458 Vol. 37, No. 3 / March 2020 / Journal of the Optical Society of America A Research Article

Characterization of the electromagnetic GaussianSchell-model beam using first-order interferenceSethuraj K. R. AND B. Kanseri*Experimental Quantum Interferometry and Polarization (EQUIP), Department of Physics, Indian Institute of Technology Delhi, Hauz Khas,NewDelhi 110016, India*Corresponding author: bkanseri@physics.iitd.ac.in

Received 21 November 2019; revised 4 January 2020; accepted 9 January 2020; posted 10 January 2020 (Doc. ID 384061);published 19 February 2020

We propose a method for the characterization of electromagnetic Gaussian Schell-model (EMGSM) beams.This method utilizes the first-order interference consisting of polarization-state projections along with the two-point (generalized) Stokes parameters. The second-order field correlations employed in this method enable us todetermine both the magnitude and the argument of the complex degree of electromagnetic coherence. We exper-imentally demonstrate this method by characterizing an EMGSM beam, which is synthesized using a laser beampassing through a rotating ground glass diffuser. This beam-characterization method is expected to be potentiallyuseful for probing the partially coherent and partially polarized beams, and have tremendous applications in broadareas of optical communication and beam propagation. ©2020Optical Society of America

https://doi.org/10.1364/JOSAA.384061

1. INTRODUCTION

Partially coherent optical beams [1–3], owing to their inherentimmunity towards diverse possible losses (such as atmos-pheric turbulence) on propagation, are potential candidatesfor free-space optical communication [4–7] and under-oceanapplications [8–11]. Their unique characteristics have enabledcritical advances in many other fields of research [12–18].Recent studies on the transfer of coherence properties from apump beam to entangled photons [19,20] have also identifiedpartially coherent beams as a potential candidate for free-spacequantum key distribution applications. Schell-model sourceshold a vital space among partially coherent optical beams due totheir mathematical simplicity and shift-invariance of the abso-lute value of the degree of spatial coherence. ElectromagneticGaussian Schell-model (EMGSM) beams are the generalized(partially polarized) form of the famous Gaussian Schell-model (GSM) beams, thus having a huge importance in theapplications just mentioned.

A mathematical description of optical coherence that rep-resents the propagation characteristics of partially coherentand partially polarized light beams in the space–time domainis the beam coherence polarization (BCP) matrix [21]. TheBCP matrix incorporates transverse coherence-polarizationmodulations of a beam in a rather simple 2× 2 matrix rep-resentation. The space–frequency domain representation ofoptical coherence of the light beams is given by the cross-spectraldensity (CSD) matrix [22]. Even though the CSD matrix isof fundamental importance at the theoretical level and can

provide wavelength-dependent properties, the BCP matrix isadequate in situations when time-dependent properties of theoptical fields are under investigation [23]. In the framework ofour experiment, as we are using a quasi-monochromatic beam,both the CSD and BCP matrices yield identical results [24].Hence, throughout this article, we use the BCP matrix for therepresentation of coherence properties of EMGSM beams. Letus consider an EMGSM beam E x x + E y y propagating in thez direction, where E x and E y are x and y field components ofthe complex electric field vector at any single space–time point.Then the elements of the BCP matrix [21] representing thebeam can be written as

Jαβ(r1, r2, z)= 〈E ∗α(r1, z)Eβ(r2, z)〉

= Iαβ(r1, r2, z)γαβ(r1, r2, z),

for (α, β = x , y ), (1)

where r1 and r2 are the two spatial points in the constant zplane, 〈·〉 represents the ensemble average, and Eα(r1, z) andEβ(r1, z) are the polarization projections of the complex elec-tric fields at the two points r1 (with polarization projection α)and r2 (with polarization projection β). A more intriguing rep-resentation of Jαβ(r1, r2, z) is the second part of Eq. (1), whereγαβ(r1, r2, z) is the polarization-dependent complex degreeof cross-coherence and Iαβ(r1, r2, z) is the geometric mean ofpolarization-dependent intensity distributions at the two points(r1 and r2) in consideration. In a Gaussian Schell-model (GSM)beam, the transverse intensity and spatial coherence profiles

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Research Article Vol. 37, No. 3 / March 2020 / Journal of the Optical Society of America A 459

of a partially coherent scalar beam are Gaussian distributionscentered at its optic axis. The EMGSM beam is an extension ofthe GSM beam model to vector fields (exchangeable with theterm ‘electromagnetic’), and hence well suited for several of thejust-mentioned explorations. Iαβ(r1, r2, z) and γαβ(r1, r2, z)for an EMGSM beam [1,25] are then defined as

