constituents that stays constant throughout the growth

process). A congruent melt is lithium–poor compared to the stoichiometric

ratio of 1:1:3 (Li:Nb:O). The lithium vacancies in the lattice act as defect

sites where dielectric breakdown of the sample can occur while poling.

3.2.2 Fabrication process

The orientation of the spontaneous polarization in LiNbO3 can be reversed

by the application of an external electric field along the crystal optical axis,

with the field direction opposite that of the original domain polarization.

The applied electric field physically shifts lithium ions to the opposite side

of the oxygen planes and niobium ions between planes. This movement of

charge can be recorded as a displacement current through the sample. The

amount of charge needed to reverse the domain is equal to twice the sponta-

neous polarization of the material (∼ 0.71 µC/mm2) multiplied by the area

to be poled. A measurement of the charge delivered to the sample gives

a good indication of the total area of the crystal that has been poled. The

spontaneous polarization of LiNbO3 is relatively large, stable below the Curie

temperature, and responsible for the large nonlinear coefficient of the mate-

rial. Unfortunately, a consequence of this is that the coercive field needed

to reverse the domain direction at room temperature is distressingly close to


the field strength sufficient to cause dielectric breakdown in the material.

The alternating domains needed for successful QPM are formed by pho-

tolithographically patterning electrodes in a grating structure on the surface

of the LiNbO3 sample. Using the electrodes to apply an electric field greater

than the coercive field to the crystal surface results in the inversion of the

domains under the electrodes (provided, of course, that the field is oriented in

the proper direction). Domain growth under the electrodes can be simplified

into four main stages:

1. domain nucleation at the electrodes edges;

2. domain tip propagation towards the opposite crystal face;

3. the solidification of the domain under the electrodes;

4. domain wall propagation away from the electrodes.

It is believed that the domain nucleation occurs primarily along the edges

of the electrodes due to the increase in field strength caused by the fringing

fields that occur at the field boundaries. The domain wall propagation away

from the electrodes edges makes it difficult to obtain a 50 % duty–cycle

(electrode width/period) between the regions of opposite domain orientation.

A 35 % duty–cycle of the electrode mask will largely compensate for the

domain wall spreading and give a final duty–cycle reasonably close to the

50 % desired for both high conversion efficiency, and for power handling

capabilities.


3.2.3 Temperature stabilization

With a given poling period of the nonlinear crystal, temperature is the main

degree of freedom to fix for phase–matching of the three waves involved in the

DFG. Therefore an effective temperature stabilization is needed. A thermo-

electric Peltier heater/cooler or a resistor–based oven (for temperatures much

higher than room temperature) can be used, depending on the temperature

range and on the accuracy required for the stabilization.

The phase–matching temperature Tpm must be first calculated, by using

the Sellmeier’s equation, that depends on the selected crystal, in our case

LiNbO3 from a congruent melt

n2e(λ, T ) = a1 + b1t

2 +a2 + b2t

2

l2 − (a3 + b3t2)2+

a4 + b4t2

l2 − a25

− a6l2 (3.3)

where the dependence on wavelength λ and absolute temperature T is hidden

in the dimensionless variables

t2 ≡ T 2 − T 2o

1 K2, To ≡ 24.5 C (3.4)

l ≡ λ

1 µm(3.5)

and the numerical coefficients are reported in Table 3.1.

The wavelength and temperature dependent phase–matching condition

can be written as

ne(λ1, Tpm)

λ1

− ne(λ2, Tpm)

λ2

− ne(λ3, Tpm)

λ3

− 1

Λ(Tpm)= 0 (3.6)

and the following energy conservation relation among wavelengths holds

1

λ1

=1

λ2

+1

λ3

(3.7)


parameter Ref. [41] Ref. [42]

a1 4.5820 5.35583

a2 0.09921 0.100473

a3 0.21090 0.20692

a4 2.1940 · 10−2 100 (fixed)

a5 0 (fixed) 11.34927

a6 0 (fixed) 1.5334 · 10−2

b1 2.2971 · 10−7 4.629 · 10−7

b2 5.2716 · 10−8 3.862 · 10−8

b3 −4.9143 · 10−8 −8.9 · 10−9

b4 0 (fixed) 2.657 · 10−5

Table 3.1: Sellmeier’s coefficients for congruent melt LiNbO3. The param-

eters set calculated by [42] is a better fit to experimental data available on

DFG than [41], since it takes also into account multiphonon contribution

near the absorption edge of the material.

To find the phase–matching temperature for three given wavelengths in-

volved in the nonlinear process, one has to solve the equation above with

respect to T . It is noticeable that we also included thermal expansion of the

crystal leaving a temperature dependence of the crystal spatial period Λ on

T of the form [43]

Λ(T ) = Λ(To)[1 + α(T − To) + β(T − To)2], To ≡ 25 C (3.8)

where the linear and quadratic coefficients for thermal expansion are

α = 1.54 · 10−5 K−1 (3.9)


β = 5.3 · 10−9 K−2 (3.10)

The calculation of Tpm can be done numerically and, for our particular

case, the result is Tpm ≈ 287 C. To reach such a high temperature we

designed a copper oven, heated by 4 cartridge heaters (basically, cylindrical–

shaped resistors) delivering a thermal power (by Joule effect) of 25 W each,

which can suitably be switched on and off by a temperature controller based

on a PID (proportional, integral, derivative) circuit (Ceam mod. 901). Ther-

mal insulation from air is granted by a PTFE box all around the copper block,

which has two rectangular holes to allow input and output of laser beams and

is roughly insulated from the copper by a glass cylinder. The temperature

is read by a K–type (chromel/alumel) thermocouple placed very close to the

crystal.

Let’s point out that our crystal could be used for generating IR radiation

in the 3.4÷ 4.5 µm wavelength range, just changing properly both the pump

beam wavelength (diode laser system) and the crystal stabilization tempera-

ture. We report in Figure 3.8 the theoretical (numerically calculated) tuning

curve for our 22.0 µm LiNbO3 crystal, keeping the signal beam (Nd:YAG)

wavelength fixed, as well as few experimental points. One possible way to

avoid operating the crystal at such a high temperature is to tilt it at a cer-

tain angle with respect to the pump and signal beams, in order to have a

longer effective poling period inside the crystal. The angular dependence of

the effective period is shown in Figure 3.9. It should be remarked that the

beam shape and the power of the generated IR radiation strongly depend

on the tilting angle. It will be shown that the optimum working angular

configuration is θ = θcut = 0.


3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6-100

0

100

200

300

Tem

pera

ture

(°C

)

Idler wavelength (µm)

4.1 4.2 4.3 4.4280

285

290

Figure 3.8: Temperature tuning of LiNbO3 crystal with generated idler wave-

length. Experimental points are superimposed on the theoretical curve.

3.2.4 Bandwidths

To evaluate the usefulness of quasi–phase–matched devices for practical ap-

plications, it is important to determine the tolerances for variations in tem-

perature, wavelength, angle, etc. by evaluating their effects on the efficiency

of the device [44]. For a device of total length l containing uniform peri-

ods, the phase–matching factor in the expression for the power conversion

efficiency is, according to 2.30, sinc2(∆kl/2), so that the QPM peak has

the same shape as that of birefringently phase–matched device, but has been

shifted by the wave vector of the periodic structure away from the bulk value.

We may use the fact that this factor goes to 1/2 when ∆kl/2 = 0.44295π

to find the full width at half maximum (FWHM) acceptance bandwidths for

several quantities which affect ∆k when they are varied. It is also of interest


22.0 22.5 23.0 23.5-60

-40

-20

0

20

40

60

Cut angle (°) 0 5 10 15 20

Effective poling period (µm)

Ang

le o

f inc

iden

ce (

°)

Figure 3.9: Dependence of the effective poling period on the tilting angle for

different cut angles θcut of the LiNbO3 crystal (our case is θcut = 0).

to calculate how rapidly the phase–matched wavelength tunes with variations

in a generic parameter ξ.

In general, ∆k(ξ) may be expanded in a Taylor series about the values ξo

which achieves QPM (so that ∆k(ξo) = 0)

∆k(ξ, λ) = (ξ − ξo)∂

∂ξ∆k +

1

2(ξ − ξo)

2 ∂2

∂ξ2∆k + · · · (3.11)

where the partial derivatives are calculated at point ξo. If ∆k has a first–

order dependence on ξ, then we usually may neglect the higher–order terms

in the expansion. The FWHM bandwidth in ξ, which we will denote by δξ,

is found by solving 3.11 (truncated at the first order) for the value of (ξ− ξo)

which satisfies ∆kl/2 = 0.44295π and then doubling it. This procedure gives

δξ =0.44295 · 4π

l

∣∣∣∣∣∂

∂ξ∆k

∣∣∣∣∣−1

(3.12)


We will use this result to calculate the bandwidths as a function of period,

wavelength and temperature.

Constant period error

Let the domain structure be perfectly periodic but with the wrong period Λ

near that required for mth–order QPM. By using

∂

∂Λ∆k =

2πm

Λ2(3.13)

and equation 3.12, we obtain

δΛ =0.8859Λ2

ml(3.14)

Spectral bandwidth

We are now interested in finding the spectral bandwidth δλ3, keeping λ2 fixed.

First we note that, because of energy conservation, the following equations

hold

∂

∂λ3

(1

λ1

)= − 1

λ23

(3.15)

∂

∂λ3

λ1 =

(λ1

λ3

)2

(3.16)

Then, from 2.49 and 3.12, we obtain

δλ3 =0.8859λ2

3

l

∣∣∣∣∣

(λ1

∂

∂λn1 − n1

)−

(λ3

∂

∂λn3 − n3

)∣∣∣∣∣−1

(3.17)

where the derivatives are evaluated at their respective wavelength. The in-

dexes and dispersion can be obtained numerically, for example, from pub-

lished Sellmeier fits for the material being used (in our case see Ref. [42]).

