resonator based measurement technique for quantification of minute birefringence

Resonator based measurement technique for quantification of minute birefringence

Tuomo von Lerber and Hanne Ludvigsen

Fiber-Optics Group, Department of Electrical and Communications Engineering, Helsinki University of Technology, P.O.Box 3000, FIN-02015 HUT, Finland

[email protected]

Albert Romann

Institute of Quantum Electronics, Swiss Federal Institute of Technology (ETH), CH-8093 Zurich, Switzerland

Abstract: We present a new method for quantification of minute birefringence in high-finesse resonators. The method is based on observing the homodyne polarization mode beat at the output of the resonator. We show that the mode beat is generated by a phase mismatch of a polarization mode in the cavity and that the magnitude of the birefringence is proportional to the beat frequency. We demonstrate the sensitivity of the technique by measuring polarization properties of a twisted 0.275 m long single-mode fiber cavity. Maximum beat length of the fiber was found to be 10.6 m, which is almost 40 times longer than the length of the studied fiber.

2004 Optical Society of America

OCIS codes: (260.1440) Birefringence; (120.2230) Fabry-Perot; (060.2430) Fibers, single-mode

References and links 1. R. Ulrich and A. Simon, “Polarization optics of twisted single-mode fibers,” Appl. Opt. 18, 2241-2251

(1979). 2. T. Chartier, A. Hideur, C. Özkul, F. Sanchez, and G. M. Stéphan, “Measurement of the elliptical

birefringence of single-mode optical fibers,” Appl. Opt. 40, 5343-5353 (2001). 3. H. Y. Kim, E. H. Lee, and B. Y. Kim, “Polarization properties of fiber lasers with twist-induced circular

birefringence,” Appl. Opt. 36, 6764-6769 (1997). 4. T. von Lerber and M. W. Sigrist, “Cavity-ring-down principle for fiber-optic resonators: experimental

realization of bending loss and evanescent-field sensing,” Appl. Opt. 41, 3567-3575 (2002). 5. S. Moriwaki, H. Sakaida, T. Yuzawa, and N. Mio, “Measurement of the residual birefringence of

interferential mirrors using Fabry-Perot cavity,” Appl. Phys. B 65, 347-350 (1997). 6. F. Brandi, F. Della Valle, A. M. De Riva, P. Micossi, F. Perrone, C. Rizzo, G. Ruoso, G. Zavattini,

“Measurement of the phase anisotropy of very high reflectivity interferential mirrors,” Appl. Phys. B 65, 351-355 (1997).

7. J. L. Hall, J. Ye, and L.-S. Ma, “Measurement of mirror birefringence at the sub-ppm level: Proposed application to a test of QED,” Phys. Rev. A 62, 013815 (2000).

8. J. Y. Lee, H.-W. Lee, J. W. Kim, Y. S. Yoo, and J. W. Hahn, “Measurement of ultralow supermirror birefringence by use of the polarimetric differential cavity ringdown technique,” Appl. Opt. 39, 1941-1945 (2000).

9. J. Morville and D. Romanini, “Sensitive birefringence measurement in a high-finesse resonator using diode laser optical self-locking,” Appl. Phys. B 74, 495-501 (2002).

10. C. Rizzo, A. Rizzo, and D. M. Bishop, “The Cotton-Mouton effect in gases: experiment and theory,” Int. Rev. in Phys. Chem. 16, 81-111 (1997).

11. D. Bakalov, et al., “Experimental method to detect the magnetic birefringence of vacuum,” Quantum Semiclass. Opt. 10, 239-250 (1998).

12. F. Maystre and R. Dandliker, “Polarimetric fiber optical sensor with high sensitivity using a Fabry-Perot structure,” Appl. Opt. 28, 1995-2000 (1989).

13. E. A. Kuzin, J. M. Estudillo, B. I. Escamilla, and J. W. Haus, “Measurements of beat length in short low-birefringence fibers,” Opt. Lett. 26, 1134-1136 (2001).

#3711 - $15.00 US Received 28 January 2004; revised 22 March 2004; accepted 26 March 2004

(C) 2004 OSA 5 April 2004 / Vol. 12, No. 7 / OPTICS EXPRESS 1363

mailto:[email protected]

1. Introduction

For many years the polarization properties of single-mode fibers have been subject to numerous studies. The complex evolution of polarization in the fiber was described already several years ago [1], but still today the subject matter continues to draw attention [2]. The motivation for the interest has been the ever-increasing demand for high-speed optical networks, whose current bottleneck is the polarization mode dispersion, which originates from the intrinsic birefringence of single-mode fiber. The polarization properties are also of keen interest in fiber-optic resonators, in fiber lasers [3], and, e.g., in fiber cavity ring-down experiments [4]. Other low-birefringence phenomena, such as mirror birefringence [5-9], the Cotton-Mouton effect [10], and the attempted measurements of vacuum birefringence [11] have also been intensively studied in conventional high-finesse optical resonators.

