Spectral shift of partially coherent twistedanisotropic Gaussian Schell-model beams in free space

Yangjian Cai, Qiang Lin*,1

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

Received 14 August 2001; received in revised form 4 January 2002; accepted 28 January 2002

Abstract

The spectral shift of partially coherent twisted anisotropic Gaussian Schell-model (GSM) beams in free space is

treated analytically by using the newly derived tensor ABCD law for partially coherent twisted anisotropic GSM beams.

This representation generalizes the previous results for isotropic GSM beams into anisotropic and twisted cases. It can

avoid numerical integral that is time consuming and commonly used in the previous study of spectral shift problems.

The results show that the spectral shift of twisted anisotropic GSM beams is closely related with the degree of co-

herence, wavefront curvature and the twist factor. The spectral shift changes from blueshift to redshift with the increase

of the transverse coordinate. � 2002 Elsevier Science B.V. All rights reserved.

Keywords: Gaussian Schell-model beams; Spectral shift; Tensor ABCD law

1. Introduction

Since Wolf first revealed that partially coherentlight beam undergoes spectral shift during itspropagation in free space [1–4], this phenomena hasbecome a very attractive topic in optics, both intheoretical [5–10] and experimental [11–13] aspects.Recently, the spectral shift and spectral switches ofGaussian Schell-model (GSM) beams passingthrough apertures were studied [14–16]. The toolsused to study the spectral shift of GSM beams are

limited to numerical integration in most cases.Palma and Cincotti [17,18] studied the spatial be-havior of the Wolf effect of GSM beams in freespace and passing through a thin lens, but confinedto the isotropic and untwisted GSM beams. Thisconfinement is partially due to the lack of appro-priate analytical methods to treat the propagationof partially coherent twisted anisotropic GSMbeams. The existed theory commonly used to studythe propagation and transformation of twisted an-isotropic GSM beams is based on the second-orderintensity moments of Wigner distribution function[19–22]. This is an equivalent method that is notappropriate to study the spectral shift phenomena.Recently, a new tensor method is developed to

study the propagation and transformation oftwisted anisotropic GSM beams [23]. In this paper,

1 April 2002

Optics Communications 204 (2002) 17–23

www.elsevier.com/locate/optcom

*Corresponding author. Fax: +86-571-8827-3086.

E-mail address: qlin@mail.hz.zj.cn (Q. Lin).1 Also with the State Key Laboratory of Modern Optical

Instrumentation, Zhejiang University.

0030-4018/02/$ - see front matter � 2002 Elsevier Science B.V. All rights reserved.

PII: S0030-4018 (02 )01206-3

we are going to apply this method to study thespectral shift phenomena of partially coherenttwisted anisotropic GSM beams propagating infree space. Explicit results have been obtainedconcerned with the dependence of the spectral shifton the degree of coherence, radius of curvature,the twist factor, lateral and longitudinal coordi-nates, etc. Our results can be easily reduced to theprevious results of isotropic GSM beams.

2. Tensor method in treating the propagation of

twisted anisotropic GSM beams

In this section, we are going to outline brieflythe tensor formulae for partially coherent twistedanisotropic GSM beams. We consider a partiallycoherent twisted anisotropic GSM beams origi-nating from a planar source located on the planeof z ¼ 0. Then the cross-spectral density of thetwisted anisotropic GSM beams takes the follow-ing form [19–22]:

Cðr1; r2;xÞ ¼ S0ðxÞ exp�� 14

rT1 ðr2I Þ�1r1

hþ rT2 ðr2I Þ

�1r2

i

� 12ðr1 � r2ÞTðr2gÞ

�1ðr1 � r2Þ

� ix2c

ðr1 � r2ÞTðR�1 þ lJÞðr1 þ r2Þ�;

ð1Þ

where S0ðxÞ is the initial spectral profile of thesource, x is the angular frequency, and c is thevelocity of light in vacuum. r1 and r2 are the posi-tion vectors of two arbitrary points in the trans-verse plane. r2I is the transverse spot width matrix,r2g is the transverse coherence width matrix, theyhave dimension of length square. R�1 is the wave-front curvature matrix with dimension of inverselength. r2I ; r

2g;R

�1 are all 2� 2 matrices withtransposition symmetry, given by

ðr2I Þ�1 ¼ r�2

I11 r�2I12

r�2I12 r�2

I22

� �;

