Optics Communications 283 (2010) 4435–4439

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Optics Communications

j ourna l homepage: www.e lsev ie r.com/ locate /optcom

Coherence and the statistics of the phase difference between partially polarizedelectromagnetic waves

Alfredo LuisDepartamento de Óptica, Facultad de Ciencias Físicas, Universidad Complutense, 28040 Madrid, Spain

E-mail address: alluis@fis.ucm.es.URL: http://www.ucm.es/info/gioq/alfredo.html.

0030-4018/$ – see front matter © 2010 Elsevier B.V. Aldoi:10.1016/j.optcom.2010.04.089

a b s t r a c t

a r t i c l e i n f o

Article history:Received 27 June 2009Received in revised form 21 November 2009Accepted 21 April 2010

Keywords:CoherencePolarizationInterference

Coherence between two vectorial harmonic light vibrations is analyzed in terms of the statistics of theirphase difference. This provides a natural and simple extension of second-order coherence to cover morecomplicate situations. In particular this assigns large coherence to quantum light states providing the mostaccurate interferometric measurements allowed by the quantum theory, even if they are incoherentaccording to the standard second-order approach.

l rights reserved.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

Coherence is a key concept in optics derived from the statisticalnature of real light beams [1–4]. We focus on two partially polarizedharmonic light vibrations with complex amplitude vectors E1,2. Fordefiniteness we can regard them as representing the light amplitudesat the two pinholes of a Young interferometer, or the amplitudes ofthe internal beams of a Mach–Zehnder interferometer.

Themotivation for thiswork is that the standard second-orderdegreesof coherence μ for scalar and vectorial waves introduced so far [1–11] areinsufficient toaddressmany relevant interferometric situations. This is thecase in quantum metrology, where light states providing the maximuminterferometric resolution allowed by the quantum theory have, maybeparadoxically, μ=0 [12–25]. It is worth noting that this holds as aconsequence of symmetry properties satisfied by the field state, and notby a large amount offluctuations, so that despite having μ=0 these statesallow the most precise interferometric measurements by using suitabledetection techniques.

This work develops an alternative measure of coherence avoidingthese difficulties. To this end we identify coherence between the twovibrations with their phase difference, so that the amount of coherenceis determined by the phase-difference fluctuations. This program hasbeen already addressed for scalar light, i.e., for coherence between fullypolarized waves [26], being extended here to partially polarized waves.

Complete information about the phase difference ϕ is provided bythe marginal distribution P(ϕ) that can be properly derived from thecomplex amplitude distribution P(E1, E2). Then the amount of

coherence can be assessed, for example, by the distance between P(ϕ)and the uniform distribution 1/(2π) associated to fully incoherent light(completely random phase difference).

This approach may be interesting in some circumstances to analyzephasefluctuations and their implications for coherence. The identificationof coherence with phase difference provides a simple and naturalextension of coherence beyond second-order optics since the phasedifference is a nonlinear function of the complex amplitudes. This isachieved without involving an infinite hierarchy of correlation functionsof arbitrary orders, that are specially difficult to handle in the vectorialcase.

2. Coherence and phase difference for partially polarized light

2.1. Coherence and phase difference

We consider transversal harmonic electromagnetic fields with twocomponents, say Ex,y, at two definite spatial points r1,2. These fielddegrees of freedom can be described with the two complex amplitudevectors

Ej =Ej;xEj;y

� �; j = 1;2: ð1Þ

In comparison with the scalar case, for vectorial light the increaseon the number of degrees of freedom allows the introduction ofseveral different second-order degrees of coherence that focus ondifferent aspects of coherence [5–11]. Accordingly, many differentrelative phases can be introduced depending on the practical schemeto be considered [27,28]. This is because, as shown in [28], one can

4436 A. Luis / Optics Communications 283 (2010) 4435–4439

combine the four field components in many different ways to produceinterference fringes with different visibilities.

In this work, for definiteness we focus on the Young schemeillustrated in Fig. 1 (or equally well any other two-beam interferom-eter), where the relevant interferometric phase difference ϕ is sharedby the two polarization components of the interfering waves. Let usconsider the following modulus-phase expressions for the complexamplitudes:

E1;x = ρ1;x exp i ξ−ϕ + δ12

� �� �; E2;x = ρ2;x exp i ξ +

ϕ−δ22

� �� �;

E1;y = ρ1;y exp i ξ−ϕ−δ12

� �� �; E2;y = ρ2;y exp i ξ +

ϕ + δ22

� �� �;

