Fraunhofer diffraction and the state ofpolarization of partially coherentelectromagnetic beamsYANGYUNDOU WANG,1,2 SHENGGANG YAN,1 XIAOFEI LI,3 XIANLONG LIU,3 YANGJIAN CAI,3,4

GOVIND P. AGRAWAL,5,6 AND TACO D. VISSER3,5,7,*1School of Marine Science and Technology, Northwestern Polytechnical University, Xi’an, China2Wellman Center for Photomedicine, Massachusetts General Hospital, Harvard Medical School, Boston, Massachusetts 02114, USA3Shandong Provincial Engineering and Technical Center of Light Manipulations, and Shandong Provincial Key Laboratory of Optics and PhotonicDevices, School of Physics and Electronics, Shandong Normal University, Jinan 250014, China4School of Physical Science and Technology, Soochow University, Suzhou 215006, China5Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA6Institute of Optics, University of Rochester, Rochester, New York 14627, USA7Department of Physics and Astronomy, Vrije Universiteit, Amsterdam, The Netherlands*Corresponding author: t.d.visser@vu.nl

Received 31 May 2019; accepted 4 June 2019; posted 6 June 2019 (Doc. ID 368716); published 26 June 2019

We generalize the concept of Fraunhofer diffraction to par-tially coherent electromagnetic beams and show how thestate of polarization is affected by a circular aperture. It isillustrated that the far-zone properties of a random beamcan be tuned by varying the aperture radius. We find thateven an incident beam that is completely unpolarized cansometimes produce a field that is highly polarized. ©2019Optical Society of America

https://doi.org/10.1364/OL.44.003330

Fraunhofer diffraction describes the far-zone properties of afield that has passed through an aperture in an opaque screen.Among its many applications are optical computing, imageprocessing, and holography. In the classical description [1],the incident field is assumed to be both scalar and spatially fullycoherent. The extension of Fraunhofer diffraction to partiallycoherent scalar fields has been considered in, for example,Refs. [2–5]. Those studies were all concerned with the effectof coherence on the intensity distribution. The influence ofan aperture on the spectrum of the far-zone field was investi-gated in Refs. [6,7]. More recently, an analysis of the role oftemporal coherence was presented in Ref. [8]. In many practicalcircumstances the state of polarization cannot be neglected, anda scalar description of the field does not suffice. Therefore, it isuseful to extend the notion of Fraunhofer diffraction to stochas-tic electromagnetic beams. Here we present a theory that allowsus to investigate how the far-zone state of polarization of suchbeams is affected by the diffraction process.

It is worth noting that when a random electromagneticbeam propagates, both its spectrum [9] and state of polarizationmay change [10], even if this propagation is through free space.These effects are referred to as “coherence-induced changes.”

As we will demonstrate, placing an aperture in the path ofthe beam will also change its far-zone properties.

Let us begin by considering a monochromatic, scalar wavewith wavelength λ. If this wave is diffracted by an aperture in aplane, opaque screen, the resulting field in the far zone is pro-portional to the truncated Fourier transform of the incidentfield U �in� ([1], Eqs. (4)–(25)):

U �∞��x,y,z�� eikz

iλzeik�x2�y2�∕2z

ZAU �in��ξ,η�e−i2π�xξ�yη�∕λzdξdη:

(1)

Here the screen is assumed to be in the xy plane, k � 2π∕λ isthe wavenumber, and A is the area of the aperture. The super-script ∞ denotes the far zone. We now let U �in� represent aCartesian component of the incident electric field vector, i.e.,

U �in��ξ, η� � E �in�j �ξ, η�, �j � x, y�: (2)

The second-order statistical properties, at frequency ω, of astochastic, electromagnetic beam at two positions r1 and r2are characterized by the cross-spectral density matrix [11]

W �r1, r2� ��W xx W xyW yx W yy

�: (3)