Iαβ(r1, r2, z)= Iαβ(0, 0, z) exp

(−

r 21

4σ 2α (z)−

r 22

4σ 2β (z)

),

γαβ(r1, r2, z)= γαβ(0, 0, z) exp

(−(r1 − r2)

2

2δ2αβ(z)

), (2)

where σα(z) and σβ(z) are the beam radii for the polarizationprojections α and β respectively, Iαβ(0, 0, z) is the geometricmean of intensities of the beam at the optic axis (r1 = r2 = 0)for the polarization-state projections α [i.e., Iαα(0, 0, z)]and β [i.e., Iββ(0, 0, z)] respectively, δαβ(z) is the coherenceradius for α and β, and γαβ(0, 0, z) is the degree of coherencebetween α and β polarization-state projections of the opticalfields at r1 = r2 = 0. The EMGSM beams have been generatedin the past using several methods [26,27]. The parametersσα(z), σβ(z), δαβ(z), γαβ(0, 0, z), and Iαβ(0, 0, z) are cru-cial in determining the properties of the field [3,28], whichalso determine the realizability condition for EMGSM beams[29–31] given as√

δ2x x (z)+ δ2

y y (z)

2≤ δx y (z)≤

√δx x (z)δy y (z)|γx y (0, 0, z)|

. (3)

Hence, precise knowledge of all the parameters is of primeimportance in any experiment involving an EMGSM beam.

The parameters governing the polarization-dependentintensity propagation (Iαβ(r1, r2, z)) can be directly mea-sured by a polarization state selector and a beam profilerarrangement, whereas the parameters determining thepolarization-dependent complex degree of cross-coherence(γαβ(r1, r2, z)) are the decisive parameters left in determiningthe second-order statistical properties of the EMGSM beam.In the past, a method proposed by Wang et al. [32] has beenused for determining the absolute value of transverse complexcross-coherence (|γαβ(r1, r2, z)|) of the EMGSM beam fromfourth-order correlation (photon coincidence) measurementsthrough Gaussian moment theorem [1]. To our knowledge, noother method has been proposed to date for the characterizationof the EMGSM beam. Considering the wide variety of applica-tions in which the EMGSM beams are used, the requirementof a less expensive, less complicated, more rugged, and precisecharacterization scheme is essential. One possible way to ensuresuch performance is by utilizing the second-order field corre-lations, which are more fundamental in statistical optics anddefine the EMGSM beam. Clearly, in this respect, there exists agap in both theory and experiment which needs to be filled.

In this paper, we propose a novel method for the characteriza-tion of EMGSM beams. This method employs the second-orderfield correlations for different polarization components of theelectric fields encapsulated in the two-point (generalized) Stokesparameters [33], which are used to determine the transverse

complex spatial coherence distribution of the electromagneticbeam. Owing to the classical nature of the measurement process,it is more robust to noise and needs less expensive off-the-shelfcomponents for implementation. Further, we experimentallydemonstrate this method by synthesizing an EMGSM beam inthe lab and then characterizing this beam through the measure-ment of parameters using the new method. This proposal, dueto its conceptual and experimental simplicity and completeness,is expected to be useful in a wide range of applications and exper-iments that utilize the EMGSM beams. Section 2 of the paperdetails our theoretical proposal, and Section 3 demonstrates thesynthesis of an EMGSM beam and the experimental implemen-tation of the proposed method by characterizing the beam. Theoutcomes of the experiment are discussed in Section 4 of thepaper.

2. PROPOSAL

The spatial coherence function, which is a complex quantity, iswell understood in the scalar theory of light and measured usinga Young’s-type interferometer [34]. Many modified versionsof this measurement scheme are available in the literature tomeasure the spatial coherence distribution across the trans-verse beam profile [34–40]. These amplitude interferometrytechniques have a clear advantage over the intensity interfer-ometric schemes, as they reveal the complete complex natureof the coherence distribution. Over the years, there have beenseveral experimental demonstrations for the measurement ofthe elements of the coherence-polarization matrix elements attwo distinct transverse spatial points using field interferometryin both space–time and space–frequency domains [23,41,42].Extension of such a matrix for different pairs of points across thebeam profile may yield the EMGSM beam coherence param-eter, which, however, involves a more complicated experimentalprocedure. We aim to use a more straightforward scheme wherethe scalar theory is employed to find the coherence parametersfor all the six standard polarization projections of the EMGSMbeam. After that, the current scheme uses two-point Stokesparameters [33], which allow us to extract the complex coher-ence details between any orthogonal polarization projections ofthe EMGSM beam for the whole transverse beam profile.