3.3. OPTICAL SCHEME 49

Temperature bandwidth

When the temperature is tuned, in addition to the change in ∆k caused

by the temperature dependence of the refractive indexes, thermal expansion

can alter both the period Λ and the total length l of the device. We must

therefore take the derivative with respect to the temperature of the product

∆kl instead of ∆k alone. So we have

∂

∂T(∆kl) = l

∂

∂T(∆k′ −K) + (∆k′ −K)

∂

∂Tl (3.18)

With the coefficient of linear thermal expansion α defined as α ≡ l−1∂l/∂T ,

we have ∂l/∂T = αl and ∂K/∂T = −αK, so the terms involving K in 3.18

cancel, leaving

∂

∂T(∆kl) = l

(α +

∂

∂T

)∆k′ = 2πl

(α +

∂

∂T

) (n1

λ1

− n2

λ2

− n3

λ3

)(3.19)

where only the refraction indexes depends on T . Finally, we get the temper-

ature bandwidth, as

δT =0.8859

l

∣∣∣∣∣

(α +

∂

∂T

) (n1

λ1

− n2

λ2

− n3

λ3

)∣∣∣∣∣−1

(3.20)

A consideration should be made upon acceptance bandwidths. The region

of the tuning curve T (λi) (see Figure 3.8) at which our DFG spectrometer

operates is quite flat. The maximum–point involves a small slope and, as a

consequence, makes the temperature and wavelength acceptance bandwidths

relatively large.

3.3 Optical scheme

The whole simple optical set–up for generating difference frequency is shown

in Figure 3.10. Starting from the Nd:YAG laser, we have a quarter–wave


CL

DM PPLN

F-PSA

slave DL

rampgenerator

digitalscope

master DL

PZT

Nd:YAG

PZT

Gefilter

l/2

l/4

l/2 l/2

l/2

wavemeter

DFGradiation

BS

chopper

Figure 3.10: Optical set–up for DFG. The following legend holds: DL=diode

laser; λ/2=half–wave plate; λ/4=quarter–wave plate; CL=cylindrical lens;

DM=dichroic mirror; BS=beam–splitter; F–P SA=Fabry–Perot spectrum

analyzer.

plate, which was inserted to obtain a polarization as linear as possible for

the signal beam: the power increase was about 11 %. An optical isolator

(Optics For Research mod. IO–5–YAG) provides a 40 dB isolation, to avoid

optical feedback onto the laser. A telescope composed of 2 AR–coated lenses

with focal lengths f1 = 50 mm and f2 = 100 mm, respectively, expands the

beam diameter a factor f2/f1=2, in order to have a good mode–matching

with the pump beam. An optical chopper (Thorlabs mod. MC1000) is placed

just in the focus position of the telescope, to get the sharpest on/off intensity

modulation of the signal beam. A half–wave plate rotates the polarization

of a particular angle to get the needed s–polarized beam. The radiation is

then transmitted through the dichroic mirror and focused by an AR–coated

(at 1064 nm) lens with f = 100 mm.

The resulting beam waist was measured to be ws ≈ 30 µm. This mea-

surement was done by observing the power transmitted by an edge cutting

3.3. OPTICAL SCHEME 51

the beam at different transverse positions, reconstructing the beam waist de-

pendence on the longitudinal position, and extrapolating the minimum beam

waist from well–known formula [45] valid for a Gaussian beam:

w(z) = wo

√1 +

(z

zR

)2

(3.21)

where zR is the Rayleigh range.

The beam from the master diode laser travels along the following path.

First a half–wave plate rotates the polarization from s–mode to p–mode, in

order to reduce reflection losses from the anamorphic prisms coming next.

This pair of prisms shrinks the elliptical beam in the horizontal direction

and let it unvaried in the vertical one, giving an approximately circular

beam cross–section. An optical isolator (Isowave mod. I–80–TH–4) pro-

vides a 30 dB isolation, to further reduce optical feedback onto the laser.

A pair of mirrors send the beam to the escape port of another 40 dB optical

isolator (Optics For Research mod. IO–5–NIR2-HP) focused by a lens with

f = 50 mm.

The beam from the slave diode laser travels along a path similar to the

master. A half–wave plate rotates the polarization from s–mode to p–mode

and another couple of anamorphic prisms shrinks the beam in the horizontal

direction. Then a cylindrical lens with horizontal focal length f = 500 mm

partially compensates for the beam astigmatism due to the prisms: this lens

is a fundamental ingredient to have a good mode–matching of the TEM00

Nd:YAG beam inside the PPLN crystal, giving an increase of DFG efficiency

of 2–3 times. The slave beam then passes through the optical isolator, which

is placed in the center of a telescope composed of two AR–coated lenses with

focal lengths f1 = f2 = 100 mm. Due to their crossed input polarizations, the


master and slave beams can travel in opposite directions inside the optical

isolator. A half–wave plate makes the pump beam s–polarized and a mirror

reflects it to the dichroic mirror, that makes the beams from the slave diode

laser and the Nd:YAG laser co–propagating.

The beam waist produced by the focusing lens was measured to be ws ≈29 µm (same procedure as for the other beam). The ξ parameters (see

equation 2.45) for the two interacting beams are ξp ≈ 1.3 and ξs ≈ 1.5, so

the mode–matching is not perfect.

3.4 Detector

To detect the generated radiation at about 4.25 µm wavelength, an indium

antimonide (InSb) detector was used (Graseby Infrared mod. IS–1.0). It is

photo–voltaic and a current is generated when exposed to IR radiation in

the 1.0÷ 5.5 µm wavelength region. For best performance, it requires 77 K

operating temperature for best performance, so it is mounted in a metal

dewar with a sapphire window.

Single crystal p–n junction technology yields high speed, low noise de-

tectors with excellent uniformity, linearity and stability. Figure 3.11 shows

the equivalent circuit for a InSb detector, including the shunt resistance RD,

junction capacitance CD and shot–noise. The shot–noise results from the

DC current IB produced by the background infrared radiation. Because IB

is proportional to the detector active area, smaller detectors have less shot–

noise and lower values of NEP. A cold 60 (full angle) field of view (FOV),

corresponding to F/1 optics, is provided. Detectivity can be improved and

IB reduced by restricting the FOV angle. The FOV cold stop angle should

3.4. DETECTOR 53

Figure 3.11: Equivalent circuit for the InSb detector.

be chosen to restrict unwanted background radiation while still accepting all

desired radiation from the optical system. Cold filters could improve detec-

tivity, as well, by eliminating background radiation in unwanted wavelength

regions.

3.4.1 Preamplifier

The InSb detector is coupled into a transimpedance gain preamplifier, (Grase-

by Infrared mod. DP–8100) which converts detector output current to volt-

age while maintaining the detector at the optimum zero volt bias (see Fig-

ure 3.12). The preamplifier is matched to the specific InSb detector to provide


maximum sensitivity, gain (10 ÷ 1000) and bandwidth (5 kHz). When se-

lecting preamp gain, choosing the largest practical value of RF results in the

lowest overall noise. However, the detector IB must be considered to avoid

DC saturation of the preamp. The characteristics of the op–amp circuit

Figure 3.12: Preamplifier circuit.

maintain the diode near 0 V bias. All the photo–current from the detector

essentially flows through the feedback resistor RF . The feedback capacitance

CF is added to control gain peaking. The value of CF depends on the detec-

tor capacitance. It is installed at the factory to provide stable preamplifier

performance with a particular detector model. The values of RF and CF ,

together with the detector characteristics RS, RD and CD, determine the

overall frequency response of the system.

3.4.2 Detection noise

Figure 3.22 shows the various noise sources of the detector/preamp system.

The preamp noise sources en, in, Vos and ib, together with the detector charac-

3.4. DETECTOR 55

teristics, determine the system noise. The effects of the various noise sources

can be summarized using the following approximation

en(ν) ≈ ZF (ν)

√√√√[

en

ZD(ν)

]2

+ i2n + 4kBT[

1

RD

+1

RF

](3.22)

This simplified noise equation provides a good approximation of the total

voltage noise density at the preamplifier output. Note that the noise is

dependent on the frequency ν, and is normalized to a 1 Hz noise bandwidth.

The four terms under the square root represent the four main sources of

current noise:

• preamplifier noise voltage en divided by the detector impedance ZD,

where

ZD(ν) =RD√

1 + (2πνCDRD)2(3.23)

• preamplifier current noise in;

• Johnson thermal current noise from the detector shunt resistance RD;

• Johnson thermal current noise from the preamp feedback resistance

RF .

The total current noise is then multiplied by the transimpedance gain ZF (ν),

where

ZF (ν) =RF√

1 + (2πνCF RF )2(3.24)

Analysis of the simplified noise equation shows that in situations where ZD

is large (> 10 kΩ) the preamplifier current noise in is more important than

the voltage noise en. This is generally the case when using high impedance

detectors (for our detector RD ≈ 500 kΩ) at moderate frequencies.


3.4.3 Responsivity calibration

In order to get an absolute measurement of the generated IR power, we must

precisely characterize our detector in terms of responsivity: this means to

know what is the detected photo–current corresponding to a certain optical

power (A/W). To accomplish this calibration we need a well–known power

reference: the best known infrared power spectrum is from a black–body

cavity source, so we used it to calibrate the detector.

First we heated the black–body cavity up to 443 C (the temperature was

measured by a K–type thermocouple), and let it thermally stabilize for a long

time (a night). The calibration set–up is shown in Figure 3.13. The employed

black-bodysource

T=443 °C

T=25 °C

InSbdetector

chopper

d

T=77 K

Figure 3.13: Geometry for the calibration of the infrared detector by means

of a black–body source.

black–body reference source (Infrared Systems Development mod. IR–563)

is designed to provide infrared radiation as an ideal black–body emitter.

The output energy from the 2.5 cm diameter cavity hole closely follows the

theoretical curve for spectral density of power flux, described by the two

3.4. DETECTOR 57

well–known Planck’s equivalent equations

Bλ(λ, T ) =2πhc2

λ5

1

ehc/λkBT − 1(3.25)

Bν(ν, T ) =2πhν3

c2

1

ehν/kBT − 1(3.26)

Using the integral aperture wheel, the infrared flux can be varied by known

amounts without disturbing critical optical set–ups, and combining apertures

and distance changes, the flux at any point can easily be determined. The 20

tapered–recessed–cone, surface emissivity, and cavity aspect ratio combine

to provide black–body radiation by multiple reflection, absorption and re–

emission of its thermal energy. The thermal energy of the cavity is provided

by a ceramic–sealed heater coil that uniformly heats the cavity cylinder to

temperatures from 50 C to 1000 C.

