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216 OPTICS LETTERS / Vol. 27, No. 4 / February 15, 2002 Tensor ABCD law for partially coherent twisted anisotropic Gaussian–Schell model beams Qiang Lin and Yangjian Cai Institute of Optics, Department of Physics, Zhejiang University, Hangzhou 310028, China Received September 27, 2001 A 4 3 4 complex curvature tensor M 21 is introduced to describe partially coherent anisotropic Gaussian– Schell model (GSM) beams. An analytical propagation formula for the cross-spectral density of partially coherent anisotropic GSM beams is derived. The propagation law of M 21 that is also derived may be called partially coherent tensor ABCD law. The analytical formulas presented here are useful in treating the propagation and transformation of partially coherent anisotropic GSM beams, which include previous results for completely coherent Gaussian beams as special cases. © 2002 Optical Society of America OCIS codes: 030.1640, 140.3300. Most beams are partially coherent. Completely co- herent and completely incoherent beams, in the strict sense, do not exist. The Collett–Wolf source, for in- stance, is a typical source that emits partially coherent Gaussian– Schell model (GSM) beams. 1 GSM beams and, later, twisted GSM beams have attracted particu- lar interest because such beams not only can be an- alyzed theoretically 2 – 12 but can also be constructed in the laboratory. 13 In the theoretical aspect, the Wigner distribution function is widely used to treat the propa- gation and imaging of both twisted and nontwisted GSM beams (see, for examples, Refs. 5–10). It is well known, however, that the transformation of completely coherent Gaussian beams with circular symmetry through axially symmetric optical system satisfies the famous Kogelnik abcd law. 14 This law was generalized in several directions such as to be applicable for more-complicated beams, such as nonorthogonal Gaussian beams, 15 non-Gaussian beams, 17 ultrashort pulsed Gaussian beams, 18 and general astigmatic beams in the spatial-frequency domain. 19 For partially coherent beams with rota- tional symmetry, a generalized abcd law was derived based on second-order moments of the Wigner distri- bution. 2 For a special kind of GSM beam, for which the real parts of the quadratic forms that arise in the exponents of the Gaussians are described by the same real, positive-definite symmetric matrix, a similar generalization that applied second-order moments of the Wigner distribution was made. 8 One may ask whether it is possible to obtain a generalized ABCD law for the most general astigmatic GSM beam with 10 physical parameters. In this Letter we show that, indeed, it is possible to obtain such a propagation law. In our approach, the partially coherent general astigmatic (or anisotropic) GSM beam is fully characterized by a 4 3 4 complex matrix M 21 , which may be called a partially coherent complex curvature tensor. Then the transformation law of M 21 through an axially nonsymmetric optical system is derived. The Wigner distribution and re- lated second-order moments, which are commonly used by many authors, are not used in our derivation. In- stead, we use the cross-spectral density of partially co- herent GSM beams directly. Our result, in contrast to the previous method, is written in terms of an 8 3 8 real matrix of the optical system instead of a 4 3 4 complex matrix as would have been obtained by use of the general n 3 n complex matrix results developed in Refs. 20–22. The cross-spectral density of the most general par- tially coherent anisotropic GSM beam is expressed as 4 Gr 1 , r 2 G 0 exp 2 1 4 r 1 T s I 2 21 r 1 1 r 2 T s I 2 21 r 2 2 1 2 r 1 2 r 2 T s g 2 21 r 1 2 r 2 2 ik 2 r 1 2 r 2 T R 21 1mJ r 1 1 r 2 æ , (1) where G 0 is a constant, k 2p l is the modulus of the wave vector, and r 1 and r 2 are position vectors of two arbitrary points in the transverse plane. s I 2 is a transverse spot width matrix and s g 2 is a transverse coherence width matrix. R 21 is a wave-front curva- ture matrix. s I 2 , s g 2 , and R 21 are all 2 3 2 matrices with transpose symmetry, given by s I 2 21 s I 11 22 s I 12 22 s I 12 22 s I 22 22 , s g 2 21 s g11 22 s g12 22 s g12 22 s g22 22 , R 21 R 11 21 R 12 21 R 21 21 R 22 21 . (2) J is a transpose antisymmetric matrix, given by J 0 1 21 0 . (3) m is a scalar real-valued twist factor. It is limited by 0 #m 2 # k 2 dets g 2 21 owing to the nonnegativ- ity requirement on the cross-spectral density of the beam. 3,5,10 All the known families of coherent and partially coherent Gaussian beams are subsets of the beam described by Eq. (1). In the general case mfi 0, the remaining three matrices s I 2 , s g 2 , and R 21 , have 0146-9592/02/040216-03$15.00/0 © 2002 Optical Society of America