A. Spatial Coherence of Individual PolarizationProjections

We select a particular polarization state α of an EMGSM beamusing a polarization state selector, and place a double-slit (ofslit-separation a ) in the transverse-plane (z= z0, Fig. 1) in sucha way that one of the slits is on the optic-axis (r1 = 0). Then, theintensity and degree of spatial coherence distributions for thepolarization components of the EMGSM beam [Eq. (2)] acrossthe two points r1 = 0 and r2 = a in the transverse plane (z= z0)are given by

Iαα(0, a , z0) = Iα(0, 0, z0) exp(−

a2

4σ 2α

),

γαα(0, a , z0) = γαα(0, 0, z0) exp(−

a2

2δ2αα

),

(4)

460 Vol. 37, No. 3 / March 2020 / Journal of the Optical Society of America A Research Article

Fig. 1. Schematic to illustrate the interferometric measurementstrategy to obtain the γαα(0, a , z0) [Eq. (1)]. α can be any of thepolarization projections of the complex electromagnetic field. The bluecurve shows the polarization-dependent Gaussian intensity profileIα(z0) of the beam at the z= z0 plane. Notations: DS, double-slit; OA,optic-axis; min, minimum; max, maximum. The superposition of thefields from the two slits at r1 and r2 (also individually from each of theslits) is obtained at the z= z1 plane.

where σα and δαα , respectively, are the beam and coherence radiiof the α polarization state projection in the transverse plane[z= z0].

The quantity γαα(0, a , z0)= |γαα(0, a , z0)| exp(iθαα(0, a , z0)), where θαα(0, a , z0) is the argument ofγαα(0, a , z0) and i =

√−1, is obtained from the double-

slit interference of the polarization projection α of the beam.Interference fringes are observed in the z= z1 plane. Using thegeneral interference law for stationary fields [34], the real valueof the complex degree of coherence of the polarization stateprojection α at any point p in the transverse plane z= z1 can beobtained as

Re .γαα(τ )=Iαα − I (1)αα − I (2)αα

2√

I (1)αα I (2)αα

, (5)

where Re . represents the real part; I (1)αα , I (2)αα , and Iαα are inten-sities at point p (Fig. 1) when only the slit at r1 is open, onlythe slit at r2 is open, and both slits are open, respectively. Thetemporal separation for light from r1 and r2 to reach the pointp is given by τ = R2−R1

c , where c is the speed of light. The argu-ment [34] of this polarization-dependent complex degree ofcoherence is given by

Ar g .γαα(τ )= θαα(τ )+2π(R1 − R2)

λ, (6)

where Ar g . represents the argument, λ is the mean wavelengthof the EMGSM beam, and R1 and R2 are the distances from thepoints r1 and r2 (in the plane z= z0), respectively, to the point p(in the plane z= z1).

When the temporal separation (τ ) is much less than thetemporal coherence (τc ) of the beam, then one can safely assumethat |γαα(τ )| ≈ |γαα(0, a , z0)| and θαα(τ )≈ θαα(0, a , z0).Then, Re .γαα(τ )≈ |γαα(0, a , z0)| cos[(θαα(0, a , z0)+2π(R1−R2)

λ)]. Therefore, |γαα(0, a , z0)| can be obtained as an

amplitude of cosine fit to the experimentally obtained value ofRe .γαα(τ ) [Eq. (5)], and the argument of γαα(0, a , z0) canbe obtained from the theoretical fit to the experimental datathrough the following:

θαα(0, a , z0)= cos−1

(Re .γαα(τ )

|γαα(0, a , z0)|

)∣∣∣∣(R1=R2)

. (7)

Thus, with the experimentally obtained values of the magni-tude (|γαα(0, a , z0)|) and argument [θαα(0, a , z0)] for the αpolarization-state projection, we can calculate the polarization-dependent complex degree of transverse spatial coherence asγαα(0, a , z0)= |γαα(0, a , z0)| exp(iθαα(0, a , z0)). As a result,Jαα(0, a , z) [Eq. (1)] for any polarization projection α of thebeam is obtained as Jαα(0, a , z0)= Iαα(0, a , z0)γαα(0, a , z0).