ðr2gÞ�1 ¼

r�2g11 r�2

g12

r�2g12 r�2

g22

!;

R�1 ¼ R�111 R�1

12

R�121 R�1

22

� �:

ð2Þ

J is a transposition anti-symmetry matrix given by

J ¼ 0 1�1 0

� �: ð3Þ

l is a scalar real-valued twist factor with inverselength dimension. Eq. (1) can be rearranged into amore compact form as follows:

Cðr;xÞ ¼ S0ðxÞ exp� ix2c

rTM�1r

; ð4Þ

where rT ¼ ðrT1 rT2 Þ ¼ ðx1 y1 x2 y2Þ, M�1 is a 4� 4complex matrix given by

M�1 ¼R�1 � ic

2x r2I� ��1 � ic

x r2g

��1icx r2g

��1þ lJT

0BB@

icx r2g

��1þ lJ

�R�1 � ic2x r2I� ��1 � ic

x r2g

��11CCA: ð5Þ

M�1 is a transposition symmetric matrix called‘‘partially coherent complex curvature tensor’’. Ingeneral case, it can be expressed as follows:

M�1 ¼M�1

11 M�112

M�112

� �T �M�111

� �� !

; ð6Þ

where M�111 is also transposition symmetric, but

M�112 is not.The general propagation formula of anisotropic

twisted GSM beams through axially non-sym-metric optical system reads [23]

Cðq;xÞ ¼ S0ðxÞ½detðAþ BM�1i Þ�1=2

� exp� ix2c

qTM�1o q

; ð7Þ

where qT ¼ ðqT1 qT2 Þ, q1 and q2 are position vectorsof two arbitrary points in the output plane. M�1

i

and M�1o denote the partially coherent complex

curvature tensor in the input and the outputplanes, respectively. They satisfy the followingtensor ABCD law for partially coherent beams:

M�1o ¼ ðC þDM�1

i ÞðAþ BM�1i Þ�1

; ð8Þ

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

18 Y. Cai, Q. Lin / Optics Communications 204 (2002) 17–23

A ¼A 0

0 A

� �; B ¼

B 0

0 �B

� �;

C ¼C 0

0 �C

� �; D ¼

D 0

0 D

� �;

ð9Þ

where A, B, C , D are all 2� 2 sub-matrices of thenon-symmetric optical system.

3. The normalized on-axis spectrum

If the optical system consists of a free space ofdistance z, the sub-matrices of the system take thefollowing form:

A ¼1 0

0 1

� �; B ¼

z 0

0 z

� �;

C ¼0 0

0 0

� �; D ¼

1 0

0 1

� �:

ð10Þ

Substituting Eq. (10) into Eq. (9) and applying Eq.(8), we can obtain the partially coherent complexcurvature tensor in the output plane:

M�1o ¼ ðM i þ BÞ�1: ð11ÞThe spectrum of the twisted anisotropic GSM

beams after passing through the optical system canbe obtained by setting q1 ¼ q2 in Eq. (7), i.e.

S q1;xð Þ ¼ C q1ð ¼ q2;xÞ: ð12ÞLet us assume the initial spectrum S0 xð Þ is of theLorentz type, i.e.

S0ðxÞ ¼ S0d2

x � x0ð Þ2 þ d2; ð13Þ

where S0 is a constant, x0 is the central frequencyof the initial spectrum, and d is the half-width athalf-maximum of the initial spectrum. In the fol-lowing, the parameters used in the numerical cal-culations are x0 ¼ 3:2� 1015 rad=s, d ¼ 0:6� 1015rad=s, S0 ¼ 1.Substituting Eqs. (9)–(11), (13) into Eq. (7) and

applying Eq. (12), we can obtain the spectrum ofthe twisted anisotropic GSM beams at any prop-agation distance. Fig. 1 shows the on-axis nor-malized spectrum SðxÞ at several propagationdistances. The parameters used in the calculationare

l ¼ 0;

ðr2I Þ�1 ¼

1 0:1

0:1 0:5

� �ðmmÞ�2;

ðr2gÞ�1 ¼

10 1

1 3:33

� �ðmmÞ�2;

R�1 ¼0:0001 0:0001

0:0001 0:0001

� �ðmmÞ�1:

From Fig. 1, we can find that the shape of nor-malized on-axis spectrum of the twisted aniso-tropic GSM beams is similar to the original oneafter propagation through a distance, but its peakposition is blue-shifted. In the following section,we will discuss the quantitative influence of thetransverse coherence width matrix, the phase frontmatrix and the twist factor on the spectral shift ofthe twisted anisotropic GSM beams.