The four matrix elements are given by the expression

W ij�r1, r2� � hE�i �r1�Ej�r2�i �i, j � x, y�, (4)

where the angular brackets indicate the average taken over anensemble of beam realizations. On substituting from Eq. (1)into Eq. (4), and interchanging the order of integration andensemble averaging, we find for the cross-spectral densitymatrix in a far-zone plane z the formula

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0146-9592/19/133330-04 Journal © 2019 Optical Society of America

W �∞�ij �x1, y1, x2, y2; z� �

1

λ2z2exp

�ik2z

�x22 � y22 − x21 − y

21��

× W̃ �in�ij

�−x1λz

, −y1λz

,x2λz

,y2λz

�,

(5)

with the four-dimensional spatial Fourier transform defined as

W̃ �in�ij �p1,q1,p2,q2� �

ZA

ZAW �in�

ij �x1, y1,x2, y2�

× e−i2π�x1p1�y1q1�x2p2�y2q2�dx1dy1dx2dy2:

(6)

Equation (5) states that each element of the cross-spectral den-sity matrix in the far zone is, up to a position-dependent phasefactor, related to its counterpart in the aperture by a Fouriertransform. This formula generalizes the notion of Fraunhoferdiffraction from monochromatic scalar fields to randomelectromagnetic beams.

If we restrict our attention to a far-zone observation pointZ � �0, 0, z� on the z axis, then Eq. (5) for two coincidentpoints simplifies to

W �∞�ij �Z ,Z � � 1

λ2z2

ZA

ZAW �in�

ij �x1, y1, x2, y2�dx1dy1dx2dy2:

(7)

It is seen that W �∞�ij �Z ,Z � is proportional to the double inte-

gral of the corresponding element in the aperture. Because, aswe will shortly discuss, the state of polarization depends onlyon W at equal points, Eq. (7) has an important consequence.For a very small aperture, over which the spatial variation ofthe matrix elements may be neglected, we get that

W �∞�ij �Z ,Z � ≈ dA2

λ2z2W �in�

ij �0, 0, 0, 0�, as dA → 0, (8)

where dA denotes the aperture area. This means that the nor-malized matrix elements at the far-zone point Z are identical totheir counterparts at the center of the aperture. Therefore, for asmall enough aperture, the state of polarization at a far-zone pointon the central axis is identical to the state of polarization of the fieldat the center of the aperture.

We illustrate the usefulness of the above formalism by con-sidering the example of an aperture field that is produced by aGaussian Schell-model source [11]. If we assume the effectivewidth of the two electric field components to be equal, wehave that

W �in�ij �ρ1,ρ2��AiAjBij exp

�−

�ρ21�ρ224σ2

��exp

�−�ρ2 −ρ1�2

2δ2ij

�,

(9)

where ρ � �x, y�. The factors Ai denote the spectral amplitudeof each Cartesian field component, Bij is the complex-valuedcorrelation coefficient between Ei and Ej, σ is the effectivebeam width, and δij is a coherence radius. These parameterscannot be chosen arbitrarily, but have to satisfy several con-straints. These follow from the definition of W�in� (Sec. 9.4.2,[11]), the realizability conditions [12], and the beam condi-tions [13].

We take the aperture to be a circle with radius a. On chang-ing variables to

ρ1�ρ1�cosϕ1, sinϕ1�, ρ2�ρ2�cosϕ2, sinϕ2�: (10)

Equation (7) can be re-written as

W �∞�ij �Z ,Z ��AiAjBij

λ2z2

Za

0

Z2π

0

Za

0

Z2π

0

ρ1ρ2 exp

�−ρ21�ρ224σ2

�

×exp�−ρ21�ρ222δ2ij

�

×exp�ρ1ρ2 cos�ϕ1 −ϕ2�

δ2ij

�dϕ1dρ1dϕ2dρ2 (11)