Therefore, with an appropriate number of double-slits of dif-ferent slit separations (a ), we can replicate the lateral coherenceprofile and lateral intensity distribution [Eq. (4)] of any polari-zation projection α, where α = {x , y , f ,m, r , l} representsthe horizontal, vertical, 45◦ (to horizontal), 135◦, right-circular,and left-circular polarization states, respectively. We note that,from Eq. (2), it is quite clear that γαβ(r1, r2, z) is shift-invariantin any constant z plane as its magnitude depends only upon therelative measure (r1 − r2)

2 and not on the individual values ofthe r1 and r2. Hence, placing one slit of the double-slits at theoptic axis (r1 = 0) is not necessary for obtaining the δx y alone.Despite this, since our study includes shift-variant quantities(Iαβ and Ar g .γαβ ), we have chosen r1 = 0 for convenience.Also, after the polarization state (α) selection, the spatial coher-ence measurement part of this method can be implemented notonly by this variable slit-separation technique but also by usingany of the earlier-mentioned spatial coherence measurementtechniques [34–40] as well. One can choose an appropriatetechnique based on beam properties such as the beam size anddivergence, as well as other conveniences. Thus, with the knowl-edge of γx x and γy y , we obtain the coherence radii δx x and δy y

(similarly, for all other polarization-state projections as well)using the Gaussian curve fitting [Eq. (4)].

B. Complex Degree of Spatial Coherence forOrthogonal Polarization Components

The complex value of J x y (0, a , z0) is obtained using the gener-alized (two-point) Stokes parameters [43–45], which bridge thepolarization projections of complex degree of transverse spatialcoherence (γαα) to γx y : The two-point Stokes parameters of theEMGSM beam across any two points (0, a ) in the transverseplane z= z0 are then measured as [33]

S0 (0, a , z0) = J x x (0, a , z0)+ J y y (0, a , z0) ,

S1 (0, a , z0) = J x x (0, a , z0)− J y y (0, a , z0) ,

S2 (0, a , z0) = J f f (0, a , z0)− Jmm (0, a , z0) ,

S3 (0, a , z0) = Jr r (0, a , z0)− J l l (0, a , z0) .

(8)

Here, parameters S2(0, a , z0) and S3(0, a , z0) are definedusing the off-diagonal elements of BCP matrix [Eq. (1)] as [33]

S2(0, a , z0) = J x y (0, a , z0)+ J y x (0, a , z0),

S3(0, a , z0) = i[J y x (0, a , z0)− J x y (0, a , z0)].(9)

Hence, J x y is obtained from the normalized generalized Stokesparameters as

J x y (0, a , z0)= 0.5[S2(0, a , z0)+ i S3(0, a , z0)]. (10)

It is worth noting a special case of Eq. (10), when Jαα for all α′shave same argument value but different magnitudes. We denote

Research Article Vol. 37, No. 3 / March 2020 / Journal of the Optical Society of America A 461

this common value of argument of Jαα as θ(0, a , z0). Then theresultant value of magnitude of J x y (0, a , z0) [Eq. (10)] willbe independent of θ(0, a , z0). In this case, the argument ofJ x y (0, a , z0) can be expressed as

Ar g .J x y (0, a , z0)= φx y (z0)+ θ(0, a , z0), (11)

where for a uniformly polarized EMGSM beam, φx y (z0)

is the phase difference between x and y field componentsat a single point. Also, if the beam having a state of polari-zation, that is either purely circular [S2(0, a , z0)= 0 andS1(0, a , z0)= 0] or purely diagonal [S3(0, a , z0)= 0 andS1(0, a , z0)= 0], J x y (0, a , z0) [Eq. (10)] will follow theremaining (non-vanishing) generalized Stokes parameter.

Using Eqs. (10), (1), and (4), one can obtain the value ofγx y (0, a , z0) from J x y (0, a , z0), as

γx y (0, a , z0)=J x y (0, a , z0)√

Ix (0, 0, z0)Iy (0, 0, z0) exp(− a2

4σ 2α (z0)

).

(12)

Thus, we can obtain |γx y (z0)| and δx y (z0) by performing aGaussian curve-fit [Eq. (2)] of the experimentally obtained val-ues of γx y (0, a , z0) [Eq. (12)] for multiple values of separations(r2 =±a ).