ð2Þ

where ρj,k=|Ej,k|≥0, so that ρj,x2 +ρj,y2 is the light intensity at eachpoint rj, π≥δj≥−π is the phase difference between the polarizationcomponents Ej,x and Ej,y at each point rj, π≥ξ≥−π is a global phase,and π≥ϕ≥−π is the phase difference between E1 and E2. Thepolarization state at each point rj is determined by ρj,x/ρj,y and δj. Inorder to avoid periodicity problems, throughout we will assume afixed−π to πwindow for all the phases. The Jacobian associated to thechange of coordinates in Eq. (2) is

d2E1;x d2E1;y d

2E2;x d2E2;y = ρ1;xρ1;yρ2;xρ2;y dρ1;x dρ1;y dρ2;x dρ2;ydξdδ1dδ2dϕ:

ð3Þ

Alternative definitions of the phase difference via other phaseparameterizations might be possible, leading in general to differentresults. The choice in Eq. (2) is motivated by its explicit symmetrybetween all field components. The price paid is that the 2π periodicityon ϕ is not explicit, being this the reason to fix once for all the phasewindows.

2.2. Probability distribution

Complete information about the phase difference ϕ is provided byits probability distribution P(ϕ). This can be derived from P(E1, E2)after the change of variables in Eq. (2) by performing the integrationover the other variables different from ϕ

P ϕð Þ = ∫π

−πdξdδ1dδ2 ∫

∞

0

dρ1;x dρ1;y dρ2;x dρ2;yρ1;xρ1;yρ2;xρ2;yP E1;E2ð Þ;

ð4Þ

where Eq. (2) should be used in P(E1, E2). Equivalently,

PðϕÞ = ∫d2E1;x d2E1;y d

2E2;x d2E2;y δ ϕ−Φ E1;E2ð Þ½ �P E1;E2ð Þ; ð5Þ

where δ is the Dirac delta, and

Φ E1;E2ð Þ = 12

arg E2;x� �

+ arg E2;y� �

− arg E1;x� �

− arg E1;y� �h i

: ð6Þ

Fig. 1. Scheme illustrating the fields considered. The dashed arrows represent therelative phases between different components.

2.3. Degree of coherence

The amount of coherent can be assessed in terms of the distance Dbetween P(ϕ) and the uniform distribution 1/(2π) representing fullyincoherent light (i. e., completely random phase difference) as [26]

D = 2π ∫2π

dϕ P ϕð Þ− 12π

� �2: ð7Þ

Since D is unbounded we can construct a normalized amount ofcoherence 0≤γ≤1 as

γ =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiD

D + 1

r: ð8Þ

This formulation is consistent with similar definitions of otheroptical properties as suitable distances [10,26–41].

Many different distance measures between phase distributionsmight be used in Eq. (7), being reasonable to expect that differentchoices will not radically change the conclusions. The choice in Eq. (7)is made for the sake of simplicity. Amore theoretically sounded choicemight be the Kullback measure, as already considered in polarization[42].

Coherence is determined by the amount of fluctuations of thephase difference. This means that D and γmight be related with somepreviously introduced measures of phase uncertainty. This is so sincewe have

D =1Δϕ

−1; γ =ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1−Δϕ

p; ð9Þ

where Δϕ is the Süssmann measure of phase uncertainty defined as

Δϕ =1

2π∫π−πdϕP

2 ϕð Þð10Þ

that has been used in different contexts as a measure of uncertaintyand information [43–55].

2.4. Invariance

The amount of coherence γ is invariant under any transformationpreserving the phase-difference statistics. Let us show some simpleexamples, such as the complex scale changes

Ej→MjEj; Mj =ηj;x 00 ηj;y

� �; ð11Þ

with ηj,k≠0 that modifies ρ1,2,x,y but preserves the phases ξ, δ1,2, ϕ upto constant phase shifts that do not modify D and γ. There is alsoinvariance under local permutation of the components

Ej→MjEj; Mj =0 ηj;xηj;y 0

� �; ð12Þ

and also under phase conjugation of both fields E1→E1⁎ and E2→E2⁎.

3. Examples

In this sectionwe apply the above formalism to some relevant fieldstates. We will show that for some classic Gaussian light the phase-difference approach essentially coincides with standard second-orderapproaches, while for quantum light providing maximum interfero-metric resolution they lead to fully opposite results.

Fig. 3. γ as a function of μx,y for the particular case μx=μy.

4437A. Luis / Optics Communications 283 (2010) 4435–4439

3.1. Thermal light

Let us consider thermal light described by a zero-mean Gaussiandistribution for the complex amplitudes

PG εð Þ = 1π4detΨ

exp −ε†Ψ−1ε� �

; ε = E1E2

� �; ð13Þ

where Ψ is a 4×4 Hermitian complex matrix

Ψ = εε†D E

=Γ1 ΩΩ† Γ2;

� �ð14Þ

Γ1,2 are 2×2 Hermitian complexmatrices containing the intensity andpolarization at each point rj, while Ω is the 2×2 complex matrixcontaining the correlation terms between the field components atpoints r1,2,

Γj = EjE†j

D E; Ω = E1E

†2

D E: ð15Þ

We assume detΨ≠0 and detΓj≠0 since otherwise we areessentially in the scalar case.