4. Relative frequency shift along the propagation

axis

The spectral shift of the partially coherent GSMbeams can be denoted by the relative central fre-quency shift. Assume xm is the central frequencyof the spectrum after propagation, then the rela-tive central frequency shift can be expressed by

Dx=x0 ¼ xmð � x0Þ=x0: ð14ÞThe on-axis relative central frequency shift of

the twisted anisotropic GSM beams with different

wno

rmal

ized

spe

ctru

m S

Fig. 1. The normalized on-axis spectrum SðxÞ on the plane of:(a) z ¼ 0; (b) z ¼ 5 m; (c) z ¼ 50 m.

Y. Cai, Q. Lin / Optics Communications 204 (2002) 17–23 19

twist factors along with the propagation axis z isdepicted in Fig. 2. The parameters used in thecalculations are

ðr2gÞ�1 ¼ 10 1

1 3:33

� �ðmmÞ�2;

ðr2I Þ�1 ¼ 1 0:1

0:1 0:5

� �ðmmÞ�2;

R�1 ¼ 0:0001 0:0001

0:0001 0:0001

� �ðmmÞ�1:

From Fig. 2, we can find that the on-axis relativefrequency shift increases with the increase of thepropagation distance z. At the far field, however,

the relative frequency shift approaches to a con-stant. We also find that the relative frequency shiftis closely related with the twist factor l. With theincrease of the twist factor, the relative frequencyshift decreases.Fig. 3 shows the on-axis relative central fre-

quency shift of the twisted anisotropic GSMbeams with different coherence length matrix ele-ments r2g11 along with the propagation axis z. Theparameters used in the calculation are

l ¼ 0;

r2g12 ¼ 1 ðmmÞ2; r2g22 ¼ 0:3 ðmmÞ2;

ðr2I Þ�1 ¼

1 0:1

0:1 0:5

!ðmmÞ�2;

R�1 ¼0:0001 0:0001

0:0001 0:0001

!ðmmÞ�1:

From Fig. 3, we can find that the relative fre-quency shift is closely related with coherencelength matrix element r2g11. With the increase ofr2g11, the relative central frequency shift decreases.That means for GSM beams with higher degree ofcoherence, the spectral shift is smaller.Fig. 4 shows the on-axis and off-axis relative

central frequency shift of the twisted anisotropicGSM beams along with the propagation axis z.The parameters used in the calculation are

rela

tive

freq

uenc

y sh

ift ∆

w/w

0

Fig. 2. The on-axis relative central frequency shift of the

twisted anisotropic GSM beams with different twist factors

along with the propagation axis z: (a) l ¼ 0; (b) l ¼ 0:0002ðmmÞ�1; (c) l ¼ 0:0004 ðmmÞ�1.

rela

tive

freq

uenc

y sh

ift ∆

w/w

0

Fig. 3. The on-axis relative central frequency shift of the

twisted anisotropic GSM beams with different coherence length

matrix elements r2g11 along the propagation axis z: (a) r2g11 ¼0:1 ðmmÞ2; (b) r2g11 ¼ 1 ðmmÞ2; (c) r2g11 ¼ 1:5 ðmmÞ2.

rela

tive

freq

uenc

y sh

ift ∆

w/w

0

Fig. 4. The on-axis and off-axis relative central frequency shift

of the twisted anisotropic GSM beams along with the propa-

gation axis z: (a) x ¼ 0; y ¼ 0; (b) x ¼ 0; y ¼ 2:5 (mm); (c)x ¼ 0; y ¼ 5 (mm).

20 Y. Cai, Q. Lin / Optics Communications 204 (2002) 17–23

l ¼ 0;

ðr2gÞ�1 ¼

10 1

1 3:33

� �ðmmÞ�2;

ðr2I Þ�1 ¼

1 0:1

0:1 0:5

� �ðmmÞ�2;

R�1 ¼0:0001 0:0001

0:0001 0:0001

� �ðmmÞ�1:

From Fig. 4, we can find that the relative frequencyshift of the off-axis points is very different from thatof the on-axis points, the relative frequency shiftsof the off-axis points decrease with the propagationdistance z in the near field, but increase with thepropagation distance z in the far field. For the off-axis points far enough away from the optical axis,the redshift of the spectrum can be observed.