We placed the mechanical chopper as close as possible to the detector, to

obtain a solid angle seen from the detector as large as possible, and operated

it at 520 Hz chopping frequency. This was necessary to extract the black–

body signal from the room background, at a lower temperature (about 20 C).

In Figure 3.14 we report the measured current signal by the InSb detector

with different apertures.

Now let us carry out the calculation to obtain the detector quantum effi-

ciency at the wavelength of interest (≈ 4.25 µm). We make use of the detector

data–sheet, reporting the relative spectral response curve r(λ) (ratio between

the quantum efficiency at a particular wavelength and its peak value), mea-

sured by the constructor in the 2.0 ÷ 5.75 µm wavelength region (probably

by a FTIR spectrometer). The optical power from a black–body source onto

the detector area A within a conical solid angle with half–aperture θ, in a


100 101 102

101

102

103

=17.8±0.3

=-0.5±0.9

Det

ecte

d ph

otoc

urre

nt (

nA)

Emitting blackbody area (mm2)

Figure 3.14: Linear dependence of black–body signal arriving at the detector,

on the emitting surface.

particular wavelength interval is given by

PBB(λ1, λ2, A, θ) =2πh

c2

(kBT

h

)4

A sin2 θ∫ x2

x1

x3

ex − 1dx (3.27)

xi =hc

λikBT(3.28)

while for the photons flux (number of photons per second) the following

equation holds

nBB(λ1, λ2, A, θ) =2π

c2

(kBT

h

)3

A sin2 θ∫ x2

x1

x2

ex − 1dx (3.29)

By a piece–wise integration on small wavelength intervals [λi, λi+1], 0.25

µm wide, and weighting each integral with the relative spectral response

at the corresponding central wavelength, we can obtain the total detected

3.4. DETECTOR 59

photo–current

iD(A, θ) = ηe∑

i

r

(λi + λi+1

2

)nBB(λi, λi+1, A, θ) (3.30)

where η is the quantum efficiency at the peak wavelength. Finally, we get

the expression for the responsivity

R(λ) ≡ λr(λ)

hcηe =

λr(λ)

hc

iD(A, θ)∑

i r(

λi+λi+1

2

)nBB(λi, λi+1, A, θ)

(3.31)

ID and θ are measured quantities, where θ is defined as

sin θ ≈ tan θ =r

d(3.32)

where r is the aperture radius and d the detector to black–body aperture

distance. The calculation of each definite integral contained in nBB (equa-

tion 3.31) is performed by the program Mathematica. We report the numer-

ical results in Table 3.2.

The value for the responsivity coming out of this calculation is R(4.25

µm) ≈ 2.7 A/W, rather different from that reported on the data–sheet,

R(4.25 µm) ≈ 3.6 A/W . It must be noted that the measured responsivity

also includes the reflection losses by the sapphire window. This can be esti-

mated by the approximated formula of the normal incidence reflectivity of a

dielectric slab with refractive index n

R(n) ≈ (n− 1)2

n2 + 1(3.33)

The approximation does not include interference effects but takes into ac-

count multiple reflections from the two plane surfaces. The calculation for

sapphire, with n = 1.66 at 4.25 µm, gives R ≈ 11.6 %, so this is not sufficient

to explain the discrepancy between the two values. Anyway, we decided to

adopt the measured value in all calculations.


i [λi, λi+1] r(

λi,λi+1

2

)nBB(λi, λi+1)

µm 1012 s−1

1 [2.00, 2.50] 0.15 0.529

2 [2.50, 2.75] 0.37 0.501

3 [2.75, 3.00] 0.52 0.678

4 [3.00, 3.25] 0.67 0.849

5 [3.25, 3.50] 0.77 1.006

6 [3.50, 3.75] 0.87 1.141

7 [3.75.4.00] 0.92 1.252

8 [4.00, 4.25] 0.95 1.338

9 [4.25.4.50] 0.98 1.400

10 [4.50.4.75] 0.98 1.442

11 [4.75, 5.00] 0.95 1.464

12 [5.00.5.25] 0.92 1.471

13 [5.25, 5.50] 0.67 1.465

14 [5.50, 5.75] 0.04 1.449

∑i [2.00, 5.75] 11.946 · 1012 s−1

Table 3.2: Numerically calculated value for the summation in equation 3.31.

3.5 Vacuum apparatus

A vacuum system is an essential element to perform spectroscopic measure-

ments on low pressure gases (about 10−3 mbar). It is mainly composed by a

pumping system, a few vacuum gauges and a gas cell. A complete schematic

drawing of the whole apparatus is shown in Figure 3.15.

3.5. VACUUM APPARATUS 61

TMP DP

DCU

CO2

gas cell

0.01

13.3

1100

Figure 3.15: The vacuum apparatus. The following legend holds:

TMP=turbo–molecular pump; DP=diaphragm pump; DCU=display con-

trol unit; 0.01=0.01 mbar vacuum gauge; 13.3=13.3 mbar vacuum gauge;

1100=1100 mbar vacuum gauge.

3.5.1 Pumping system

A turbo–molecular drag pumping station (Pfeiffer mod. TSH 071 E) is suf-

ficient for our purposes. It comprises 3 main parts:

• a turbo–molecular drag pump (Pfeiffer mod. TMH 071 P) with elec-

tronic drive unit (Pfeiffer mod. TC 600): 90000 turns/min rotation

speed, 33 l/s volume flow rate for nitrogen, 5 ·10−10 mbar final pressure

(after baking out);

• a diaphragm vacuum pump (Pfeiffer mod. MVP 015): 15 l/min volume

flow rate at 1000 mbar, 3.8 l/min at 10 mbar, 3.5 mbar final pressure;

• a display control unit (Pfeiffer mod. DCU 100).


Actually, we require a vacuum level of ∼ 10−3 mbar, and by this pump we

can easily reach ∼ 10−5 mbar or even less, without baking out.

3.5.2 Vacuum gauges

Different kinds of vacuum gauges can be used, that cover different pres-

sure ranges. We use a cold cathode gauge and 2 capacitance diaphragm

gauges with different measuring ranges: one (Balzers mod. CMR 261) has a

10−1 ÷ 1100 mbar range, the other one (Varian mod. VCMT11TDA) 1.33 ·10−3 ÷ 13.3 mbar. Our applications require an initial evacuation of a spec-

troscopic cell to a low pressure, followed by gas filling to a desired measuring

pressure. Since repeatability and precision are required, a capacitance di-

aphragm gauge is a good choice to monitor pressure. Capacitance diaphragm

gauges are highly accurate transducers that give pressure measurements in-

dependent of gas species. Typical accuracy values are 0.12÷1.5 % of reading.

In our case, the high pressure gauge has 0.2 % accuracy, the low pressure one

0.25 %. As high vacuum gauge, a cold cathode type was selected (Officine

Galileo mod. OG 950), with a measuring range of 10−7 ÷ 10−2 mbar and an

accuracy of about 30 %.

3.5.3 Gas cell

As shown in Figure 3.16, the gas cell is, essentially, a glass tube 21 cm long

and with 24 mm diameter, closed at its ends by two CaF2 windows (25 mm

diameter and 3 mm thickness), which are held by O–ring–sealed mountings.

A 6 mm diameter pipe has been fused to the middle part of the cell, so

allowing both the evacuation and filling of the cell. A shut–off valve allows

3.5. VACUUM APPARATUS 63

CaF

window2

mirror

gasinlet

O-ring

Figure 3.16: Schematic drawing of the gas cell. The internal mirror can be

moved or even removed, if transmission operation is required.

to fill the cell at the selected CO2 pressure. The CO2 is contained in a 13.4 l

cylinder, equipped with a pressure regulator.

3.5.4 Flushing box

CO2 relative concentration in air, in normal atmospheric conditions, is about

3.3 · 10−4, corresponding to a partial pressure of 33 Pa. Considering the

linewidth (air–broadening coefficient ≈ 22 kHz/Pa) and intensity of the

strongest lines of the ν3 ro–vibrational band (linestrength ≈ 3.5 · 10−18 cm),

the 1/e transmission path–length is about 5.8 cm. In order to minimize op-

tical power losses, it is necessary to remove as much CO2 as possible from

the air–path of IR radiation.

For this purpose we built a wooden box, all around the IR section of the

experimental apparatus, in which the CO2 concentration can be significantly

reduced by flushing nitrogen into the box (see Figure 3.17). We used soft

rubber as sealant between the bottom part of the box itself and the optical

table. Using this box, the total air path for the 4.25 µm beam reduces to


N in2

N

out2

pumpsout pumps+IR

in

InSb

Sapphirewindow

liquid N in2

sealingrubber

a

c

b

Figure 3.17: Wooden box for nitrogen flow. External dimensions are a ×b × c = 50 × 22 × 21 cm, corresponding to 15.55 l internal volume. Holes

for nitrogen and radiation input/output are shown. The dewar of the InSb

detector is partly standing out of the box.

about 2.5 cm (distance from the output facet of LiNbO3 crystal to the input

sapphire window of the box). A nitrogen flux was used to remove CO2 also

from the PTFE covered crystal oven, without any observable degradation of

temperature stabilization.

In this way, the observed IR power reduction from the PPLN crystal

to the box was limited to about 35 %. Considering that the typical daily

nitrogen consumption is about 30 bar from a 51 l cylinder, it turns out that

the box is flushed by a nitrogen quantity corresponding to about 100 times

its volume in about 10 hours, and that the flow is about 2.5 l/min.

Chapter 4

Measurements

4.1 Noise analysis

Since we could not find any reference in literature, neither theoretical nor

experimental, about noise properties of a DFG source, before switching to

spectroscopic measurements, a preliminary noise analysis was made for our

novel device [46]. A radiation source can be always characterized in terms

of frequency and intensity noise. The former is responsible for coherence

time and linewidth, the latter determines properties such as sensitivity and

frequency–locking stability.

