Diffraction properties of partially coherent electromagnetic four-petal Gaussian beams

Optics Communications 283 (2010) 3105–3114

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Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m

Diffraction properties of partially coherent electromagnetic four-petal Gaussian beams Jia Li ⁎, Yanru Chen, Yu Xin, Muchun Zhou, Shixue Xu Department of Optical Engineering, Nanjing University of Science and Technology, Nanjing, Jiangsu Province 210094, China

a r t i c l e

i n f o

Article history: Received 25 January 2010 Received in revised form 19 March 2010 Accepted 6 April 2010 Keywords: Coherence length Four-petal Gaussian Polarization

a b s t r a c t The concept of existing scalar coherent four-petal Gaussian (FPG) beam is extended to the more general partially coherent electromagnetic four-petal Gaussian (PCFPG) beam, and the cross-spectral density matrix of the latter is derived in analytical expressions. Intensity distributions and degree of polarization in free space are investigated by numerical examples. Results show that propagation properties of the PCFPG beam are different from their scalar coherent and scalar partially coherent counterparts. Both intensity distributions and polarization strongly depend on transversal coherence lengths. When the propagation distance increases to a large value, intensity distributions of PCFPG beams convert into the Gauss profile. Polarization of PCFPG beams shows an oscillating characteristic upon propagation, and the oscillation would enhance with the increase of coherence lengths. These results may find potential applications in beam shaping and free space optical communications. © 2010 Elsevier B.V. All rights reserved.

1. Introduction In recent years beam pattern formations and beam shaping have attracted more and more attentions as well as their propagation properties [1–3]. Since then, methods of generating various beam patterns have found wide applications in optical resonators. For example, the dark hollow beams [4,5] and flat-topped beams [6–8] have been frequently utilized in last two decades. More recently, beams carrying Flower-like patterns [9] and hexagonal patterns [10] have been synthesized in the laboratory, respectively. Daisy patterns carrying three or more petals around a single circle have also been generated by manipulating ring cavity devices [11]. Very recently, a new form of laser beams called the four-petal Gaussian (FPG) beam is proposed in analytical expressions, and its propagation properties in free space are investigated in detail [12]. Due to its significant advantages in micro-optics and beam splitting techniques [13], many works have been carried out investigating its propagation properties in various media [14–18]. Although these works referred above are valuable, they are restricted to the concept of scalar coherent [12,15,16,18] or vectorial linearly polarized FPG beams [14,17]. On the other hand, fully coherent laser beams are more sensitive to properties of propagation media than other beams. Subsequently researchers suggested that partially coherent beams have less sensitivity to some random media than coherent cases. Since then, many works have been carried out characterizing beam shaping and

⁎ Corresponding author. E-mail address: [email protected] (J. Li). 0030-4018/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.04.009

polarization of partially coherent beams [19–21]. Considering that in practical optical fields for experimental applications, most laser beams should be considered in the electromagnetic case, so their electric components in x and y directions should be simultaneously considered here. In this situation, it is necessary to acquire information about propagation properties of partially coherent electromagnetic four-petal Gaussian (PCFPG) beams, and also it is important to know its propagation properties such as intensity distributions and polarization etc., which are induced by variable transversal coherence lengths. To the best of our knowledge, these problems have not been taken into account so far. In this paper, the existing concepts of the scalar coherent four-petal Gaussian beam [12,14–18] is extended into the more general PCFPG beam, and the cross-spectral density matrix of the latter is derived in an analytical form. Changes of intensity distributions and polarization induced by transversal coherence lengths are investigated by numerical examples. Results show that propagation properties of the partially coherent electromagnetic four-petal Gaussian beam is different from its scalar coherent or scalar partially coherent counterpart, which indicates that intensity distributions and polarization strongly depend on source coherence lengths. It is also shown that, when the propagation distance increases, intensity distributions of PCFPG beams would convert into the Gauss profile. Polarization should be influenced by transversal coherence lengths and show oscillating characteristic as z increases. This oscillation would enhance with the increase of transversal coherence length. These phenomenon and their causations are given by analytical explanations in Appendix A. These effects have been shown constructive for applications related to PCFPG beams propagating in free space.