3. EXPERIMENTAL DEMONSTRATION

To demonstrate the aforesaid methodology, we conducted anexperiment (Fig. 2) in which we have synthesized an EMGSMbeam in the laboratory and characterized its properties usingamplitude interferometry, the details of which are explained inthe following section.

A. Synthesis of an EMGSM Beam

We synthesize the EMGSM beam from a randomly polarized,spatially coherent light beam obtained from a He–Ne laser(Newport make; wavelength, 632.8 nm) through a two-stepprocess: First, we introduce partial polarization to the laser beamand then create a Gaussian transverse spatial coherence profile[Eq. (4)]. The randomly polarized laser beam is converted intoa partially polarized beam using Brewster’s law [46,47]. Whenan unpolarized light field is incident on a glass slab at Brewster’sangle, the reflected light is fully polarized, and the transmit-ted field becomes partially polarized. The field equation,E x x + E y y , 〈E ∗x E y 〉 = 0, and |E x | = |E y |, represents the ran-domly polarized (P = 0) laser beam, where E x and E y are thehorizontal and vertical field amplitudes, and angular bracketsrepresent the ensemble average. The degree of polarization (P )of this beam can be elevated either by increasing the correlationbetween the orthogonal fields (〈E ∗x E y 〉> 0) or by increasing thedifference of magnitude between orthogonal field amplitudes(|E x | 6= |E y |). These changes make 4|J |> (Tr .J )2 and hence,1≥ P ≥ 0. By placing a thin glass plate in the beam path at near-Brewster’s angle, one can selectively fetch out a portion of theE y , resulting in |E x |> |E y |, and hence a partial polarization(P > 0) is introduced in the transmitted beam [Fig. 2(a)]. Wehave used a standard polarimetric scheme [42] to estimate thevalue of P , which is P = 0.21 at z= z0 [Fig. 2(b)].

Fig. 2. Experimental schemes (a) to generate an EMGSM beam,and (b) to measure the second-order coherence properties of theEMGSM beam. Planes z= z0 and z= z1 have the same meaningas in Fig. 1. (c) Beam profile at z= z0 plane; intensity profile of thebeam, obtained from a CCD image, is shown in violet color; the solidred curve is the Gaussian curve fit (σ = σx = σy = 12.4 mm). Insetof (c) shows the lateral intensity profile of the beam after focusingwith a lens. Notations: M, mirror; GP, thin glass plate; θB , Brewster’sangle; BD, beam dump; HWP, half-wave plate; P, degree of polari-zation; L, lens; RGGP, rotating ground glass plate; IA, iris aperture;PP, polarization projector; QWP, quarter-wave plate; PR, polarizer;DS, double-slit; TS, translation stage; CCD; charge-coupled devicecamera; OA, optic-axis; and a.u., arbitrary unit.

To conveniently demonstrate the measurement ofγx y (0, a , z0) [Eq. (4)] of the beam requires a high value ofthe degree of coherence [γx y (0, 0, z0)] associated with thepolarization matrix J , which can be tuned as [48]

|γx y (0, 0, z0)| =

√P 2 − D2

1− D2, (13)

where D= Ix − Iy ; Ix = J x x and Iy = J y y are normalizedintensities corresponding to orthogonal field orientations.Hence, |γx y (0, 0, z0)| reaches a maximum value (equal to P )when Ix = Iy . We have used a half-wave plate (as a polariza-tion rotator) to equalize the intensities Ix and Iy . The meanvalue of |γx y | [obtained at plane z= z0 (Fig. 2) using standardpolarimetry] is |γx y | = 0.20(≈ P ). Now, the full spatial coher-ence needs to be reduced to partial spatial coherence. A spatiallycoherent beam can be converted into a partially coherent beamby using either a ground-glass plate [49] or a spatial light modu-lator (SLM) [27]. We have introduced a Gaussian transversecoherence profile by focusing this partially polarized beam ofGaussian transverse intensity profile to a rotating ground glassplate (RGGP) using a lens [L1]. Hence, the spatial coherencegenerated after the RGGP will follow the Van Cittert–Zerniketheorem [34]. Therefore, the transverse profile of complexdegree of cross coherence formed at the z0 plane, which is at adistance z0(= 21 cm) from the RGGP, is expressed as [34,35]

γ (0, a , z0)= exp

(−π2w2

λ2z20

a2

)exp

(−i

π

λz0a2

), (14)