4.1.1 Frequency noise

We measured the frequency relative stability of our DFG source by using

a side of a Fabry–Perot spectrum analyzer fringe as frequency discrimina-

tor. Figure 4.1 shows the measured fluctuations in the center frequency of

the injected laser as compared to the same slave laser without any injec-

65

66 CHAPTER 4. MEASUREMENTS

tion [47]. For all time scales, there is a significant reduction in the injected

0 10 20 30 40-10

-5

0

5

10

not injected injected

Fre

quen

cy s

hift

(MH

z)

Time (s)0 1 2 3 4

-10

-5

0

5

10

Fre

quen

cy s

hift

(MH

z)

Time (s)

0.0 0.1 0.2 0.3 0.4-10

-5

0

5

10

Fre

quen

cy s

hift

(MH

z)

Time (s)0.00 0.01 0.02 0.03 0.04

-10

-5

0

5

10

Fre

quen

cy s

hift

(MH

z)

Time (s)

Figure 4.1: Fluctuations in the center frequency of the high-power diode laser

under conditions of injection lock and not injection locked.

laser’s frequency fluctuations. The extent of frequency fluctuations on all

time scales is below 1 MHz, confirming that the injection–locked diode laser

has a linewidth of a few hundreds kilohertz. This also indicates that the

thermal and mechanical stabilization of the extended–cavity master diode

laser is good, if compared with the performance of similar lasers reported in

literature [36].

4.1.2 Intensity noise

We analyzed the noise in our system by measuring the intensity noise of the

two pump sources. As expected, at low Fourier–frequencies the dominant

noise was from the Nd:YAG laser, even if it was equipped with a noise sup-

pressing circuitry designed to reduce the relaxation oscillation peak at about

4.1. NOISE ANALYSIS 67

500 kHz. When the slave diode laser was injection locked, we measured the

spectral density of the intensity noise on the 4.25 µm beam and observed a

noise reduction of up to 30 dB for Fourier frequencies between 1 and 4 kHz.

The measured intensity noise spectral density of the 4.25 µm output beam is

shown in Figure 4.2. The red trace was recorded when the slave diode laser

0 1 2 3 4 5-110

-100

-90

-80

-70

-60

not injected injected

Inte

nsity

noi

se p

ower

(dB

V)

Fourier frequency (kHz)

Figure 4.2: Intensity noise measured on the 4.25 µm beam with the InSb

detector. Noise spectra were obtained as fast Fourier transform (FFT) of

time domain data, acquired by a digital oscilloscope.

was not injected from the master laser and significantly higher noise can be

seen when compared to the black one, where injection from the master was

present. All lasers had fixed injection current and temperature during the

recording of the two traces. We attribute the higher noise in the red trace

to mode competition within the not injected slave laser, which can produce

strong intensity fluctuations. Mode jumps were also evident when analyzing


the emission with the 10 MHz linewidth FPSA cavity. Due to the intrinsic

instability of the slave laser, the red trace was not reproducible at all, while

the black one was perfectly stable as long as injection was present.

We then investigated the predominant intensity noise sources on our DFG

signal. Noise at the output of the detector can be expressed as a function

of the current flowing in the detector (which is proportional to the incident

power) in the following way

N = P0 + P1i + P2i2 (4.1)

where i is the detected photo–current, N is the noise power, and the con-

stant term takes into account detector intrinsic noise and thermal background

noise. The linear term results from shot–noise and the quadratic term cor-

responds to the classical intensity fluctuations of the laser.

Figure 4.3 shows experimental measurements of N as a function of detec-

tor current i. Each point is the average of 620 values taken in the 2÷ 5 kHz

interval, where the noise is spectrally flat. The IR intensity (and the detected

photo–current) were varied by use of different optical attenuators, without

any change of the drive currents for the three lasers. The small P2 term, in

the fit to the experimental data, confirms that, according to equation 4.1, our

DFG IR source operates in a near shot–noise limited regime. Indeed, at the

maximum power allowed by the detector dynamic range, the departure from

the shot–noise corresponds to 4.7 dB. We attribute the residual quadratic

term to Nd:YAG noise transferred to the IR. For a better determination

of the quadratic term, the P1 value was fixed, in the fit, at the calculated

shot–noise slope. The non–zero intersection with the ordinate axis is then a

measure of the P0 term.

4.1. NOISE ANALYSIS 69

0 1 2 3 4 5 60

1

2

3

4

5

6detector

saturation

background

shot-noise

fit

=2.007±0.016 =0.3024(fixed) =0.040±0.004

Noi

se p

ower

(pA

2 /Hz)

Detected photocurrent (µA)

Figure 4.3: Experimental points and parabolic best fit curve for intensity

noise density relative to detected photo–current. Data were taken using a

spectrum analyzer with 30 Hz resolution bandwidth, 100 s acquisition time,

5 kHz frequency span.

Two terms contribute to this constant background: one is the intrinsic

detector–noise and the other is the thermal fluctuation of the background

radiation reaching the detector. We determined that about 60 % of the noise

floor at zero current comes from thermal fluctuations and the remaining 40 %

depends on internal detector noise. Only 9 % additional noise contribution

comes from the hot oven: this is due to the small solid angle seen by the

detector. Figure 4.4 shows the spectral density of the intensity noise from

the InSb detector [47]. The 3 traces shown were taken with the detector

looking at: a mirror in front of it (which means the detector is looking at a

background close to its own temperature of 77 K); a 25 C background (room


1 2 3 4 5

-80

-75

-70Backgroundtemperature

77 K 25 °C 284 °C

Noi

se p

ower

(dB

m)

Fourier frequency (kHz)

Figure 4.4: Dependence of the InSb detector intensity noise spectral density

on the background temperature.

temperature); the same background except with the crystal oven heated at

the operating temperature of 284 C and occupying part of the field of view.

This means that, with a reduction of thermal fluctuations, e. g. by use of

cold narrow–bandwidth optical filters, the source could be only 2.9 dB above

shot–noise.

4.2 Efficiency analysis

Parameters often used to qualify the nonlinear process are the conversion ef-

ficiency η, or the normalized efficiency η/l. The former is normally measured

in %/W units and does depend on the nonlinear crystal length. The latter is

measured in %/(W·cm) units and, according to equation 2.42, its value does

not depend on the nonlinear crystal length (in this sense it is normalized).

4.3. SENSITIVITY ANALYSIS 71

In the following calculations the normalized DFG efficiency will be used.

A maximum IR power of 12 µW was generated with pump and signal pow-

ers, incident onto the PPLN crystal, of 80 mW and 710 mW respectively.

This translates into a normalized efficiency η/l ≈ 0.012 %/(W·cm), which

compares well with the previous results of 0.011 %/(W·cm) at this wave-

length [18]. The theoretical limit for the generated IR power is about 40 µW

(η/l ≈ 0.040 %/(W·cm), calculated from equation 2.42, with deff=14 pm/V

and taking into account the Fresnel’s reflection losses from crystal surfaces,

13.8 %, 13.4 % and 11.8 % respectively for 850 nm, 1064 nm and 4.25 µm

beams.

We attribute the discrepancy between experimental and calculated effi-

ciency to non–optimized focusing and/or spatial overlap for input beams.

Moreover, it is likely that the duty–cycle of the poling period is not pre-

cisely 50 %, which also reduces the efficiency of the nonlinear mixing (see

equation 2.54). Lastly, the absorption of 4.25 µm radiation by LiNbO3 it-

self, not taken into account for the calculation of this theoretical value, also

contributes to limit the efficiency (see Figure 3.7).

4.3 Sensitivity analysis

From the previous analysis of intensity noise, several considerations on the

ultimate, achievable sensitivity can be made. In direct–absorption experi-

ments using this liquid–nitrogen–cooled InSb detector, saturating at about

1.9 µW incident power, a limiting sensitivity ∆I/I of 2.5×10−7 (1 Hz band-

width and 2.7 A/W responsivity) under shot–noise–limited conditions could

be obtained. Considering that our DFG source is 4.7 dB above shot noise the


sensitivity would go to 4×10−7. At the maximum IR power generated with

our set–up (12 µW), an excess noise of about 7.3 dB can be extrapolated

from Figure 4.3. In this case, if one could use a non–saturating detector with

the same responsivity and intrinsic noise, the sensitivity would be 2.3×10−7.

However, in real world applications these theoretical sensitivities are sel-

dom achieved. Unluckily, these systems tend to be limited by excess techni-

cal noise due to optical interference fringes and to thermal and mechanical

instabilities. In Table 4.1 we compare the sensitivity and miminum absorp-

tion that can be obtained for CO2 detection in air for this set–up and with

semiconductor–lasers based spectrometers [48, 47].

CO2 IR transition (2ν1 + 2ν2 + ν3) P(8) (ν1 + 2ν2 + ν3) R(24) (ν3) R(16)

λ (µm) 1.577 2.002 4.235

S (cm) 1.2 · 10−23 1.0 · 10−21 3.5 · 10−18

Sensitivity (Hz−1/2) 7 · 10−8 7 · 10−7 4 · 10−7

CO2 detection (ppb·m·Hz−1/2) 1000 100 0.01

Reference [49] [49] this work

Table 4.1: Comparison of sensitivities for CO2 detection using different ab-

sorption lines and methods.

4.4 Spectroscopy

In this section we deal with spectroscopic measurements on CO2 transitions,

performed with two different detection techniques: direct absorption and

wavelength modulation.

4.4. SPECTROSCOPY 73

4.4.1 Wavelength measurements

To perform spectroscopic measurements, a wave–meter is necessary to pre-

cisely determine the wavelength of each radiation source. In our wave–meter

(NIST mod. LM–10) a frequency stabilized He–Ne laser is used as reference

source with known wavelength. This instrument, which is a Michelson–type

interferometer operated in air, with a maximum arm length unbalance of

2.1 m, provides an accuracy of a few parts in 107 with an update period of a

few seconds [50]. Since the He–Ne laser is stabilized onto two orthogonally

polarized modes, the “red” frequency, at νR=473.612298 THz or the “blue”

one, at νB=473.613085 THz, can be selected.

4.4.2 Direct–absorption spectroscopy

The simplest way to perform spectroscopy on a molecular gas is to sweep the

laser frequency through the absorption profile (see Appendix A). This simple

technique was used to measure relative intensity and pressure–broadening

coefficients of CO2 lines. Some preliminary direct–absorption recordings of

three CO2 transitions, all belonging to the ν3 band, but with different J

numbers, were taken, to test the quality of our spectrometer and to compare

the expected signal–to–noise ratio (S/N) with the experimental one. The IR

beam was amplitude modulated by the mechanical chopper at frequencies in

the 1÷ 2 kHz range. The results are shown in Figure 4.5, 4.6, 4.7 and 4.8.