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2. Analysis of theory It is assumed that the PCFPG beam propagates towards the half space z ≥ 0, and the z-axis is taken to be the propagation axis. Here we assume that each electric field component of PCFPG beams at the source plane z = 0 takes the following electromagnetic form [12,14–18]:

Ei ðx;y;0Þ =

xy σi2

!2m

i

exp −

! x2 + y2 → ei ; σi2

i = x;y;

ð1Þ

where σi is the waist width component of Gaussian beams, mi is the beam order, e i is the transversal unit vector in plane z = 0. When mi = 0, Eq. (1) reduces to the expression of fundamental electromagnetic Gaussian beams. Eq. (1) is different compared to the existing scalar coherent counterpart [12,14–18] in forms that, here the existing scalar coherent beam has been extended into the more general partially coherent electromagnetic case described by Eq. (1). So components of the electric field in x and y directions must be taken into account simultaneously. For a partially coherent electromagnetic beam generated by the Gaussian Schell-model source [22], its cross-spectral density matrix in plane z = 0 can be expressed in the following form [20] Wij ðx1 ;y1 ;x2 ;y2 ;0Þ =

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ii ðx1 ;y1 ;0ÞIj ðx2 ;y2 ;0Þμij ðx1 −x2 ;y1 −y2 ;0Þ;

ð2Þ

where Ii(x, y; z) is the spectral density component represented by D E 4 Ii ðx;y;zÞ = Ei ðx;y;zÞEi ðx;y;zÞ ;

ð3Þ

where the asterisk denotes the complex conjugate, bN denotes taking the ensemble average. µij is the correlation function which corresponds to the Gaussian Schell-model source [20] "

# ðx1 −x2 Þ2 + ðy1 −y2 Þ2 μij ðx1 −x2 ;y1 −y2 ;0Þ = exp − ; 2α2ij

ð4Þ

where αij is the transversal coherence length in the initial plane. x1, y1, x2 and y2 are position coordinates in the plane z = 0. Substituting Eqs. (1), (3), and (4) into Eq. (2), the elements of cross-spectral density matrix of the PCFPG beam in the initial plane is expressed as

Wij ðx1 ;y1 ;x2 ;y2 ;0Þ =

x1 y1 σi2

!2m

i

x2 y2 σj2

!2m

(

j

exp −

) x21 + y21 x22 + y22 ðx1 −x2 Þ2 + ðy1 −y2 Þ2 − − ; i;j = x;y; σi2 σj2 2α2ij

ð5Þ

Under the condition aij → ∞, Eq. (5) reduces to the formula for coherent electromagnetic FPG beams. When mi = mj = 0 is satisfied, Eq. (5) reduces to the expression for partially coherent electromagnetic Gaussian Schell-model beams. In this paper diffraction properties of PCFPG Gaussian beams are investigated. Under the paraxial approximation, the cross-spectral density matrix of a partially coherent electromagnetic beam through an aligned ABCD optical system is formulated by the generalized Collins formula [23]

 Wij ðp1 ;q1 ;p2 ;q2 ;zÞ =

k 2πjBj

2

+ ∞ + ∞ + ∞ + ∞

∫

∫

∫

∫ Wij ðx1 ;y1 ;x2 ;y2 ;0Þ

−∞ −∞ −∞ −∞

   ik  4 2 ik  4 2 4 2 4 2 × exp − 4 A x1 −2x1 p1 + D p1 − 4 A y1 −2y1 q1 + D q1 2B 2B     ik ik  2 2 2 2 Ax2 −2x2 p2 + Dp2 + Ay2 −2y2 q2 + Dq2 dx1 dy1 dx2 dy2 ; × exp 2B 2B

ð6Þ

where p1, q1, p2, and q2 are position coordinates in the output reference plane z. A, B, C and D are matrix elements of optical systems, k = 2π/λ is the wave number of light in vacuum. Substituting Eq. (5) into Eq. (6) and expanding the power exponent into the Hermite polynomial as the following form [24]

2m

x

=

pffiffiffi  ð2mÞ! m 1 H2s 2x ; ∑ 3m 2 s = 0 ðm−sÞ!ð2sÞ!

ð7Þ

J. Li et al. / Optics Communications 283 (2010) 3105–3114

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then Eq. (6) can be rewritten as 

k 2πjBj

Wij ðp1 ;q1 ;p2 ;q2 ;zÞ =

h   i 2 ð2m Þ! 2m ! 2 i j

mi

6mi + 6mj

2 mj

mi

   ik 4  2 ik  2 2 2 D p2 + q2 exp − 4 D p1 + q1 + 2B 2B

mj



× ∑ ∑ ∑ ∑

s = 0 t = 0 h = 0 l = 0 ðm −sÞ!ðm −t Þ! i i

"

+ ∞ + ∞

∫ exp

× ∫

−

−∞ −∞

1    mj −h ! mj −l !ð2sÞ!ð2t Þ!ð2hÞ!ð2lÞ!