Tensor ABCD law for partially coherent twisted anisotropic Gaussian—Schell model beams

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216 OPTICS LETTERS / Vol. 27, No. 4 / February 15, 2002

Tensor ABCD law for partially coherent twisted anisotropicGaussian–Schell model beams

Qiang Lin and Yangjian Cai

Institute of Optics, Department of Physics, Zhejiang University, Hangzhou 310028, China

Received September 27, 2001

A 4 3 4 complex curvature tensor M21 is introduced to describe partially coherent anisotropic Gaussian–Schellmodel (GSM) beams. An analytical propagation formula for the cross-spectral density of partially coherentanisotropic GSM beams is derived. The propagation law of M21 that is also derived may be called partiallycoherent tensor ABCD law. The analytical formulas presented here are useful in treating the propagationand transformation of partially coherent anisotropic GSM beams, which include previous results for completelycoherent Gaussian beams as special cases. © 2002 Optical Society of America

OCIS codes: 030.1640, 140.3300.

Most beams are partially coherent. Completely co-herent and completely incoherent beams, in the strictsense, do not exist. The Collett–Wolf source, for in-stance, is a typical source that emits partially coherentGaussian–Schell model (GSM) beams.1 GSM beamsand, later, twisted GSM beams have attracted particu-lar interest because such beams not only can be an-alyzed theoretically2 – 12 but can also be constructed inthe laboratory.13 In the theoretical aspect, the Wignerdistribution function is widely used to treat the propa-gation and imaging of both twisted and nontwistedGSM beams (see, for examples, Refs. 5–10).

It is well known, however, that the transformationof completely coherent Gaussian beams with circularsymmetry through axially symmetric optical systemsatisfies the famous Kogelnik abcd law.14 Thislaw was generalized in several directions such asto be applicable for more-complicated beams, suchas nonorthogonal Gaussian beams,15 non-Gaussianbeams,17 ultrashort pulsed Gaussian beams,18 andgeneral astigmatic beams in the spatial-frequencydomain.19 For partially coherent beams with rota-tional symmetry, a generalized abcd law was derivedbased on second-order moments of the Wigner distri-bution.2 For a special kind of GSM beam, for whichthe real parts of the quadratic forms that arise in theexponents of the Gaussians are described by the samereal, positive-definite symmetric matrix, a similargeneralization that applied second-order moments ofthe Wigner distribution was made.8 One may askwhether it is possible to obtain a generalized ABCDlaw for the most general astigmatic GSM beam with10 physical parameters.

In this Letter we show that, indeed, it is possible toobtain such a propagation law. In our approach, thepartially coherent general astigmatic (or anisotropic)GSM beam is fully characterized by a 4 3 4 complexmatrix M21, which may be called a partially coherentcomplex curvature tensor. Then the transformationlaw of M21 through an axially nonsymmetric opticalsystem is derived. The Wigner distribution and re-lated second-order moments, which are commonly usedby many authors, are not used in our derivation. In-stead, we use the cross-spectral density of partially co-herent GSM beams directly. Our result, in contrast tothe previous method, is written in terms of an 8 3 8

0146-9592/02/040216-03$15.00/0

real matrix of the optical system instead of a 4 3 4complex matrix as would have been obtained by use ofthe general n 3 n complex matrix results developed inRefs. 20–22.