� 4π2AiAjBij

λ2z2

Za

0

Za

0

exp

�−ρ21 � ρ222β2ij

�

× ρ1ρ2I 0

�ρ1ρ2δ2ij

�dρ1dρ2, (12)

where I 0 denotes the modified Bessel function of order zero,and where we introduced the new parameter

1

β2ij� 1

δ2ij� 1

2σ2: (13)

The limit of this expression, as the aperture radius a → ∞, canbe derived by considering the identity [14]Z

∞

0

xe−x2∕�2b2�I 0�px�dx � b2eb2p2∕2: (14)

On making use of this in Eq. (12), we find that

W �∞�ij �Z ,Z � � 4π2AiAjBij

λ2z2β4ijδ

4ij

δ4ij − β4ij, as a → ∞: (15)

For a finite aperture radius, Eq. (12) is easily evaluated numeri-cally. Having thus established the far-zone cross-spectral densitymatrix, the polarization properties of the field there can be ex-pressed in terms of the four spectral Stokes parameters, denotedSn�Z �, with n � 0, 1, 2, 3. Their expectation values are givenby the formulas [11]

S0�Z � � W xx�Z ,Z � �W yy�Z ,Z �, (16a)

S1�Z � � W xx�Z ,Z � −W yy�Z ,Z �, (16b)

S2�Z � � W xy�Z ,Z � �W yx�Z ,Z �, (16c)

S3�Z � � i�W yx�Z ,Z � −W xy�Z ,Z ��: (16d)

The normalized version of the Stokes parameters is defined as

sm�Z � � Sm�Z �∕S0�Z �, �m � 1, 2, 3�: (17)

The degree of polarization (DOP), the ratio of the spectral den-sity of the fully polarized part of the beam, and its total spectraldensity, follow from the relation (Sec. 6.3.3, [15])

DOP�Z � �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis21�Z � � s22�Z � � s23�Z �

q: (18)

The DOP is bounded by zero and one. The lower boundrepresents a completely unpolarized beam, whereas the upperbound represents a fully polarized beam. We next investigatehow the state of polarization of the far-zone field, characterized

Letter Vol. 44, No. 13 / 1 July 2019 / Optics Letters 3331

by the DOP and the Stokes parameters, can be tuned by vary-ing the size of the aperture.

The dependence of the three normalized Stokes parameters,at the far-zone axial point Z , on the aperture radius a is shownin Fig. 1. The first parameter, s1�Z �, remains constant at 0.6.Its value does not depend on a, because we have set δxx � δyy.The two other parameters, s2�Z � and s3�Z �, increase signifi-cantly when the aperture radius a becomes larger. Notice thatthis tendency persists, even when a is larger than the effectivebeam width σ � 7 mm, idem dito for δxx, δyy, and δxy.Eventually, the parameters s2�Z � and s3�Z � tend to their asymp-totic value which can be calculated from Eq. (15) and which is0.30 and 0.52, respectively. Furthermore, as remarked belowEq. (8), in the limit a → 0, the Stokes parameters are equalto their value at the center of the aperture. For example, it iseasily verified from Eq. (9) and definition (18) that everywherein the aperture s2 � 2AxAyjBxyj cos ϕ∕�A2

x � A2y � � 0.08.

Similarly, the DOP is also strongly affected by the aperturesize. This is illustrated in Fig. 2 in whichDOP�Z � is plotted forselected values of the amplitude Ax . Just as the Stokes param-eters, the DOP on the far-zone axis is seen to be very sensitiveto the aperture radius. As noted in connection with Fig. 1, thisinfluence persists, even when a > σ. In addition, in agreementwith the statement below Eq. (8), when the aperture radius isvery small, i.e., a < σ, δij, we find that DOP�Z � is equal to itsvalue at the center of the aperture.