5. Relative frequency shift in the far field

In this section, we are going to study the relativefrequency shift of the twisted anisotropic GSMbeams under various beam parameters in the farfield. The influences of the coherence length matrixon the relative frequency shift in the far field aredepicted in Fig. 5. The lateral coordinate is thecoherence length matrix element r2g11, the longitu-dinal coordinate is the on-axis relative frequencyshift in the far field. The parameters used in thecalculation are

r2g12 ¼ 1 ðmmÞ2; r2g22 ¼ 0:3 ðmmÞ2;

ðr2I Þ�1 ¼

1 0:1

0:1 0:5

� �ðmmÞ�2;

R�1 ¼0:0001 0:0001

0:0001 0:0001

� �ðmmÞ�1:

From Fig. 5, we can see that the relative fre-quency shift of the twisted anisotropic GSMbeams in the far field is strongly dependent on thecoherence length matrix and twist factor. With theincrease of the lateral coherence width, the relativefrequency shift decreases. It implies that for higherdegree of coherence of the twisted anisotropicGSM beams, the relative frequency shift is smaller.With the increase of twist factor, the relative fre-quency shift decreases.Next let us analyze the influence of the wave-

front matrix on the frequency shift in the far field.Fig. 6 shows the relative on-axis frequency shiftDx=x0 versus the radius of wavefront curvaturematrix element of the twisted anisotropic GSMbeams with several twist factors. The parametersused in the calculation are

R12 ¼ 10000 mm; R22 ¼ 10000 mm;

ðr2I Þ�1 ¼

1 0:1

0:1 0:5

� �ðmmÞ�2;

ðr2gÞ�1 ¼

1 0:1

0:1 0:33

� �ðmmÞ�2:

rela

tive

freq

uenc

y sh

ift ∆

w/w

0

σ

Fig. 5. The relative on-axis frequency shift Dx=x0 versus co-

herence length matrix element r2g11 of the twisted anisotropicGSM beams with different twist factors: (a) l ¼ 0; (b)l ¼ 0:0001 ðmmÞ�1; (c) l ¼ 0:0002 ðmmÞ�1.

rela

tive

freq

uenc

y sh

ift ∆

w/w

0

Fig. 6. Relative on-axis frequency shift Dx=x0 versus the ra-

dius of wavefront curvature matrix element R11 in the far field:(a) l ¼ 0; (b) l ¼ 0:0001 ðmmÞ�1; (c) l ¼ 0:0002 ðmmÞ�1.

Y. Cai, Q. Lin / Optics Communications 204 (2002) 17–23 21

From Fig. 6, we can find that with the increaseof the radius of wavefront curvature matrix ele-ment, the far-field relative frequency shifts of thetwisted anisotropic GSM beams increase, but aftera certain value of R11, the relative frequency shiftapproaches to a constant. With the increase of thetwist factor, the growing speed of the relative fre-quency shift along with the wavefront curvaturematrix element decreases.The on-axis relative frequency shift versus the

twist factor is given in Fig. 7. The parameters usedin the calculation are

ðr2I Þ�1 ¼

1 0:1

0:1 0:5

� �ðmmÞ�2;

ðr2gÞ�1 ¼

10 1

1 3:33

� �ðmmÞ�2;

R�1 ¼0:0001 0:0001

0:0001 0:0001

� �ðmmÞ�1:

From Fig. 7, we can see that the on-axis frequencyshift of the twisted anisotropic GSM beams isclosely related with the twist factor l. The relativefrequency shift in the far field decreases with theincrease of the twist factor. After a certain value ofl, the relative frequency shift approaches to zero.Finally, we calculate and compare the fre-

quency shift of the twisted anisotropic GSMbeams for the on-axis and off-axis points in the farfield. The relative frequency shift Dx=x0 versus thetransverse coordinate y in the far field is shown inFig. 8. The parameters used in the calculation are

x ¼ 0;

ðr2I Þ�1 ¼

1 0:1

0:1 0:5

� �ðmmÞ�2;

ðr2gÞ�1 ¼

10 1

1 3:33

� �ðmmÞ�2;

R�1 ¼0:0001 0:0001

0:0001 0:0001

� �ðmmÞ�1:

From Fig. 8, we can find that the frequency shiftsfor the on-axis and off-axis points are very differ-ent. With the increase of the transverse coordinatey, the spectral shift changes from blueshift to red-shift. For a certain value of the transverse coordi-nate, the spectral shift is zero. The results also showthat with the increase of the twist factor, the de-creasing speed of the relative frequency shift alongwith the transverse coordinate becomes slower.