The simplest way to measure birefringence of an optical medium is to let linearly polarized light propagate through it and split the output with a high extinction-ratio polarizing beam splitter. The measured output intensities are proportional to the imposed ellipticity. An accurate measurement of intensity, however, is a non-trivial task and drifts in the photodetector sensitivities or amplifier gains may deteriorate the result. Often the determination of time, or consequently frequency, can be made with higher accuracy than the reading of the voltage or current of a photodetector. Therefore, it is usually advantageous to perform the measurement in the time domain instead of relying on absolute measurements of voltage or current.

The present work describes a sensitive method to observe and quantify minute birefringence in optical fiber in a cavity configuration by detecting the homodyne polarization mode beat effect. The beat frequency is proportional to the magnitude of the birefringence. We present a simple theoretical model for the effect and compare its predictions with experimental findings.

2. Theory

We consider an optical system composed of a linearly polarized light source, a Fabry-Perot type resonator, and a polarizing beam splitter (see Fig. 1 below).

Fig. 1. Schematic illustration of the studied optical system, consisting of a linearly polarized light source, a Fabry-Perot type resonator, and a polarizing beam splitter.

Light circulating in the resonator accumulates a phase-shift after a single roundtrip of

4π

rtn

tϕ ωλ

= =l, (1)

where n is the refractive index of the cavity medium, l the cavity length, λ the wavelength of light, ω = 2πc/λ the angular frequency, 2rtt n c= l the roundtrip time, and c the speed of light in vacuum. After a roundtrip, the electric field in the cavity can be expressed as

( )0 expE r iϕ− , where E0 represents the initial field amplitude in the cavity and r denotes the

fractional amplitude attenuation factor by the cavity mirrors and by absorption and scattering in the cavity medium. By continuously injecting monochromatic light of constant amplitude



into the cavity (build-up phase), the complex cavity field amplitude after N accumulated roundtrips will be

( ) ( )( )

1

0 0 00

1 exp 1exp ,

1 exp

NNp

N Np

r i NE E r i p E E a

r i

ϕϕ

ϕ

+

=

− − + = − = =− −∑ (2)

When the injection of light is ceased (start of the ring-down phase) and the remaining light is left to circulate within the resonator for M additional roundtrips, the electric field can be written as

( )0 0exp ,MM N N ME E a r i M E a bϕ= − = (3)

where ( )expMMb r i Mϕ= − .

Next we assume that the optical resonator exhibits some birefringence so that the cavity has two, possibly non-degenerate, linear orthogonal polarization eigenmodes [12], which result in different aN and bM terms for the transversal x and y components of the field. When the cavity output is coupled to a polarizing beam splitter placed after the resonator at an azimuthal angle of β with respect to the effective axis of birefringence of the resonator (see Fig. 2), the output field is divided into two parts Ek (k = 1, 2), each part containing a contribution from both cavity modes

01

02

coscos sin

sinsin cosNx Mx

Ny My

E a bE

E a bE

αβ βαβ β

∝ −

, (4)

where the incident electric field polarization is assumed to make an angle α with respect to the effective axis of birefringence. The intensity at each arm is thus

2 2 2 2

10

2 2 2 22

1cos cos sin sin sin 2 sin 2

4 ,1

cos sin sin cos sin 2 sin 24

x

y

B

II

I II

I

α β α β α β

α β α β α β

= − ′

(5)

where * *x Nx Nx Mx MxI a a b b= , * *

y Ny Ny My MyI a a b b= , * * * *B Nx Ny Mx My Nx Ny Mx MyI a a b b a a b b′ = + , and

20 0I E∝ .

Resonator

α x

y

PBS Arm1 Arm2

β

Fig. 2. Schematic illustration of the relevant angles. The direction of polarization makes an angle α with respect to the effective axis of birefringence of the fiber resonator (x- and y-axis). The polarizing beam splitter (PBS) makes an angle β with respect to the fiber coordinates.