These particular J quantum numbers were chosen because the span a wide

range (4 orders magnitude) linestrengths. The R(16) line is so strong that

the relatively small room concentration is sufficient to saturate completely

the absorption profile.

74 CHAPTER 4. MEASUREMENTST

rans

mis

sion

sig

nal (

a.u.

)

Relative frequency (1 GHz/div)

Figure 4.5: Direct–absorption recording of R(16) air–broadened CO2 transi-

tion, recorded with the following experimental conditions: absorption length

L = 27 cm, frequency span ∆ν=11.5 GHz, lock–in time constant τ=30 ms,

scan time T = 50 s.

Relative intensity measurements

To measure the relative intensities of two or more different molecular tran-

sitions it is of help to record them in a single, continuous frequency span.

If the transitions belong to different isotopic species of the molecule, a di-

rect information about isotopic ratios can be obtained by relative intensity

measurements. Looking at the CO2 spectrum (see Figure 2.3), two different

interesting regions are found, having several lines of comparable intensities in

a small frequency interval. Their spectral positions are around 2296.0 cm−1

and 2304.7 cm−1 and the recorded absorption spectra are plotted in Fig-

ure 4.9 and 4.10.


Tra

nsm

issi

on s

igna

l (a.

u.)


Figure 4.6: Direct–absorption recording of R(58) air–broadened CO2 tran-

sition, recorded with the following experimental conditions: L = 27 cm,

∆ν=11.5 GHz, τ=10 ms, T = 20 s.

Pressure broadening

Another useful measurement that can be easily done with direct–absorption

spectroscopy is pressure–broadening coefficient for different CO2 transitions.

With a given cell length, only a few transitions, distributed in a relatively

narrow intensity range, are good candidates for such kind of measurements.

Therefore the R–side of the 12C16O2 (|0, 00, 0〉 → |0, 00, 1〉) band was chosen,

also due to the absence of other interfering lines. The optimum linestrength

for these measurements is about 5 · 10−21 cm−1, corresponding to the inten-

sities of R(64), R(66), R(68), R(70) transitions.

Figures 4.11, 4.12 and 4.13 show the absorption profiles that we experi-

mentally recorded with the corresponding fits, to determine the linewidths.

76 CHAPTER 4. MEASUREMENTST

rans

mis

sion

sig

nal (

a.u.

)

Relative frequency (100 MHz/div)

Figure 4.7: Direct–absorption recording of R(58) self–broadened CO2 tran-


P = 22.7 Pa, ∆ν=2.9 GHz, τ=10 ms, T = 20 s.

Some fringes are evident, intrinsically due to the simple optical set–up, with a

spacing of about 400 MHz, corresponding to a ≈40 cm long interfering cavity.

We do not think these fringes may affect the determination of the pressure–

broadening coefficients, since they only add a weak and “short–wavelength”

oscillating component to the signal.

A Lorentzian lineshape is assumed at these pressure regimes (5÷90 kPa).

In fact, for CO2 molecule at λ ≈ 4.25 µm, the Doppler width, calculated

by equation B.6, is much smaller, γD=65.7 MHz (HWHM), and the natural

linewidth, given by

γN =2

3

k3µ2

4πεoh(4.2)

is completely negligible, giving γN=108 Hz (HWHM).


Tra

nsm

issi

on s

igna

l (a.

u.)


Figure 4.8: Direct–absorption recording of R(82) self–broadened CO2 tran-


P = 1.33 kPa, ∆ν=1.4 GHz, τ=10 ms, T = 20 s.

We made a preliminary estimation of the background, taking into account

the Lorentzian contribution due to the absorption by CO2 in the room, as-

suming it of the following form

yb = m(x− xb) + q +Ab

1 +(

x−xb

wb

)2 (4.3)

where yb stands for the recorded background transmission signal, x for the

frequency, and Ab, xb, wb are the absorption amplitude, center frequency,

and halfwidth, respectively. The linear coefficient m takes into account the

residual amplitude modulation due to the frequency sweep on the diode laser,

and q is an offset representing the transmission signal at center frequency

(x = xb), supposing the absence of any absorption.


(b)(a)

(c)

Tra

nsm

issi

on s

igna

l (a.

u.)


Figure 4.9: Direct–absorption spectrum of CO2 around 2296.0 cm−1,

recorded with the following experimental conditions: P = 600 Pa,

∆ν=11.5 GHz, τ=10 ms, T = 20 s. The assignments of isotopic

species and quantum numbers are: (a)=13C16O2 (|0, 00, 0〉 → |0, 00, 1〉)R(16); (b)=13C16O2 (|0, 11, 0〉 → |0, 11, 1〉) R(35); (c)=12C16O2 (|0, 00, 0〉 →|0, 00, 1〉) P(56).

Then we added the absorption contribution due to the pure CO2 inside

the cell, assuming it of the simple Lorentzian form

yg =Ag

1 +(

x−xg

wg

)2 (4.4)

where yg stands for the contribution to the signal coming from the gas cell,

and Ag, xg, wg are the correspondent absorption amplitude, center frequency,

and halfwidth, respectively. It must be noted that, in general, the following

inequalities hold: Ab 6= Ag, xb 6= xb, and wb 6= wg. The first and third one


(d)

(e)

(c)

(b)

(a)T

rans

mis

sion

sig

nal (

a.u.

)


Figure 4.10: Direct–absorption spectrum of CO2 around 2304.7 cm−1,

recorded with the following experimental conditions: P = 430 Pa,

∆ν=11.5 GHz, τ=10 ms, T = 20 s. The assignments of isotopic

species and quantum numbers are: (a)=12C16O2 (|0, 22, 0〉 → |0, 22, 1〉)P(23); (b)=12C16O2 (|0, 11, 0〉 → |0, 11, 1〉) P(36); (c)=13C16O2 (|0, 00, 0〉 →|0, 00, 1〉) R(30); (d)=12C16O18O (|0, 00, 0〉 → |0, 00, 1〉) P(33); (e)=12C16O2

(|0, 00, 0〉 → |0, 00, 1〉) P(48).

are due to the different CO2 concentrations in the room and in the cell. The

second one is due to the different pressure shifts occurring for the background

CO2 (mixed with other species and at room pressure) and for the CO2 in the

cell (pure and at variable pressure).

Merging these two main contributions (equation 4.3 and 4.4) to the total


Pressure (kPa) 0 5.6 12.9 25.2 37.6 51.2 63.1 75.2 87.7


Tra

nsm

issi

on s

igna

l (a.

u.)

Figure 4.11: Direct–absorption recordings of R(64) CO2 transition at dif-

ferent pressure values and with the following experimental conditions: L =

4 mm, ∆ν=11.5 GHz, τ=10 ms, T = 20 s.

transmission signal, we get the final expression for the fitting function

y = yb + yb = m(x− xb) + q +Ab

1 +(

x−xb

wb

)2 +Ag

1 +(

x−xg

wg

)2 (4.5)

It must be remarked that the three parameters Ab, xb, and wb are kept fixed

to the previously determined values. In addition, to simplify the fit function,

we have implicitly done an approximation: we are neglecting the exponential

nature of absorption (see Appendix B), which is the more accurate the smaller

is the absorption. This condition is not satisfied for all three lines, since the

absorption is about 40 % for R(64), but it is well fulfilled by R(70), which

has only a 15 % absorption.

Finally, we obtained the pressure–broadening coefficients by linear fits of


Pressure (kPa)5.712.525.437.950.262.975.1


Tra

nsm

issi

on s

igna

l (a.

u.)



4 mm, ∆ν=11.5 GHz, τ=10 ms, T = 20 s.

the linewidths with respect to the pressures, which are shown in Figures 4.14,

4.15 and 4.16. The values of the three different pressure–broadening coef-

ficients are very similar and it can be noted that the worst fit is that cor-

responding to the R(70) transition. This is not surprising, since this line is

the weakest one and, in this case, the disturbance coming from fringes and

background absorption of room CO2 have a greater relative weight.

4.4.3 Saturated–absorption spectroscopy

The absorption linestrength is about 3 · 10−18 cm for the strongest lines, so

one can expect to saturate these transitions with relatively low optical power,


Pressure (kPa)0 6.212.9 2637.6 50.462.5 75.587.9


Tra

nsm

issi

on s

igna

l (a.

u.)



4 mm, ∆ν =11.5 GHz, τ=10 ms, T = 20 s.

as for our DFG source [51]. The saturation intensity is expressed by

IS ≡ cεoh2

2

γ‖γ⊥µ2

(4.6)

where γ‖ and γ⊥ are the populations and coherences linewidths (HWHM),

respectively. At low pressures, where the transit time broadening mechanism

is dominant, γ‖ = γ⊥ = γ. At those regimes, a saturation intensity IS ≈1 mW/mm2 can be expected for a linewidth γ=1 MHz (HWHM), and a

transition dipole moment µ = 7.69 · 10−31 C·m. Therefore, I ≈ IS can be

achieved with only 4 µW of DFG power, if the beam is focused to a waist of

35 µm.

Our simple experimental set–up for saturation spectroscopy is shown in

Figure 4.17. The dichroic mirror combines the two s–polarized laser beams,


0 20 40 60 80 1000.0

0.5

1.0

1.5

2.0

2.5

=0.0227±0.0005

=0.031±0.016

Hal

fwid

th (

GH

z)

Pressure (kPa)

Figure 4.14: Linear fits of the pressure–broadening coefficient of R(64) CO2

transition, which results to be 22.7(5) kHz/Pa.

which are focused into the PPLN crystal by a lens (f = 100 mm) to a waist of

about 30 µm. A second lens (f = 100 mm) follows the oven and collimates the

IR beam, while the input beams are blocked by the AR–coated germanium

filter. The IR beam is then focused by a third lens (f = 50 mm) to a waist

of 35 µm onto the reflecting mirror, that is inside the cell containing CO2.

The mirror reflects the beam back onto itself, and part of it is directed by

the beam–splitter to the InSb detector.

This design provides a very simple saturated–absorption set–up, with the

incoming beam serving as the pump and the reflected beam as the probe.