No simple general expressions are available unless for the mostsimple cases such as

Ψ =

Ix 0ffiffiffiffiffiffiffiIxI

′x

qμx 0

0 Iy 0ffiffiffiffiffiffiffiIyI

′y

qμyffiffiffiffiffiffiffi

IxI′y

qμ x 0 I′x 0

0ffiffiffiffiffiffiffiIyI

′y

qμy 0 I′y

0BBBBBBB@

1CCCCCCCA; ð16Þ

where 1≥μ x, μ y≥0 are the intrinsic degrees of coherence introducedin [9].

The details of the derivation of the phase-difference distributionand the associated coherence γ are in Appendix A. In Figs. 2 and 3 wehave plotted γ as a function of μ x,y for large values of μ x,y, showingthat γ and μ x,y provide equivalent information. In Appendix A it can beseen that this also holds for small μ x,y.

The main conclusion is that for this thermal Gaussian example thephase-difference approach essentially coincides with standard sec-ond-order approaches.

3.2. Twin photon states

As a more sophisticated example let us consider a field state withquantum metrological applications. This is four-mode twin photon-number state jn⟩1;x jn⟩1;y jn⟩2;x jn⟩2;y with n photons in each mode Ej,k

Fig. 2. γ as a function of μx,y.

[12–22,26]. More specifically we focus on its coherence propertiesafter passing through the input 50% isotropic lossless beam splitter ofa Mach–Zehnder interferometer illustrated in Fig. 4.

As a suitable field distribution P(E1, E2) we consider the Q function,defined by projection of the field state on coherent states (assumingE1,2 dimensionless by a suitable scaling) [2,56]. This means that thephase-difference distribution P(ϕ) is given by the quantum phase-difference formalism based on the Q function [57–64].

The problem factorizes as the product of the distributions for the xand y components

Pn εð Þ = Pn E1;x;E2;x� �

Pn E1;y;E2;y� �

; ð17Þ

where [26]

Pn E1;E2ð Þ = 1πn!

� �2 E21−E22 2n

22n exp − E1j j2− E2j j2� �

: ð18Þ

Since Pn(ε) is an even function of its arguments Pn(−E1, E2)=Pn(E1, −E2)=Pn(E1, E2), we get E1;jE42;k

D E= 0 and Ω = E1E

†2

D E= 0,

so that all second-order approaches for vectorial coherence predictcomplete lack of coherence. As mentioned above this holds becausesymmetry and not because of randomness.Moreover, these states arealso second-order unpolarized Γj = EjE

†j

D E∝ identity although they

are highly polarized beyond second-order optics [29,36].Since Ω=0 the average values of the output intensities in Fig. 4

vanish, so that they are insensitive to phase. A simple way to obtainphase-dependent measurements is to consider averages of higher-order magnitudes, such as the square of differences of outputintensities or their products.

Fig. 4. Scheme of a Mach–Zehnder interferometer illuminated by the twin photon-number state jn⟩1;x jn⟩1;y jn⟩2;x jn⟩2;y .

4438 A. Luis / Optics Communications 283 (2010) 4435–4439

Performing the change of variables in Eq. (2) and using Eq. (4) weget to

Pn ϕð Þ = 12π

∫π

−πdδ1dδ2Pn ϕ− δ2−δ1

2

� �Pn ϕ +

δ2−δ12

� �; ð19Þ

where Pn(ϕ) is the phase difference distribution for the x and ycomponents [26]

Pn ϕð Þ = ∑n

k=0ck cos

2k ϕ; ck =2n + 1ð Þ!ffiffiffiπ

p4n + 1n!

−1ð Þkn−kð Þ!Γ k + 3= 2ð Þ : ð20Þ

It can be seen that Pn(ϕ) consists of two equal peaks at ϕ=±π/2whose width decreases as n increases.

In order to proceed further let us take into account that theinterferometric usefulness of these states requires n≫1, so that thepeaks in Pn(ϕ) are very narrow. Then, if we approximate them byGaussians

Pn ϕð Þ≃ 12ffiffiffiffiffiffi2π

pσ

exp − ϕ−π=2ð Þ22σ2

" #+ exp − ϕ + π=2ð Þ2

2σ2

" #( ); ð21Þ

with σ2⋍2/n we get that Eq. (19) leads to

Pn ϕð Þ≃ 1

4ffiffiffiffiffiffiffiffiffiπσ2

p f exp − ϕ−π=2ð Þ2σ2

" #+ exp −ϕ2

σ2

!