462 Vol. 37, No. 3 / March 2020 / Journal of the Optical Society of America A Research Article

where w is the size of the Gaussian beam at the RGGP. A cir-cular aperture is used immediately after the RGGP to selectthe beam from scattered light. Note that |γ (0, a , z0)| is shift-invariant but the argument of γ (0, a , z0) is not. The argumentof γ (0, a , z0) is shift-variant as the value of a2 in the argu-ment term is obtained as a2

= r 21 − r 2

2 . Also, it is evident fromEq. (14) that the transverse distribution of the argument valuesis independent of the beam size (w) at the RGGP, but inverselyproportional to the longitudinal distance (z0) from the RGGP.Thus, at a far zone (z0� a ), the beam will have a high spatialcoherence with a small rate variation of the argument valuesin the transverse plane. Both the magnitude and argument [ofγ (0, a , z0)] follow a peak function which peaks at the opticaxis; the magnitude follows a Gaussian distribution, and theargument values form a parabolic distribution in any transverseplane.

Scalar description of the spatial coherence [Eq. (14)] is alsovalid for all the polarization-state projections of the complexelectromagnetic beam. Hence, the cross-coherence formedbetween the points r1 = 0 and r2 = a is expressed as

γαβ(0, a , z0)= |γαβ(0, 0, z0)| exp

(−π2w2

αβ

λ2z20

a2

)

× exp

(i[−π

λz0a2+ φαβ

]),

for (α, β)= (x , y ), (15)

where wαβ is the size of the transverse intensity distribution inthe plane of the RGGP made by the polarization state projection(α, β) of the partially polarized Gaussian beam. In addition tothe properties possessed by γ (0, a , z0) [Eq. (14)], the argu-ment of γαβ(0, a , z0) [of Eq. (15)] has an additional term [φαβ ,Eq. (11)], and its magnitude has a new multiplication factor(|γαβ(0, 0, z0)|). φαβ and |γαβ(0, 0, z0)| are the phase differ-ence and degree of coherence, respectively, between the α and βpolarization-state projections of the beam at any single point inthe z0 plane.

B. Measurement

The detection process, as illustrated in Fig. 2(b), includes apolarization-state projector (PP), a double-slit, and a CCDcamera to capture the fringe images. The PP consists of aquarter-wave plate (QWP) and a polarizer (PR). If I (ζ, η) is theoutput intensity after PP, where ζ is the angle of QWP and η isthe angle of PR, then we obtain the different polarization stateprojections as

Ix = I (0◦, 0◦), Iy = I (0◦, 90◦),I f = I (45◦, 45◦), Im = I (45◦, 135◦),Ir = I (0◦, 45◦), Il = I (0◦, 135◦),

(16)

where Iα is the intensity of a polarization-state projectionα, andα = {x , y , f ,m, r , l} [Eq. (8)].

Each of these polarization-state projections is allowed tointerfere at the double-slit [placed in z0 plane, Fig. 2(b)] toobtain Jαα(0, a) [when the DS is placed such that each of theslits at r1 = 0 and r2 = a , respectively] and Jαα(0,−a) (when

r1 = 0, and r2 =−a ), and is mounted on a translation stage tomove to these two positions. We have used double-slits of slitwidth 0.15 mm and different slit-separations a = 1, 0.75, 0.5,and 0.25 mm, enabling us to get the coherence at eight distinctsets of positions across the z0 plane. The intensity obtained afterthe DS is focused using a thin lens to a CCD camera. Figure 3details the process of obtaining the γαα(0, a , z1). From theCCD images, we have obtained the horizontal variation of thetotal intensity [Iα(0, a , z1), Fig. 3(e)], and also the variationsof the diffraction pattern from individual slits [Iα(0, 0, z1) andIα(a , a , z1), Fig. 3(f )]. In order to reduce stray noise, we haveisolated the CCD sensor from ambient light by enclosing it ina black box. Also, the dark value present in the CCD measure-ment is subtracted from each of the images. Re .γαα(0, a , z1)

is then determined from Figs. 3(e) and 3(f ) using Eq. (5),and is shown in Fig. 3(g). Both the magnitude and phase ofγαα(0, a , z0) are obtained using a cosine fit [Eq. (7)]. The maxi-mum value of each of the diffraction patterns (Fig. 3) is obtainedusing a sinc2 fit [47]; hence, from individual maximum valuesand γαα(0, a , z0), we obtain Jαα(0, a , z0) using Eqs. (1) and