Next, we consider an incident field that is completelyunpolarized. One possible example (see also Ref. [16]) is abeam given by Eq. (9) while setting Ax � Ay � A, andBxy � Byx � 0. In that case,

W �in�xx �ρ, ρ� � W �in�

yy �ρ, ρ� ≠ 0, (19)

W �in�xy �ρ, ρ� � W �in�

yx �ρ, ρ� � 0: (20)

From Eq. (12), it readily follows that the far-zone elementsW �∞�

xy �Z ,Z � � W �∞�yx �Z ,Z � � 0, and hence s2�Z � �

s3�Z � � 0. However, if δxx ≠ δyy, the two diagonal matrixelements,W xx and W yy, will evolve differently on propagationand, thus, give rise to a field that is partially polarized. An ex-ample is presented in Fig. 3. The larger the difference between

δxx and δyy, the more W �∞�xx and W �∞�

yy will diverge. This thencauses the value of s1 to increase, leading to a growing DOP.We emphasize that this effect does not occur for all unpolarizedaperture fields, but hinges on the fact that the two Cartesiancomponents of the electric field have different coherence radii,i.e., δxx ≠ δyy.

We have restricted our analysis to the field on the far-zoneaxis. For non-axial points the more general expression Eq. (5)must be used. From the reciprocity between a function and itsFourier transform, we can get a qualitative idea of the off-axisfield. As the width of the elementsW �in�

ij gets less, their far-fieldcounterparts will become broader, and the off-axis and theon-axis state of polarization will be more similar.

In conclusion, we have extended the concept of Fraunhoferdiffraction from scalar fields to stochastic electromagneticbeams. This allowed us to study the polarization propertiesof the on-axis field in the far zone. The Stokes parametersand the DOP were both found to be quite sensitive to the

s(Z)

a [cm]

s (Z)

s (Z)

s (Z)

Fig. 1. Three normalized Stokes parameters on the far-zone axis as afunction of the aperture radius a. The red curve corresponds to s1�Z �,the blue curve � s2�Z �, and the green curve � s3�Z �. In this exampleAx � 2.0, Ay � 1.0, σ � 7 mm, λ � 632.8 nm, Bxy � 0.2eiπ∕3,δxx � δyy � 2 mm, and δxy � 4 mm.

a [cm]

DOP (Z)

Ax = 1

Ax = 3

Ax = 2

Fig. 2. DOP on the far-zone axis as a function of the apertureradius a, for selected values of Ax , the amplitude of Ex . The red curveis for Ax � 1, the blue curve is for Ax � 2, and the green curve is forAx � 3. The amplitude Ay is kept fixed at 1, and all other parametersare as in Fig. 1.

DOP (Z)

a [cm]

Fig. 3. Far-zone DOP produced by a completely unpolarized inci-dent beam, as a function of the aperture radius a, for selected values ofδxx , the coherence radius of Ex . The lower (red) curve is forδxx � 3 mm, the middle (green) curve is for δxx � 4 mm, and thetop (blue) curve is for δxx � 5 mm. Here Ax � Ay , σ � 7 mm,δyy � 2 mm, and λ � 632.8 nm.

3332 Vol. 44, No. 13 / 1 July 2019 / Optics Letters Letter

aperture size. An example was given in which an incident beamthat is completely unpolarized becomes highly polarized in thefar zone. The enhancement of the DOP is due to the fact that,in general, the four elements of the cross-spectral density matrixevolve differently on propagation. If, for example, W xx be-comes dominant, the beam will become essentially x polarized.This enhancement of the DOP will be useful for applicationsthat use incoherent illumination, but require some coherencein the far field. An example would be where the x and y po-larized components are forced to interfere after passing themthrough a polarizer oriented at 45 deg.

Funding. Northwestern Polytechnical University (NPU)National Natural Science Foundation of China (NSFC)(11525418, 11804198, 91750201); Air Force Office ofScientific Research (AFOSR) (FA9550-16-1-0119).

Acknowledgment. Y. Wang is supported by theExcellent Doctorate Foundation of NPU, and Y. Cai is sup-ported by the NSFC. T. Visser acknowledges funding bythe AFOSR.

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