6. Conclusions

The spectral shift of partially coherent twistedanisotropic GSM beams propagating in free spaceis investigated by using the tensor ABCD law forpartially coherent beams. The results show that thespectral shift is closely related with the parametersof twisted anisotropic GSM beams, such as thedegree of coherence, wavefront curvature matrixand the twist factor. The spectral shift of thetwisted anisotropic GSM beams decreases with theincrease of the degree of coherence, the radius of

twist factor m (mm-1)

rela

tive

freq

uenc

y sh

ift ∆

w/w

0

Fig. 7. The relative frequency shift Dx=x0 versus the twist

factor l in the far field.

rela

tive

freq

uenc

y sh

ift ∆

w/w

0

Fig. 8. The relative frequency shift Dx=x0 versus the transverse

coordinate y in the far field: (a) l ¼ 0; (b) l ¼ 0:0001 ðmmÞ�1;(c) l ¼ 0:0002 ðmmÞ�1; (d) l ¼ 0:0003 ðmmÞ�1.

22 Y. Cai, Q. Lin / Optics Communications 204 (2002) 17–23

wavefront curvature and the twist factor. The on-axis spectrum is always blue-shifted along with thepropagation axis z, but the spectrum changes fromblueshift to redshift with the increase of thetransverse coordinate. The method used in thispaper is a convenient way to study the spectralshift problems of partially coherent GSM beamsthat avoids the time-consuming numerical integraland can be applied to any paraxial optical systems.

Acknowledgements

This work is supported by National NaturalScience Foundation of China (60078003), and HuoYing Dong Education Foundation of China(71009).

References

[1] E. Wolf, Phys. Rev. Lett. 56 (1986) 1370.

[2] E. Wolf, Nature 326 (1987) 363.

[3] E. Wolf, Opt. Commun. 62 (1987) 12.

[4] E. Wolf, Phys. Rev. Lett. 58 (1987) 2646.

[5] G.M. Morris, D. Faklis, Opt. Commun. 62 (1987) 5.

[6] Z. Dacic, E. Wolf, J. Opt. Soc. Am. A 5 (1988) 1118.

[7] F. Gori, G.L. Marcopoli, M. Santarsiero, Opt. Commun.

81 (1991) 123.

[8] M. Santarsiero, F. Gori, Phys. Lett. 167 (1992) 123.

[9] A. Gamliel, J. Opt. Soc. Am. A 7 (1990) 1591.

[10] F. Gori, G. Guattari, C. Palma, C. Padovani, Opt.

Commun. 66 (1988) 1.

[11] H.C. Kandpal, J.S. Vaishya, K.C. Joshi, Opt. Commun. 73

(1989) 169.

[12] H.C. Kandpal, J.S. Vaishya, K. Saxena, D.S. Mehta, K.C.

Joshi, J. Mod. Opt. 42 (1995) 455.

[13] F. Gori, G. Guattari, C. Palma, Opt. Acta 30 (1983) 1075.

[14] Q. Lin, Z. Wang, J. Chen, S. Wang, J. Opt. (Paris) 26

(1995) 177.

[15] J. Pu, H. Zhang, S. Nemoto, Opt. Commun. 162 (1999) 57.

[16] J. Pu, S. Nemoto, IEEE. J. Quantum Electron. 36 (2000)

1407.

[17] C. Palma, G. Cincotti, Opt. Lett. 22 (1997) 671.

[18] C. Palma, G. Cincotti, J. Opt. Soc. Am. A. 14 (1997) 1885.

[19] R. Simon, N. Mukunda, J. Opt. Soc. Am. A 10 (1993) 95.

[20] D. Ambrosini, V. Bagini, F. Gori, et al., J. Mod. Opt. 41

(1994) 1391.

[21] R. Simon, A.T. Friberg, E. Wolf, Pure Appl. Opt. 5 (1996)

331.

[22] G. Ding, X. Yuan, B. Lu, J. Mod. Opt. 47 (2000) 1483.

[23] Q. Lin, Y. Cai, Opt. Lett. 27 (2002) 216.

Y. Cai, Q. Lin / Optics Communications 204 (2002) 17–23 23