A drawback of this scheme is fringe formation, due to radiation propagating

back and forth. Some additional fringes were in fact observed, but they did

not significantly degrade the S/N for scans of a few MHz, due to their much


0 20 40 60 80 1000.0

0.5

1.0

1.5

2.0

=0.02302±0.00034

=0.0424±0.01113

Hal

fwid

th (

GH

z)

Pressure (kPa)



wider period. The interaction region of 0.5 mm length was delimited by the

CaF2 window and by the mirror. This short interaction length is not so

detrimental since it corresponds to the Rayleigh range for the 35 µm spot

size on mirror, so that to have an homogeneous field along the cell. The

remaining part of the cell (21 cm total length and 65 cm3 internal volume)

was not necessary for this saturated–absorption experiment but did serve as

a gas ballast, to maintain a stable CO2 pressure during the measurements.

We tuned the IR radiation to resonance with the R(14) CO2 line at

4.237 µm, which is one of the strongest lines in the band, with a linestrength

of 3.5·10−18 cm.


0 20 40 60 80 1000.0

0.5

1.0

1.5

2.0

=0.0199±0.0016

=-0.01±0.03

Pressure (kPa)

Hal

fwid

th (

GH

z)



BSInSbdet.

lock-inamplifier

digitalscope gas

cell

PA

DFGradiation

Figure 4.17: Experimental set–up for saturation spectroscopy. The following

legend holds: BS=beam–splitter; PA=preamplifier.


First–derivative signals

Figure 4.18 shows a first–derivative recording of the R(14) absorption pro-

file [48]. A strong saturated–absorption lineshape is visible in the center of


Firs

t der

ivat

ive

sign

al (

a.u.

)

Figure 4.18: Saturated–absorption dip of the CO2 R(14) line recorded as

a first–derivative wavelength–modulation spectrum, at the following exper-

imental conditions: pressure P = 8.0 Pa, frequency span ∆ν=285 MHz,

lock–in time constant τ=10 ms, scan time T = 20 s, modulation frequency

νm=2 kHz, modulation amplitude ∆νm=1.8 MHz.

the line. The Lamb dip feature is shown in Figure 4.19 with an expanded

frequency axis [47]. To record this lineshape, the frequency of the master

diode laser was scanned by feeding a voltage ramp to the PZT, while the

Nd:YAG frequency was sinusoidally modulated with frequency νm ≈ 2 kHz

(wavelength modulation regime) and modulation depth ∆νm, defined by the



Firs

t der

ivat

ive

sign

al (

a.u.

)

Figure 4.19: Saturated–absorption dip of the CO2 R(14) line recorded as

a first–derivative wavelength–modulation spectrum, at the following experi-

mental conditions: P = 6.7 Pa, ∆ν=28.5 MHz, τ=3 ms, T = 5 s, νm=2 kHz,

∆νm=1.8 MHz.

following equation

ν = νo + ∆νm sin 2πνmt (4.7)

Since we did not observe the Lamb dip in direct absorption, we used the

size of the derivative signal to estimate that the contrast in direct absorption

would have been about 2 %. This estimation was done by a numerical inte-

gration of the derivative Doppler profiles recorded with different modulation

widths. From the saturated–absorption contrast value it can be inferred that

power broadening can contribute about 2 % to the linewidth, which is com-

parable with the experimental uncertainty of the linewidth measurements.

A residual sloping background that is evident in Figure 4.19 is due to the


broader Doppler profile, which has a linewidth of about 66 MHz (HWHM).

One of the most common requests for a first–derivative signal is to opti-

mize its slope [52], in order to get the best results when using it as a frequency

error signal (e.g. in a frequency–locking experiment). Therefore we measured

the dependence of this slope on the modulation depth ∆νm, and we got the

results of Figure 4.20. So, the optimum modulation depth seems to be about

0.5 1.0 1.5 2.0 2.5 3.0

Firs

t der

ivat

ive

slop

e (a

.u.)

Modulation depth (MHz)

Figure 4.20: First–derivative slope dependence on the modulation depth.

1.7 MHz and we adjusted it to this value when looking for the best signals.

Third–derivative signals

For precise frequency measurements, third–derivative detection could be used

to achieve a flatter baseline. Indeed, a flatter baseline is obtained for the

third–derivative recording of the R(14) line shown in Figure 4.21. As for the

previous case, we measured the dependence of the slope of third–derivative



Res

idua

ls

Thi

rd d

eriv

ativ

e si

gnal

(a.

u.)

Figure 4.21: Saturated–absorption dip of the CO2 R(14) line recorded as a

third–derivative wavelength–modulation spectrum, at the following experi-

mental conditions: P = 10.7 Pa, ∆ν=28.5 MHz, τ=3 ms, T = 5 s, νm=2 kHz,

∆νm=4 MHz. Experimental points, fit curve and residuals are shown.

signal on the modulation depth, and we got the results of Figure 4.22. In

this case the optimum modulation depth seems to be about 4 MHz, about

2 times higher than for the first–derivative case. This is not surprising at

all: it is well–known that the optimum modulation depth increases with the

order of the derivative.

We also looked for the best pressure conditions, observing the dependence

of the slope of third–derivative signal on gas pressure [47]. The results are

plotted in Figure 4.23. Accordingly, all saturated–absorption recordings were

taken with CO2 pressures in the 6÷10 Pa range, which was found to maximize

the signal to noise ratio.


2 4 6 8

Thi

rd d

eriv

ativ

e sl

ope

(a.u

.)

Modulation depth (MHz)

Figure 4.22: Third–derivative slope dependence on the modulation depth.

0 10 20 30

Thi

rd d

eriv

ativ

e sl

ope

(a.u

.)

Pressure (Pa)

Figure 4.23: Third–derivative slope dependence on the gas pressure.


The experimental data of Figure 4.21 are fitted with a Gaussian profile

(solid line) and the residuals from the fit are also shown in the figure. The

good quality of the fit is consistent with the main contribution to the mea-

sured linewidth of 2.56(2) MHz (HWHM) resulting from the transit time

of CO2 molecules through the IR beam and the wavefront curvature of the

focused beam itself. The transit–time/wavefront of curvature width of a

saturated–absorption signal, for a beam waist wo, a molecular mass M , and

temperature T , is [53, 54, 55]

γTT ≈ 1

8

√2kBT

M

1

w(z)

√√√√1 +

[πw2(z)

R(z)λ

]2

=1

8

√2kBT

M

1

wo

(4.8)

where w(z) and R(z) are, respectively, waist and radius of curvature at po-

sition z, give by [45]

w(z) = wo

√1 +

(z

zR

)2

(4.9)

R(z) = z

[1 +

(zR

z

)2]

(4.10)

It is notable that the dependence on the position z, for a Gaussian beam,

completely disappears. This gives an expected linewidth γ ≈ 1.2 MHz

(HWHM). The additional width measured experimentally is due to mod-

ulation broadening and third–derivative detection. The pressure broadening

contribution to the linewidth is expected to be only about 320 kHz, if the

value of 30.5 kHz/Pa [35] is used. The natural linewidth is much smaller, con-

tributing only 108 Hz. This suggests that reduction of transit–time broaden-

ing for these lines could make them good candidates for frequency references

in the IR.


4.5 Beam–shape analysis

Both in the visible range (for obvious reasons) and in the near IR range up

to about 2 µm (thanks to special fluorescent IR sensor cards), it is possible

to directly see the spot corresponding to the intensity profile of a laser beam.

In the mid IR (and in particular at our wavelength of 4.25 µm), instead, it

is not easy to perform a beam–shape analysis. A device made of a matrix of

separate InSb chips of very small size could be used, though quite expensive.

A Nipkow disk is a much cheaper device and, as it will be shown, is also

suitable for this purpose. It simply consists of a rotating disk with circu-

lar holes placed at different radial distances. The holes, with a diameter of

20 µm each, act as a vertically moving diaphragm for the incoming radiation.

Properly triggering the rotation speed, it is then possible to record the frac-

tion of transmitted radiation for each radial distance (beam x–coordinate)

and for each vertical position (beam y–coordinate). The resolution of the

Nipkow disk available in our laboratory is 60×64 points on a square section

with 10 mm side. In our set–up the disk was placed about 10 cm away from

the collimating lens after the oven.

After a proper background subtraction, we were able to record well–

resolved beam–shapes, which are shown in Figure 4.24, with the two input

beams collinear and perpendicular to the crystal front facet. Let us define

θi as the angle between the incidence direction of the input beams and the

normal to the poling planes of the crystal. In particular, images with the

oven on, but with the two lasers off were subtracted to similar images with

both lasers on (IR radiation at 4.25 µm present). Each picture is colored

with its own grey–scale palette, which is normalized to the intensity range

4.5. BEAM–SHAPE ANALYSIS 93

Figure 4.24: Far–field beam–shape patterns recorded for θi = 0 mrad. The

corresponding temperatures are, from left to right and top to bottom: 280,

282, 284, 286, 288, 290 and 292 C.

spanned by the different beam points. In these images the brightest beam

points are represented by the darkest pixels and vice versa. It is evident that,

for a temperature lower than the phase–matching temperature, the beam is

ring–like, while for higher temperature it becomes more and more narrow,

till it disappears and a much weaker one rises out.

This can be explained as an effect of the focusing and of the cylindrical

symmetry which involve non–collinear phase–matching for a class of wave

vectors ki with similar module, but with slightly different orientations. This

phenomenon depends, as expected, on crystal temperature. The weaker rings

coming out at temperatures higher or lower than the phase–matching tem-

perature are probably due to the secondary peaks of the focusing function

h(µ, σ, ξ) of equation 2.46. This function can be numerically calculated for

our experimental conditions and the resulting curve is shown in Figure 4.25.


The shape is, roughly speaking, similar to a sinc2, but with a strong deform-

-4 -2 0 2 4 6 8 10

0

0.1

0.2

0.3

0.4

0.5

sigma

h

Figure 4.25: Focusing function h(µ, σ, ξ) for µ= 0.8 and ξ ≈ 1.4. The maxi-

mum point is h ≈ 0.4 for σ ≈ 1.2 (Mathematica calculation).

ing left/right asymmetry.