We assume that the signal is detected only when at least one polarization mode is

properly excited. In the following derivation, frequency is chosen so that the x-mode is at the



peak of the resonator transmission and the y-mode is not resonant. A simulated signal with arbitrarily chosen parameter values (r2 = 0.99, ϕx = 0º, ϕy = 5º, α = 45º, β = 45º, N = 0..1000, and M = 0..1000) is depicted in Fig. 3.

Fig. 3. Signal intensities in the two arms (I1 and I2), their sum (I1 + I2), and the beat term IB with tenfold magnification. The maximum of the beat signal IB,max is marked with the dashed envelope-line.

The figure also includes a ten-fold magnified intensity beat term

( )4 sin 2 sin 2B BI I α β′= , which shows a decrease of the beat amplitude when the

electromagnetic field inside the cavity approaches steady-state conditions. The phenomenon occurs in both the build-up and the ring-down phase.

During the build-up phase, when N is growing, M = 0, bMx = bMy = 1, and Nxa is real valued. The beat function is then given by

( )

( ) ( ){ }1

1,

2 Re

12 cos 1 cos 1 ,

1

B Nx Ny

N

y y y yy

I a a

rN r

r dξ ϕ δ ϕ

+

′ =

− = − + + − + −

(6)

where ( )1tan sin 1 cosy y yr rδ ϕ ϕ− = −

, ( )2, 2 cos 1p p

p y yd r r pϕ= − + , and

11,

Ny yr dξ += . In the ring-down phase, when N is constant and M is growing, the beat

function becomes

Build-up Ring-down

I1 + I2 I

nten

sity

0

0

0

0

0 500 1000 1500 2000

I1

I2

N Nmax + M

10 IB



( ) ( ) ( ) ( )( ) ( ) ( ) ( )

( )

11 2

1, 2

1,

2 Re Re Re Re

2 Re Im Re Im

12 cos ,

1

B Nx Ny Mx My

Nx Ny Mx My

NN y M

y yy

I a a b b

a a b b

drM r

r dϕ

++

′ =

−

− = + ∆ −

(7)

where

( )

( )1

sin sin 1tan

1 cos cos 1

y y y yy

y y y y

r N

r N

ϕ ξ ϕ δ

ϕ ξ ϕ δ− − + + ∆ =

− − + +

. (8)

When the build-up phase is sufficiently long ( 1 0Nr + → ), the ring-down beat function can be approximated as

( )

2

1,

12 cos

1M

B y yy

I M rr d

ϕ δ ′ = + − . (9)

The decrease of the beat amplitude follows the decay of the signal and the ratio between the beat amplitude maximum and the sum of the signals, ( ),max 1 2B BA I I I= + remains constant

with time. The one-pass loss of intensity equals the roundtrip loss of the electromagnetic field in the absence of non-reciprocal elements in the cavity, and the time constant can be directly written as ( )lnn c rτ = −l . The homodyne beat frequency generated by the cosine term (Eqs.

(6), (7), and (9)) is

4πB

B

c cf

n L n

ϕ∆= =l

, (10)

where ∆ϕ is the phase difference of the modes after a roundtrip, LB = λ/∆n is the beat length with ∆n denoting the difference in the refractive indices of the two polarizations (∆n << n). Since the derivation is performed in discretized time domain, the beat frequency appears to be redundant for multiples of 2π of the phase difference. By use of continuous time one can prove the unambiguous character of the beat frequency and the applicability of Eq. (10) for any ∆ϕ.

3. Experimental setup

The experimental setup (see Fig. 4) is composed of a light source, a high-finesse fiber-optic cavity, a polarizing beam splitter, and a pair of photodetectors. The light source is an External Cavity Diode Laser (ECDL, EOSI/Newport 2010), whose wavelength and intensity are modulated using a function generator and a pulse generator, respectively. The function generator sweeps the ECDL emission wavelength over two free spectral ranges (∆ν ≈ 1 GHz) with a 20-Hz saw-tooth signal. The pulse generator modulates the intensity with a 3.5 µs period and 50% duty cycle. The rise-up and fall-down times of the laser intensity with the current setup were about 10 ns. The pulse generator sends a triggering signal to the oscilloscope at the falling slope of the pulse. The high-finesse fiber-optic cavity is constructed by attaching standard FC/PC connectors to a conventional single-mode telecom fiber (Corning SMF-28) and depositing high reflectivity dielectric coatings (Evaporated Coatings Inc; nominal R > 0.999 at 1550-1560 nm) directly onto the connector end facets. Short (about 20 cm) patch cables are attached to both ends of the cavity. The construction of the fiber cavity and its properties are discussed in greater detail in Ref. [4]. The output of the cavity is directed to a polarization controller (Thorlabs, FPC031) and thereafter to the polarizing beam splitter (OZ Optics, FOBS-22P). The signals in the output arms of the beam splitter are