The cross-spectral density of the most general par-tially coherent anisotropic GSM beam is expressed as4

Gr1, r2 G0 expΩ2

14

r1T sI

221r1 1 r2T sI

221r2

212

r1 2 r2T sg221r1 2 r2

2ik2

r1 2 r2T R21 1 mJ r1 1 r2æ

, (1)

where G0 is a constant, k 2pl is the modulus ofthe wave vector, and r1 and r2 are position vectors oftwo arbitrary points in the transverse plane. sI

2 is atransverse spot width matrix and sg

2 is a transversecoherence width matrix. R21 is a wave-front curva-ture matrix. sI

2, sg2, and R21 are all 2 3 2 matrices

with transpose symmetry, given by

sI221

∑sI11

22 sI1222

sI1222 sI22

22

∏,

sg221

∑sg11

22 sg1222

sg1222 sg22

22

∏,

R21

∑R11

21 R1221

R2121 R22

21

∏. (2)

J is a transpose antisymmetric matrix, given by

J

∑0 1

21 0

∏. (3)

m is a scalar real-valued twist factor. It is limitedby 0 # m2 # k2 detsg

221 owing to the nonnegativ-ity requirement on the cross-spectral density of thebeam.3,5,10 All the known families of coherent andpartially coherent Gaussian beams are subsets of thebeam described by Eq. (1). In the general case m fi 0,the remaining three matrices sI

2, sg2, and R21, have

© 2002 Optical Society of America

February 15, 2002 / Vol. 27, No. 4 / OPTICS LETTERS 217

different principal axes (the axes along which thematrices are diagonal), and the beam is a generalastigmatic beam. If m 0 and if all these threematrices have the same principal axes, and they arealigned (or rotated) with respect to the laboratoryaxes x and y, the beam is an aligned simple astigmatic(rotated simple astigmatic) beam. If m 0 and ifall three matrices are proportional to the identitymatrix, the beam is stigmatic and has rotationalsymmetry.10

Equation (1) can be rearranged into a more compactform, as follows:

Gr G0 expµ2ik2

rTM21r∂, (4)

where rT r1T 2 r2

T x1, y1, x2, y2.M21 is a 4 3 4 complex matrix given by

M21

2664 R21 2

i2k

sI221 2

ik

sg221 i

ksg

221 1 mJik

sg221 1 mJT 2R21 2

i2k

sI221 2

ik

sg221

3775 , (5)

is a transpose symmetric matrix, and is called a par-tially coherent complex curvature tensor in what fol-lows. In the most general case, the partially coherentcomplex curvature tensor can be expressed as follows:

M21

"M11

21 M1221

M1221T 2M11

21

#, (6)

where M1121 is also transpose symmetric but M12

21 isnot. In the general case, M11

21 contains six real pa-rameters (three complex parameters) and M12

21 con-tains four real parameters; therefore M21 contains tenindependent real parameters. It is easy to get the re-lations between beam parameters m, sI

2, sg2, and R21

and the submatrices of M21 from Eqs. (5) and (6).The diffraction integral formula for a beam passing

through a general axially nonsymmetric (or generalastigmatic) optical system reads as18,19,23,24

Er1 2il

detB212ZZ

Er1exp2ikldr1 , (7)

where l is Hamilton’s point characteristic function be-tween the incident and output planes, given by19,23

l l0 112

µr1

r1

∂TV

µr1

r1

∂,

V

∑B21A 2B21

C 2 DB21A DB21

∏. (8)

l0 is the optical path along the axis, V is the 4 3 4point characteristic matrix, and A, B, C, and D are all2 3 2 submatrices of the 4 3 4 ray transfer matrix thatdescribes the axially nonsymmetric optical system.