To check all these hypotheses and to seek new phenomena that could be

possibly observed experimentally, we wrote a numerical simulation program,

in Fortran language. The program performs numerical integration of the

differential equation governing the difference–frequency generation inside the

nonlinear PPLN crystal [56]

[∂

∂z− i

2k1

(∂2

∂x2+

∂2

∂y2

)]E1(x, y, z)

=iω1d(z)

cn1

E3(x, y, z)E∗2(x, y, z)ei∆k′z (4.11)

which differs from equation 2.26 only for the extra diffraction term at the

first member, represented by the partial second derivatives with respect to

the transverse coordinates x and y. The three electric fields are supposed of


the form:

ej(x, y, z, t) ≡ Ej(x, y, z)ei(kjz−ωjt) (j = 1, 2, 3) (4.12)

Therefore, the complex amplitudes Ej(x, y, x) include the slowly–variable

part of the fields, while the complex exponentials include the oscillating part.

The pump and signal fields are supposed of the Gaussian form:

Ej(x, y, z) =Ejo

1 + iτj

e− x2+y2

w2oj

(1+iτj) (j = 2, 3) (4.13)

where τj ≡ z/zRj.

The program takes several experimental conditions into account, for the

pump and signal beams:

• Rayleigh range;

• angle of incidence with respect to the poling planes of the crystal;

• horizontal tilting angle of one beam with respect to the other (in case

of non–parallel beams);

• horizontal shift of one beam with respect to the other (in case of parallel

beams).

The two beams are supposed, in any case, to be symmetrically placed with

respect to the crystal center.

The program calculates partial derivatives by the finite–differences nu-

merical approximation and reconstructs the field profile E1(x, y, z) of the

idler beam generated, at discrete steps along the propagation direction z. An

integration step of the value Λ/20 was chosen, in order to get a sufficiently

good numerical precision. After the propagation inside the crystal, far–field

distribution of the idler beam is calculated by the Fraunhofer’s diffraction


integral (using the FFT algorithm). The total IR power generated is also

determined, by numerical integration of the intensity over the transverse co-

ordinates.

Selection of appropriate boundary conditions proved to be a difficult task,

due to spurious reflections of the fields from the boundaries of the square

integration region. A mathematical method employing a proper set of “ab-

sorbing” functions [57] was used to efficiently eliminate these undesirable

effects.

In Figure 4.26 the graphical results for collinear input beams and normal

incidence (θi = 0 mrad) are shown. We can immediately compare these

intensity profiles with the experimental ones of Figure 4.24. The similarity

is quite amazing if we select the seven simulated profiles at temperatures

5 C lower than the experimental ones, i. e. 275, 277, 279, 281, 283, 285 and

287 C. This temperature shift is partly due to the approximate model [42]

that gives the dependence of n on T and λ. Experimental images and their

calculated counterparts are shown in Figure 4.27, for a direct comparison.

A slightly “square–like” shape can be noted in the experimental recordings,

while calculated images have perfectly circular symmetry. This discrepancy

could be attributed to the residual astigmatism of the slave laser beam, even

with the compensation by the cylindrical lens.

Next, in Figures 4.28 and 4.29 it is shown the case of two input beams

that, though overlapped, are tilted at small angles (θi =5 and 10 mrad,

respectively) with respect to the normal to the poling planes. As it can

be observed from the simulations, the cylindrical symmetry is broken and

“moon–like” intensity profiles of the idler beam come out. This asymmetry is


Figure 4.26: Far–field beam–shape patterns calculated for θi = 0 mrad. The


275, 277, 279, 281, 283, 285, 287, 289 and 291 C.

particularly evident in Figure 4.29 and can be explained as a “walk–off” effect

of the idler beam with respect to the pump and signal beams [44]. While

the effects of this walk–off on the efficiency of the nonlinear interaction are

similar to those of a walk–off in a conventional birifrengently phase–matched

interaction (see Appendix A), it is important to note that, in the case of

QPM tye walk–off is related to the phase velocities of the interacting waves

(wave vectors) and can occur in isotropic media, while the walk–off in the

conventional case is related to the group velocities (Poynting vectors) and

occurs only in anisotropic media.


Figure 4.27: Comparison between recorded and calculated beam–shape pat-

terns in the far–field, for θi = 0 mrad.

Since it is clear that the far–field beam–shape strongly depends on the

tilting angle, an interesting question is how the generated IR power depends

on this angle. The answer is shown in Figure 4.30. Two things are especially

worth be noted:

1. the maximum IR power is generated for normal incidence (θi = 0 mrad)




274, 276, 278, 280, 282, 284, 286, 288, 290 and 292 C.

and even a relatively small tilting with respect to normal incidence (e. g.

θi = 10 mrad) can strongly reduce the nonlinear conversion efficiency

(a factor 2 or even more);




274, 276, 278, 280, 282, 284, 286, 288, 290 and 292 C.

2. as the tilting angle gets larger, the phase–matching temperature Tpm

rises and the temperature acceptance bandwidth gets wider.

Whenever a tilted crystal is used to change the effective poling period


270 275 280 285 290 2950

10

20

30

40

Tilting angle 0 mrad 5 mrad 10 mrad

Idle

r po

wer

(µW

)

Temperature (°C)

Figure 4.30: Dependence of the idler power on crystal temperature, for three

different tilting angles.

[20, 21, 29, 30], these implications should be carefully considered. How-

ever, some more experimental work has to be made in the next future, to

be compared with results from calculation in tilted geometries. We are con-

fident that their validity will be demonstrated, as for the normal incidence

case. Finally, it should be remarked that this simulation program can be of

importance to design appropriate set–ups, whenever the idler beam–shape

pattern and its intensity are crucial experimental parameters. This is cer-

tainly the case for high resolution saturated–absorption spectra, as discussed

in section 4.4.3. Even more importance assumes the beam–shape parameters

when narrow linewidths have to be recorded, with spectroscopic techniques

like beam expansion [58] or cold molecule selection, first introduced by the

group of Chebotayev [59].

Chapter 5

Summary

Let us now summarize the main results of this work. A novel diode–laser

based DFG spectrometer was built, that generates radiation tunable around

the strong ν3 absorption band of CO2 at 4.25 µm. This apparatus proved

to be a useful tool for high sensitivity measurements of CO2 concentrations,

due to the intrinsically low intensity noise of this source and to its narrow

linewidth. The broad tunability of this spectrometer, encompassing also iso-

topic bands of CO2, and its extremely high sensitivity for molecular detection

in this wavelength region, may also find application in studies of atmospheric

chemistry (global warming etc.), biological systems, process monitoring and

fundamental physics experiments (e. g. test of Pauli’s symmetrization prin-

ciple).

Sub–Doppler resolution was also demonstrated with a simple saturation

set–up, not involving build–up cavities and with powers as low as 10 µW. This

demonstration opens new possibilities for very high resolution spectroscopy

with a broadly–tunable and convenient source of IR. Taking advantage of the

very wide spectral coverage of present–day DFG spectrometers, the rich and

103

104 CHAPTER 5. SUMMARY

strong vibro–rotational spectra possessed by many simple molecules in this

wavelength region can be studied in detail. This can provide new spectro-

scopic data for molecular Hamiltonians, molecular sensing and monitoring

systems, and potentially for frequency references in the IR. Frequency lock-

ing onto narrow resonances in the infrared could be useful to bridge the gap

between near– and mid–IR regions for metrological purposes.

An original analysis of the beam intensity profiles of the generated IR

beam was experimentally performed and compared with calculation. This

latter allowed simulation of many different experimental conditions, that can

be of great help for the design of future set–ups.

Appendix A

Optics of uniaxial crystals

In uniaxial crystals a special direction exists called the optic axis (Z axis).

The plane containing the Z axis and the wave vector k of the light wave

is termed the principal plane. The light beam whose polarization is nor-

mal to the principal plane is called an ordinary beam or an o–beam. The

beam polarized in the principal plane is known as extraordinary beam or

e–beam. The refractive index of the o–beam does not depend on the propa-

gation direction, whereas for the e–beam it does. Thus, the refractive index

in anisotropic crystals generally depends both on light polarization and prop-

agating direction.

The difference between the refractive indices of the ordinary and extraor-

dinary beams is known as birefringence ∆n. The value of ∆n is equal to zero

along the optic axis Z and reaches a maximum in the direction normal to this

axis. The refractive indices of the ordinary and extraordinary beams in the

plane normal to the Z axis are termed the principal values of the refractive

index and are denoted by no and ne respectively. The refractive index of the

extraordinary wave is, in general, a function of the polar angle θ between the

105

106 APPENDIX A. OPTICS OF UNIAXIAL CRYSTALS

Z axis and the vector k. It is determined by the equation:

ne(θ) = no

√√√√ 1 + tan2 θ

1 + (no/ne)2 tan2 θ(A.1)

If no > ne the crystal is negative; if no > ne, it is positive. The quan-

tity ne(θ) does not depend on the azimuthal angle φ. The dependence of

the refractive index on light propagation direction inside the uniaxial crystal

(index surface) is a combination of a sphere with radius no (for an ordinary

beam) and an ellipsoid of rotation with semi-axes no and ne (for an extraor-

dinary beam, the axis of the ellipsoid of rotation is the Z axis). In the Z

axis direction the sphere and ellipsoid are in contact with each other. In a

negative crystal the ellipsoid is inscribed in the sphere, whereas in a positive

crystal the sphere is inscribed in the ellipsoid.

When a plane light wave propagates in a uniaxial crystal, the direction

of propagation of the wave phase (vector k) generally does not coincide with

that of the wave energy (vector S). The direction of S can be defined as the

normal to the tangent drawn at the point of intersection of vector k with the

n(θ) curve. For an ordinary wave the n(θ) dependence is a sphere with radius

no. Therefore, the normal to the tangent coincides with the wave vector k.

For an extraordinary wave the normal to the tangent (with the exception of

the cases θ = 0 and θ = π/2) does not coincide with the wave vector k, but

is rotated from it by the birefringence or walk–off angle:

ρ(θ) = ± arctan[(no/ne)

2 tan θ]∓ θ (A.2)

where the upper signs refer to a negative crystal and the lower signs to a

positive one.

A.1. PHASE–MATCHING CONDITIONS 107

A.1 Phase–matching conditions

Under usual conditions all optical media are weakly nonlinear, i.e. the in-

equalities χ(3)E2 ¿ χ(2)E ¿ χ(1) are valid. Noticeable nonlinear effects can

be observed only when light propagates through fairly long crystals and the

so–called phase–matching conditions are fulfilled:

k3 = k2 + k1 (SFG)

k3 = 2k1 (SHG)

k4 = k2 − k1 (DFG)

(A.3)

where ki are the wave vectors corresponding to the waves with frequencies

ωi (i = 1, 2, 3, 4):

|ki| = ki =ωini

c=

2πni

λi

(A.4)

where the quantities ni = n(ωi) and λi are the refractive index and wave-

length at the frequency ωi, respectively.