detected with InGaAs-photodetectors (Thorlabs, FC400) followed by transimpedance amplifiers (FEMTO, 40 MHz, 100 kV/A). The amplifier gains are set to be as symmetric as possible for both arms. The signals are displayed on an oscilloscope (Tektronix TDS620, 8 bit, 500 MHz). The oscilloscope starts the data acquisition when it receives the triggering signal from the pulse generator and when the intensity of one arm exceeds a predefined level. The oscilloscope averages 20 signals before the data are sent to a computer for further analysis.

Fiber cavity PBS

ECDL

Functiongenerator

Pulsegenerator

Fixed Twisted

Photo-diode

Photo-diode

Amplifier

Amplifier

Oscilloscope

3.5 µs

Trigger

∆ν = 1 GHz20 Hz Polarization

controller

Fig. 4. Experimental arrangement for polarization mode beat and fiber cavity birefringence measurements. The setup includes a light source, a function generator and a pulse generator. The wavelength is swept over 1 GHz range in 0.05 s (20 Hz) period and the intensity is switched on and off in a 3.5 µs cycle. The setup further contains a twisted high finesse fiber optical cavity, a polarization controller, and a polarizing beam splitter. The data are acquired with an oscilloscope. Abbreviations: ECDL – external cavity diode laser, PBS – polarizing beam splitter.

The polarization mode beat effect is generated by the interaction of two cavity-modes.

Since optical fibers in practice always exhibit some birefringence [1], the two polarization modes of the injected optical light experience different roundtrip path lengths within the resonator. In the current measurement setup, the data are acquired only when at least one of the modes is properly mode-matched, i.e. the phase-shift of the mode is an integer of π. It may be that the other mode is not phase-matched and some residual light energy will circulate within the resonator. While the output electric field is divided by the beam splitter, each arm contains a contribution from both the mode-matched and the non-mode-matched electric fields, which together generate the beating signal. The beat has a homodyne nature, because the electric fields originate from the same light source and thus the angular frequency is equal for both modes.

The observed beat has the following characteristics: when the polarization controller is set to divide the signal between the two arms, the sum of the channels will not show the beat, since the waveforms are in opposite phase (Figs. 3 and 5(a)). The beat is absent also when the polarization controller is adjusted to direct all energy to one arm (β = 0 or β = π/2). The same happens when the polarization of the incoming electric field coincides with the effective axis of birefringence (α = 0 or α = π/2), or both cavity modes are phase-matched (ϕx, ϕy = 2πq, q = 0, 1, 2, …). Furthermore, the beat is present only in the beginning of the ring-up or ring-down events, and disappears when the field inside the cavity approaches steady-state. These observations can easily be explained within the framework of the theory presented in Section 2.



4. Measurement results

The acquired signals show the cavity ring-up and ring-down events in the presence of polarization mode beats, provided that the requirements for the beat are fulfilled. A signal with strong polarization mode beat in a 0.6 m long fiber cavity is presented in Fig. 5(a). As predicted by Eqs. (7) and (9), the beat amplitude follows in the ring-down phase the general decay of the cavity intensity and the normalized signal amplitude stays constant (Fig. 5(b)). Fitting of Eq. (5) to the data (solid line) yields the following parameter values: r = 0.9942, α = –39º, β = 35º, ϕx = 0º, ϕy = 14.8º, fB = 6.97±0.61 MHz, which refers to a beat length LB of 29.3 m.

Fig. 5. Signals with strong polarization mode beat effect acquired from a 0.6 m long fiber cavity. (a) Build-up and ring-down phases from the two measurement arms and their calculated sum. (b) The oscillations from arm 1 normalized against the sum intensity.