Assume that the optical fields at the two arbitrarypoints r1 and r2 in the incident plane are Er1 and

Er2 and that the optical fields at the two arbitrarypoints r1 and r2 in the output plane are Er1 andEr1, respectively. The cross-spectral density in theincident and output planes will be

Gr Er1Er2, Gr Er1Er2 , (9)

where denotes an ensemble average, rT r1T ,r2

T ,and rT r1

T ,r2T . Using Eq. (7), we can get the

propagation formula of cross-spectral density of a par-tially coherent beam through the axially nonsymmetricoptical system as follows:

Gr 1

l2 detB

ZZZZGrexp2ikl1 2 l2dr , (10)

where

lj l0 112

µrj

rj

∂TV

µrj

rj

∂, j 1, 2 . (11)

From Eqs. (8) and (11) we have

l1 2 l2 12

µrr

∂T

"B

21A 2B

21

C 2 DB21

A DB21

# µrr

∂,

(12)

where A, B, C, and D are defined as follows:

A

∑A 00 A

∏, B

∑B 00 2B

∏,

C

∑C 00 2C

∏, D

∑D 00 D

∏. (13)

Because l1 2 l2 is a scalar quantity, we get directlyfrom Eq. (12) the following relations:

B21AT B21A , DB21T DB21,

C 2 DB21A 2B21T . (14)

These are equivalent to the famous Luneburg rela-tions25 that describe the symplecticity of the axiallynonsymmetric optical system. Substituting Eqs. (4)and (12) into Eq. (10), we get

Gr G0

l2detB12

ZZZZexp

µ2ipl

L∂dr , (15)

where

218 OPTICS LETTERS / Vol. 27, No. 4 / February 15, 2002

L rT B21A 1 Mi21r 2 2rTB21

r 1 rTDB21r

jB21A 1 Mi

2112r 2 B21A 1 Mi

21212B21

rj2

1 rT C 1 DMi21 A 1 BMi

2121r . (16)

In the above derivation, the transpose symmetric prop-erty of M21 and Eq. (14) have been used. SubstitutingEq. (16) into Eq. (15), we have

Gr G0detA 1 BMi21212 exp

µ2ik2

rTMo21r

∂,

(17)

where Mi21 and Mo

21 denote the partially coherentcomplex curvature tensor in the input and the outputplanes, respectively. They satisfy the following for-mula (valid only for the phase factor under the expo-nential above):

Mo21 C 1 DMi

21 A 1 BMi2121. (18)

It should be noted that there is an additionalconstant factor detA 1 BMi

21212 in the outputcross-spectral density [Eq. (17)] in comparison withthe input density [Eq. (4)]. This factor, which has noinf luence on the relative intensity distribution of GSMbeams, ensures the conservation of energy. Thishappens also with the regular ABCD law for coherentGaussian-beam propagation.18,19

In the derivation of Eq. (17), the integral formulaR`

2` exp2ax2dx p

pa has been used. Equa-tion (17) is the general propagation formula of ageneral astigmatic GSM beam passing through anaxially nonsymmetric optical system. Equation (18)is the transformation law of the partially coher-ent complex curvature tensor M21 through axiallynonsymmetric optical systems, which may be calledthe tensor ABCD law for general partially coherentbeams.

The method of Wigner distribution,2 – 10 which we donot use in this Letter, and the tensor ABCD law de-rived in this Letter are two different and equivalentapproaches. Both methods are applicable to generalastigmatic GSM beams, but we believe that the tensorABCD law is a more direct approach because it usescross-spectral density instead of the second momentsof the Wigner distribution to describe the beam. Thetensor ABCD law is a direct generalization of Kogel-nik’s abcd law.

In conclusion, we have introduced a 4 3 4 matrixM21 to characterize the most comprehesive generalastigmatic (or anisotropic) partially coherent GSMbeams, including twist. The transformation law ofM21 through an axially nonsymmetric optical systemis derived, and this is called the partially coherenttensor ABCD law. This propagation law is elegant

in form and can be used in various steps in the propa-gation and transformation of partially coherent GSMbeams. More importantly, all the previous resultsfor coherent Gaussian beams can be converted intopartially coherent GSM beams, as the propagationlaw for them is exactly the same in form.

This study is supported by the National Natu-ral Science Foundation of China (60078003), theHuo-Ying-Dong Education Foundation of China(71009), and the Zhejiang Provincial Natural ScienceFoundation of China (RC98029). Q. Lin’s e-mailaddress is [email protected].

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