The relative location of the wave vectors under phase–matching can be

either collinear (scalar phase–matching) or non-collinear (vector phase–ma-

tching). Under scalar phase–matching we have

ω3n3 = ω2n2 + ω1n1 (SFG)

n3 = n1 (SHG)

ω4n4 = ω2n2 − ω1n1 (DFG)

(A.5)

The physical sense of phase–matching conditions A.3 is the space res-

onance of the propagating waves, namely between the wave of nonlinear

dielectric polarization at the frequency ω3 (or ω4 for DFG) and produced by

the light wave at the same frequency ω3 (or ω4 respectively). Note that in

the optical transparency region in isotropic crystals (and also in anisotropic


crystals for identically polarized waves), the equality A.5 for SHG is never

fulfilled because of normal dispersion (n1 < n3). The use of anomalous

dispersion is almost impossible since energy absorption is very high. The

phase–matching conditions are fulfilled only in anisotropic crystals under

interaction of differently polarized waves.

Combination of nonzero square nonlinearity of an optically transparent

crystal with phase–matching is the necessary and sufficient condition for an

effective three–wave interaction.

A.2 Types of phase–matching

To fulfill the phase–matching condition in three–frequency interaction, dif-

ferently polarized wave should be used. Let us consider the case of SFG. If

the mixing waves have the same polarization, the radiation at sum frequency

(SF) will be polarized in the perpendicular direction; in this case type I

phase–matching is realized. In negative crystals,

ko1 + ko2 = ke3 (A.6)

(this is called “ooe” phase–matching or type I(−) phase–matching). In posi-

tive crystals,

ke1 + ke2 = ko3 (A.7)

(“eeo” phase–matching or “eeo” interaction or type I(+) phase–matching).

Here and below for SFG the first symbol in the expression ooe, eeo, eoe, and

so on, refers to the wave with the lower frequency, the third symbol to the

wave with the higher frequency.

A.3. CRYSTAL SYMMETRY AND EFFECTIVE NONLINEARITY 109

If the mixing waves are of orthogonal polarizations, type II phase–ma-

tching takes place and the SF wave corresponds to an extraordinary wave in

negative crystals:

ko1 + ke2(θ) = ke3(θ) (A.8)

(“oee” phase–matching or “oee” interaction or type II(−) phase–matching)

or

ke1(θ) + ko2 = ke3(θ) (A.9)

(“eoe” phase–matching or “eoe” interaction or type II(−) phase–matching);

and to an ordinary wave in positive crystals:

ko1 + ke2(θ) = ko3 (A.10)

(“oeo” phase–matching or “oeo” interaction or type II(+) phase–matching)

or

ke1(θ) + ko2 = ke3 (A.11)

(“eoo” phase–matching or “eoo” interaction or type II(+) phase–matching).

Note that, in general, the non-collinear or vector phase–matching takes

place. In practice, however, collinear or scalar phase–matching, which is the

special case, is widely used.

A.3 Crystal symmetry and effective nonlin-

earity

For anisotropic media the dielectric susceptibility coefficients χ(1) and χ(2)

are, in general, tensors of the second and third rank, respectively. In the


dielectric reference frame X,Y, Z, where Z is the optic axis, the tensors χ(1)

and ε(1) are diagonal. The following components:

ε(1)XX = ε

(1)Y Y ≡ n2

o (A.12)

ε(1)ZZ ≡ n2

e (A.13)

are nonzero components of the linear dielectric polarization tensor ε(1). In

practice the tensor dijk is used instead of tensor χ(2)ijk, the two tensors being

interrelated by the equation

χ(2)ijk ≡ 2dijk (A.14)

Unlike tensor ε(1), tensors χ(2) and d can be given only in a three dimensional

representation. Usually a “plane” representation of tensor dijk in the form

dil is used, where i = 1 corresponds to X, i = 2 to Y , i = 3 to Z, and l takes

the following values:

XX Y Y ZZ Y Z = ZY XZ = ZX XY = Y X

l = 1 2 3 4 5 6(A.15)

The expression 2.2 can be rewritten in a reduced form (with respect to the

components):

Pi = εo

[χ

(1)ij Ej + 2dilE

2l + . . .

](A.16)


where E2l is the six–dimensional vector of the field products (summation over

the repeating indices is carried out). In matrix form we have:

PX

PY

PZ

= 2εo

d11 d12 d13 d14 d15 d16

d21 d22 d23 d24 d25 d26

d31 d32 d33 d34 d35 d36

·

E2X

E2Y

E2Z

2EY EZ

2EXEZ

2EXEY

(A.17)

The total number of the components of the square nonlinearity tensor dil is

18. In centrosymmetrical crystals (where the center is a symmetry element)

all the components of the square nonlinearity tensor d are equal to zero. The

non-centrosymmetrical crystals comprising 21 crystallographic classes out of

32 usually have one or more symmetry elements (axes or plane of different or-

ders), which considerably decrease the number of independent components of

the tensor dil. Kleinman [60] has established additional symmetry conditions

for the case of no dispersion of electron nonlinear polarizability. When the

Kleinman symmetry conditions are valid (in the great majority of practical

cases), the number of independent components of the tensor Dil decreases

from 18 to 10 because

d21 = d16; d24 = d32; d31 = d15;

d13 = d35; d12 = d26; d32 = d24; (A.18)

d14 = d36 = d25

Since any linearly polarized wave in a uniaxial crystal can be represented as

a superposition of two waves with “ordinary” and “extraordinary” polariza-

tions, we provide the components of a unit polarization vector p given in


polar coordinates θ and φ along the dielectric axes X,Y, Z, where Z is the

optic axis and |p| = 1:

poX = − sin φ, peX = cos θ cos φ

poY = cos φ, peY = cos θ sin φ

poZ = 0, peZ = − sin θ

(A.19)

The equations for calculating the conversion efficiency use the effective non-

linearity deff , which comprises all the summation operations along the po-

larization directions of the interacting waves:

deff = p1dp2p3 = p2dp3p1 = p3dp1p2 (A.20)

The quantity deff represents a scalar product of the first vector in A.20 and

a tensor–vector product of the dpp type, which is also a vector. Depending

on the type of interaction (ooe, oee, and so on), the vector components pi

are calculated by equations A.19, and the product in equation A.20 is found

by the known rules of vector algebra. Table A.1 illustrates the values of deff

determined in this way for nonlinear uniaxial crystals of 13 point groups.


Point Type of interaction

group ooe, oeo, eoo eeo, eoe, oee

D2d d36 sin θ sin 2φ d36 sin 2θ cos 2φ

C3v d31 sin θ − d22 cos θ sin 3φ d22 cos2 θ cos 3φ

C4v d31 sin θ 0

C6 d31 sin θ 0

C6v d31 sin θ 0

S4 (d36 sin 2φ + d31 cos 2φ) sin θ (d36 cos 2φ− d31 sin 2φ) sin 2θ

C3 (d11 cos 3φ− d22 sin 3φ) cos θ + d31 sin θ (d11 sin 3φ + d22 cos 3φ) cos2 θ

D3 d11 cos θ cos 3φ d11 cos2 θ sin 3φ

C3h (d11 cos 3φ− d22 sin 3φ) cos θ (d11 sin 3φ + d22 cos 3φ) cos2 θ

D3h d22 cos θ sin 3φ d22 cos2 θ cos 3φ

D4 0 0

D6 0 0

Table A.1: Expression for deff in uniaxial crystals of different point groups

when Kleinman symmetry relations are valid.

Appendix B

Direct–absorption lineshape

The transmission of monochromatic light through an absorbing gas is ex-

pressed by the Beer–Lambert law as:

I(ν, Pg, L) = Ioe−α(ν,Pg)L (B.1)

where α(ν, Pg) is the linear absorption coefficient defined in cm−1, ν is the

frequency in cm−1, Pg is the partial pressure of the absorbing gas, L is the

optical pathlength in cm, Io is the initial light intensity, and I(ν, Pg, L) is

the transmitted intensity. The absorption coefficient can be related to the

molecular line intensity S by

α(ν, Pg) ≡ NPgSg(ν) (B.2)

where g(ν) is the normalized lineshape function in cm and N is the total

number density of molecules (absorbing and not) in cm−3. The normal atmo-

spherical value of N at 296 K is “Loschmidts’ number” NL = 2.479·1019 cm−3.

As it can be seen, σ(ν) ≡ Sg(ν) is the absorption cross section per molecule

in cm2, and n ≡ NPg is the absolute density of absorbing molecules in cm−3.

115

116 APPENDIX B. DIRECT–ABSORPTION LINESHAPE

The Boltzmann’s population factor and the value of the isotopic abundances

are contained within S.

There are three main lineshapes profiles, depending on the measuring

pressure ranges: Lorentzian (high pressures), Gaussian (low pressures) and

Voigt (intermediate pressures). The Lorentzian (pressure broadened) profile

is given by

gL(ν) ≡ 1

πγL

1

1 +(

ν−νo

γL

)2 (B.3)

where γL is the pressure–broadened linewidth (HWHM) in cm−1, given by

the expression

γL = gPt (B.4)

where g is the pressure–broadening coefficient, and Pt is the total pressure.

The Gaussian (Doppler broadened) profile is expressed by

gD(ν) ≡ 1

γD

√ln 2

πe− ln 2

(ν−νoγD

)2

(B.5)

where γD is the pressure–broadened linewidth (HWHM) in cm−1, given by

γD = νo

√2 ln 2kBT

Mc2(B.6)

where M is the molecular mass of the absorbing gas. The Voigt profile is

a convolution integral over both the Doppler and Lorentzian profiles, and is

used when both mechanisms are present in approximately equal amounts.

The expression for this curve is:

gV (ν) ≡ 1

γLγD

√ln 2

π3

∫ +∞

−∞e− ln 2

(ν′−νo

γD

)2

1 +(

ν−ν′γL

)2 dν ′ (B.7)

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a tunable and narrow–linewidth diﬀerence–frequency ... · nonlinear crystals, having...

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