Fig. 3. Image of interference fringes for the y -projection of theEMGSM beam obtained using the CCD camera (Fig. 2) for double-slit separations at a = (a) 1 mm, (b) 0.75 mm, (c) 0.5 mm, and(d) 0.25 mm. Parts (e)–(g) demonstrate the calculation of γαα fromdouble-slit interference (for a = 1 mm): (e) horizontal variation ofintensity distribution obtained in the CCD camera when both the slitsare open; (f ) intensity variation at measurement plane when individualslits are blocked; the black squares and red circles are the experimen-tal values when r2 and r1 (Fig. 1) respectively are blocked; the solidlines are the theoretical fit using the single-slit diffraction equation{Eq. (5.19) of [47]}, and the vertical magenta dashed lines indicate thepeak position of the plots. The green dashed vertical line shows theequidistant line (R1 = R2). (g) Modulation of Re .γαα ; green spheresare experimentally obtained values, and the solid line is the theoreticalfit [Eq. (7)]. Plots (e)–(g) have the same horizontal scale, which is therelative transverse distance.

Research Article Vol. 37, No. 3 / March 2020 / Journal of the Optical Society of America A 463

(7). A total of 30 sets of measurements are taken for each of theexperimental data points to determine the statistical error.

The method is repeated for all polarization-state projections[Eq. (16)] and all slit separations (±a ). With the knowledgeof γαα(0,±a , z0) and Jαα(0,±a , z0) for all polarizationprojections (α), J x y (0,±a , z0) and hence γx y (0,±a , z0) aredetermined using Eqs. (8) and (10).

4. RESULTS AND DISCUSSION

The synthesized partially polarized beam has a Gaussiantransverse intensity profile [Fig. 2(c)], and the beam is madepartially polarized. Also, |γαβ(0, a , z0)| has a Gaussian coher-ence profile [Eq. (15)]. Therefore, this is an EMGSM beam ifthe realizability condition [Eq. (3)] is met. Since we are usinga single lens L1 [Fig. 2(a)] to focus the beam to RGGP, thebeam-waist wαβ [Eq. (15)] is the same for all polarization-stateprojections. Thus, from Eqs. (4), (12), and (15), we find thatδx x = δy y = δx y , so the beam meets the realizability conditionand is EMGSM for all the values of |γαβ(0, 0, z0)|. The char-acteristics of the beam have been measured using the proposedmethod, and are detailed in Figs. 4–6.

Figure 4 shows all the experimentally obtained values ofγx x (0, a , z0), γy y (0, a , z0), and γx y (0, a , z0) and theoreti-cal Gaussian fit [Eq. (4)] to all the three sets of experimentalvalues. Error bars in the plot show the experimental uncer-tainties in the measurements. The curve fits of |γx x (0, a , z0)|,|γy y (0, a , z0)|, and |γx y (0, a , z0)| with theory have 97.1%,97.7%, and 83.8% fit quality (adjacent R2 value obtainedusing the software Origin), respectively. The fact that the fit-ted curves are within the experimental uncertainties (errorbars) of most of the measured data points infers a reason-ably good match between the theory and the experiment.From the curve fittings [Eqs. (4) and (12)], we have foundδx x (= 1.03 mm)≈ δy y (= 1.03 mm)≈ δx y (= 1.04mm) with|γx y (0, 0, z0)| = 0.20, confirming the realizability condition[Eq. (3)].

Fig. 4. Transverse distribution of |γαβ(0, a , z0)| plotted withrespect to a . Black squares, red circles, and blue triangles represent themeasured values of |γx x (0, a , z0)|, |γy y (0, a , z0)|, and |γx y (0, a , z0)|

(= |γy x (0, a , z0)|), respectively. The Gaussian curve fits to experimen-tal data to determine the respective polarization-dependent coherenceparameters [Eqs. (4) and (12)] are correspondingly shown by black,red, and blue lines.

Fig. 5. Cyclic variation of the argument of γαβ(0, a , z0) [i.e.,θ , Eq. (6)] in plane z= z0 (Fig. 2). x = cos(θ) and y = sin(θ).x = 1, y = 0 is the zero-argument line. The solid red line shows theγ = 0 line, and the solid blue line and solid green lines show the theo-retical variation of argument [Eqs. (14) and (15)] between (0, a) forγαα and γx y , respectively. The blue and green spheres represent (samecolor code of theoretical plot is followed) experimentally obtainedmean values of argument for different values of a . Uncertainties inexperimental data points are represented using cyan (of γαα) and black(of γx y ) lines, which are (an average of 0.07 rad for each data point) toosmall to be visible.