We also made measurements with a 0.275 m long single-mode fiber cavity, twisted from

one end around its axis. The build-up and ring-down signal forms for this fiber were recorded for increasing twist angle. By fitting the theoretical model to the acquired signals the relevant parameter values, i.e., the angle between the incoming state of polarization and the effective axis of birefringence α, the beat frequency fB, the time constant τ, and the normalized beat amplitude AB could be determined. As discussed in Ref. [3] the twist of a short fiber cavity results in a revolution of the effective axis of birefringence, which with our setup would manifest itself as a change in α. Another anticipated effect of the twist is the periodic change of the beat length. The results of the twist experiment are represented in Fig. 6. The change of the effective axis of birefringence (Fig. 6(a)) follows closely the findings of Ref. [3] for a partially twisted fiber cavity. Our fiber cavity is short and there exist untwisted parts at both ends where the ferrules keep a portion of the fiber straight. A π/2 jump in α occurred whenever triggering in the setup was changed from arm 1 (solid line) to arm 2 (dashed line) or vice versa. As expected, the beat frequency varies periodically with the twist angle (Fig. 6(b)).

Sum Arm1

Arm2

0 500 1000 1500 2000 2500

900 1000 1100 1200 1300 1400 1500 1600

Time (ns)

Time (ns)

Nor

mal

ized

bea

t int

ensi

ty

Sig

nal

(mV

) a)

b)

-20

0

20

40

60

80

-0.1

-0.05

0

0.05

0.1



The lowest measured beat frequency, 19.3 MHz, refers to 10.6 mBL = , i.e., the measurement was performed with a fiber, whose length is less than 3% of the beat length. A measurement of fiber birefringence with a ratio of similar magnitude was demonstrated recently by Kuzin et al. [13]. The beat amplitude (Fig. 6(c)) relates to the angle of the birefringence α and therefore at some twist angles, where sin 2α is close to zero, the beat frequency cannot reliably be determined. As seen in Fig. 6(c), at some twist angles the time constant drops by about 10 ns,

which results in an increase of 80 ppm ( 43.5 10−× dB) in the fractional intensity loss per one pass inside the resonator. The increase of the loss coincides with the maxima of the beat amplitude AB, which are related to the angle of the effective axis of birefringence α. The increase of the transmission loss is so small that it would be unnoticed in traditional measurement setups. This minor, though interesting phenomenon has no adequate explanation within the theory presented above.

Fig. 6. Measurement results of the twisted 0.275 m long fiber cavity. Angle of the effective axis of birefringence α (a), the beat frequency fB (b), the resonator time constant τ (c) and the normalized beat amplitude AB are plotted for varying twist angles. In (a) a π/2 jump occurs whenever the triggering is changed from arm 1 (solid line) to arm 2 (dashed line).

5. Conclusions

The work presents a measurement technique for quantification of minute birefringence in an optical resonator arrangement. The method is based on observation of the homodyne beat of the matched and unmatched resonator polarization eigenmodes. In the paper we derive a theoretical framework, which shows that the beat frequency is directly linked to the phase mismatch of the suppressed cavity mode, and demonstrate the sensitivity of the technique by using high-finesse single-mode fiber-optic cavities. The experimental observations were found to support the theoretical predictions, namely: 1) the beat of the two output arms have equal amplitude and reversed phase, and is thus absent when the signals are summed, 2) the beat may be observed during the beginning of the build-up, or ring-down phase, but is absent when the cavity has reached the steady state, 3) the amplitude of the beat follows the decay of

0 50 100 150 200 250 300 350 Twist angle (deg)

0 50 100 150 200 250 300 350 Twist angle (deg)

0 50 100 150 200 250 300 350 Twist angle (deg)

a)

b)

c)

-90

0

90

180

270

20

25

30

380

390

400

410

420

370 0

2

4

6

8

10.2

8.2

6.8

τ (

ns)

f B (

MH

z)

α (

deg)

LB (

m)

AB

(%)



the ring-down signal and normalized beat amplitude stays constant, and 4) occurrence of the beat requires suitable linear polarization from the incoming light and proper alignment of the polarizing beam splitter. The measurements were performed in a normal laboratory environment where the fluctuation of air was not eliminated. Therefore the modes of the fiber cavity wandered randomly forth and back, but while the polarization modes were affected equally by temperature-induced changes, the refractive index difference ∆n remained constant and the fluctuation was not found to affect the measurements. The beat length of a 0.275 m long twisted single-mode fiber cavity was measured to be maximum of 10.6 m, which is several times the length of the fiber itself. We believe the technique is suitable also for other low birefringence setups, like for measurement of the Cotton-Mouton effect.

Acknowledgments

The experimental part of the work was conducted in collaboration with Prof. M. W. Sigrist in the Laboratory for Laser Spectroscopy and Sensing (Institute of Quantum Electronics) at Swiss Federal Institute of Technology (ETH). The authors wish to express their thankfulness to him.



resonator based measurement technique for quantification of minute birefringence

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