+ exp − ϕ + π=2ð Þ2σ2

" #+ exp − ϕ−πð Þ2

σ2

" #g:ð22Þ

The two-peak structure of Pn(ϕ) leads to a Pn(ϕ) distribution withfour identical and equally spaced peaks with half of the width of thepeaks of Pn(ϕ).

Finally we get

γ≃ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1− 4ffiffiffiffiffiffi

πnp

s: ð23Þ

This shows that γ approaches its maximum value γ→1 as nincreases, in agreement with the interferometric usefulness of thesestates. This is because both interferometric resolution and γ dependon the width of the peaks (this is its sensitivity to phase shifts), butnot on the number and position of the peaks or any other symmetryproperty.

4. Conclusions

We have developed a measure of phase coherence betweenpartially polarized waves by identifying coherence with phasedifference. This provides a very simple extension of coherence beyondsecond-order optics that includes in a single formalism both classicand quantum light states.

The phase-difference approach and the standard approachesessentially coincide when applied to simple Gaussian light, but leadto opposite results when applied to quantum light providing themaximum resolution in interferometric measurements. Such stateshave vanishing second-order degrees of coherence, while they arehighly coherent in the alternative formalism presented in this work, inagreement with their interferometric usefulness.

Acknowledgment

This work has been supported by Project No. FIS2008-01267 of theSpanish Dirección General de Investigación del Ministerio de Ciencia eInnovación.

Appendix A. Phase-difference distribution for thermal light

Here we derive the phase-difference distribution for thermal light inEq. (13) in the particular case in Eq. (16), computing then the coherenceγ. In the particular case in Eq. (16) the probability distribution for thecomplex amplitude PG(ε) factorizes into the product of two distributionsfor the pairs of scalar components (E1,x, E2,x), and (E1,y, E2,y), withsecond-order scalar degrees of coherence given by the intrinsic degreesof coherence of the vectorial waves μx and μy, respectively

PG εð Þ = PG;x E1;x;E2;x� �

PG;y E1;y;E2;y� �

; ðA:1Þ

where PG, j for j=x, y, are

PG; j E1; j;E2; j� �

=1

π2detΨjexp −E†

jΨ−1j Ej

� �; Ej =

E1;jE2;j

� �; ðA:2Þ

with

Ψj = EjE†j

D E=

IjffiffiffiffiffiffiIjI

′j

qμ jffiffiffiffiffiffi

IjI′j

qμ j I′j

0@

1A; ðA:3Þ

and we have assumed detΨj≠0 [26].The above factorization inEq. (A.1) leads to this convolution formula for the phase-differencedistribution PG(ϕ)

PG ϕð Þ = 12π

∫π

−πdδ1dδ2PG;x ϕ− δ2−δ1

2

� �PG;y ϕ +

δ2−δ12

� �; ðA:4Þ

where [26]

PG;j ϕð Þ = ∑∞

n=0cn cos

n ϕ; cn =1−μ2

j

2π

2nΓ2 n2+ 1

� �n!

μnj ; ðA:5Þ

and Γ is the gamma function. It can be seen [26] that PG, j(ϕ) has asingle peak at ϕ=0whose width decreases as μ j increases.In order toproceed further let us consider that μ j is large enough so that the peakof PG, j is narrow and we can approximate it by a Gaussian

PG; j ϕð Þ≃ 1ffiffiffiffiffiffi2π

pσj

exp − ϕ2

2σ2j

!; ðA:6Þ

with

σj≃ffiffiffiπ

p1−μ2

j

� �ðA:7Þ

we get from Eq. (A.4)

PG ϕð Þ≃ 1ffiffiffiffiffiffi2π

pσexp − ϕ2

2σ2

!; σ2 =

14

σ2x + σ2

y

� �ðA:8Þ

leading to

γ≃ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1− σffiffiffi

πp

r: ðA:9Þ

On the other hand, for small μ x,y we get from Eq. (A.5) up to μ x,y2 ,

PG; j ϕð Þ≃1−μ2j

2π+

μ j

4cosϕ +

μ2j

πcos2 ϕ; ðA:10Þ

leading via Eq. (A.4) to

PG ϕð Þ≃ 12π

− π16

μxμy +μx + μy

π2 cosϕ +π8μxμy cos

2 ϕ; ðA:11Þ

4439A. Luis / Optics Communications 283 (2010) 4435–4439

so that

γ≃ffiffiffi2

p

πμx + μy

� �: ðA:12Þ

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