Fig. 6. Transverse distribution of the absolute value of the BCPmatrix elements [|Jαβ(0,a ,z0)|, Eq. (1)] in the z0 plane (Figs. 1 and 2).The black square, red circle, and blue triangle symbols represent theexperimentally determined values of |J x x (0, a , z0)|, |J y y (0, a , z0)|,and |J x y (0, a , z0)|, respectively. The solid lines (with color codes assimilar to the experimental data) show the theoretical fit [Eq. (1)] tothe experimentally determined values.

Since the distance from the RGGP to plane z= z0 is thesame for all the polarization-state projections, the argument ofγαβ(0, a , z0) [Eq. (15)] would be same for both α and β. Theconstant value of the argument in this case makes |γx y (0, a , z0)|

independent of the argument of γαβ(0, a , z0) [Eq. (7)] forall polarization projections. It is worth mentioning here thatthe coherence parameters (δx x , δy y , δx y , and |γx y (0, 0, z0)|)

464 Vol. 37, No. 3 / March 2020 / Journal of the Optical Society of America A Research Article

are obtained from Fig. 4 alone. Figures 5 and 6 discuss theadditional information provided by this method.

The theoretical plot of cyclic variation of argument of bothγαα(0, a , z0) and γx y (0, a , z0), along with experimentallydetermined values, are shown in Fig. 5. The maximum deviationof the mean value of any experimental data point from the theo-retical value is 0.21 rad, which is 3.3% of one cyclic variation(2π rad) of the argument value, where 62.5% of the experimen-tal data points (mean values) come below a deviation of 0.07 rad.Ar g .γαα(0, a , z0) is zero for a = 0 and changes symmetri-cally on both sides, while the argument of γx y (0, a , z0) varieswith a constant phase difference of φx y (0, 0, z0)=−0.4 rad[Eqs. (11) and (15); obtained from the polarization-matrixmeasured at z= z0] from the argument ofγαα(0, a , z0).

The experimentally obtained values of the magnitude ofnormalized BCP matrix elements with the theoretical curve fit[Eq. (4)] are depicted in Fig. 6. Error bars in the plot quantify theexperimental uncertainties in measurements. The curve fits of|J x x (0, a , z0)|, |J y y (0, a , z0)| and |J x y (0, a , z0)| have 97.9%,98.4%, and 88.7% fit quality (adjacent R2 value obtained usingthe software, Origin), respectively, showing a reasonably goodmatch between the theory and the experiment. Therefore, theexperimental results detailed in Figs. 4–6 validate the effective-ness of this method in characterizing the second-order statisticalcharacteristics of the EMGSM beam.

The main advantage of our method is that it does not needspatial separation between x and y polarization-state pro-jections of the optical beam. In addition, for obtaining thecross-polarization coherence parameter, interference of spa-tially separated orthogonal polarized beams is not required.In existing approaches for determining the cross-polarizationspatial coherence [23,42,45], it is generally necessary to put anx polarization projection of the optical beam in one slit and ypolarization projection of the beam in the other slit, which is anexperimentally cumbersome task. We bypass this difficulty byusing the two-point (generalized) Stokes parameters. We alsonote that, even though our method is experimentally easy toimplement, it involves an increased number of measurements inorder to separately find the transverse spatial coherence for all sixpolarization-state projections of the EMGSM beam.

5. CONCLUSION

In conclusion, we propose a measurement scheme for thecharacterization of the second-order coherence features of theelectromagnetic Gaussian Schell-model beams. In this method,we propose the use of field interferometry as a tool for characteri-zation, which has been used extensively in the past for studyingcoherence features of electromagnetic fields. Additionally, thismethod helps to reveal the polarization-dependent argumentinformation of the complex degree of spatial coherence of thebeam. The experimental demonstration involves the realizationof a source of an electromagnetic Gaussian Schell-model beam,and the use of field interferometry and two-point (generalized)Stokes parameters to characterize the beam, showing goodagreement with the theoretically predicted characteristics of thesynthesized beam proving the efficacy of the method. We believe

in the applicability of this method in characterizing electro-magnetic fields having different coherence and polarizationfeatures.

Funding. Science and Engineering Research Board(YSS/2015/000743); Council of Scientific and IndustrialResearch, India (03(1401)/17/EMR-II).

Disclosures. The authors declare no conflicts of interest.

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