! ik 4 1 1 2 A − − x1 + 2B4 σi2 2α2ij

! ik x2 p + x1 + 1 B4 α2ij

! # ik 1 1 ik 2 A− 2 − 2 x2 − p2 x2 2B B σj 2αij

" ! pffiffiffi ! pffiffiffi ! + ∞ + ∞ 2x1 2x2 ik 4 1 1 2 H2h dx1 dx2 × ∫ ∫ exp − 4 A − 2 − 2 y1 + σi σj 2B σi 2αij −∞ −∞

× H2s

"

ik y q1 + 22 B4 αij

ð8Þ

!#

! # pffiffiffi ! pffiffiffi ! 2y1 2y2 ik 2 1 1 ik 2 Ay2 − 2 − 2 y2 − q2 y2 × H2t H2l dy1 dy2 ; 2B B σi σj σj 2αij

× exp

Recalling the following equations [24]    pffiffiffi    ay 2 2 2 m π 1−a 2 Hm pffiffiffiffiffiffiffiffiffiffiffiffiffi ; ∫ exp −x + 2xy Hm ðaxÞdx = exp y 1−a2 −∞ + ∞

Hm ðx + yÞ =

1 m=2

2

 ∑

f =0

ð9Þ

 pffiffiffi  pffiffiffi  m Hf 2x Hm−f 2y ; f

ð10Þ

integrating over variables x1,and y1, Eq. (8) can be further rewritten as the following form  Wij ðp1 ;q1 ;p2 ;q2 ;zÞ =

k pffiffiffi 2 πjBj

h   i 2 ð2m Þ! 2m ! 2 i j 26mi

" × exp −

+ 6mj

   ik 4  2 ik  2 2 2 D p2 + q2 exp − 4 D p1 + q1 + 2B 2B

# mj mj mi mi k2 p21 k2 q21 1     4 2 − 4 2 × ∑ ∑ ∑ ∑ s = 0 t = 0 h = 0 l = 0 ðm −sÞ!ðm −t Þ! m −h ! m −l ! 4Mij B 4Mij B i i j j

1 × ð2sÞ!ð2t Þ!ð2hÞ!ð2lÞ!

1 1 − 2 Mij σi2 Mij

!s + t 2s

2t

∑∑

f = 0g = 0

2s

!

2t

f

g

0

!

1

ikp1 B C ffi A H2s−f @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Mij2 σi2 −2Mij B4

0

1 " ! # + ∞ ikq1 ik 1 1 1 2 B C A− 2 − 2 + × H2t−g @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A × ∫ exp x2 2B σj 2αij 4Mij α4ij −∞ Mij2 σi2 −2Mij B4 " × exp

0 1 ! # pffiffiffi ! 2x2 ikp1 ik x2 B C − p2 x2 H2h Hf @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Adx2 σj 2Mij B4 α2ij B M2 σ 2 −2M α2 ij

+ ∞

"

× ∫ exp −∞

ð11Þ

i

ij

ij

! # " ! # pffiffiffi ! 2y2 ik 1 1 1 ikq1 ik 2 A− 2 − 2 + q − y exp y H 2 2 2l 2 2B σj 2Mij B4 α2ij B σj 2αij 4Mij α4ij

0

1

y2 B C ffi Ady2 ; × Hg @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 Mij σi −2Mij αij where the factor Mij in Eq. (11) is represented by Mij =

ik 4 1 1 A + 2 + ; 2B4 σi 2α2ij

ð12Þ

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J. Li et al. / Optics Communications 283 (2010) 3105–3114

In order to integrate over x2, and y2 in Eq. (11), the following two equations are utilized [24]

    m 2 2 −m pffiffiffi πHm ðiγÞ; ∫ x exp −x + 2γx dx = exp γ ð2iÞ

+ ∞

ð13Þ

−∞

½f = 2

u

Hf ðxÞ = ∑ ð−1Þ u=0

f! f −2u ð2xÞ ; u!ðf −2uÞ!

ð14Þ

Substituting Eqs. (13) and (14) into Eq. (11) and integrating over variables x2, and y2, after tedious but straightforward calculations, Eq. (11) can be arranged in the following analytical form  Wij ðp1 ;q1 ;p2 ;q2 ;zÞ =

k 2jBj

h   i 2 ð2m Þ! 2m ! 2 i j 26mi

( × exp −

mi

"

k2 Nij

mi

+ 6mj

mj

   ik 4  2 ik  2 2 2 D p2 + q2 exp − 4 D p1 + q1 + 2B 2B

!2 q1 q2 − + 4Nij B4 α2ij 2B

mj

× ∑ ∑ ∑ ∑

s = 0 t = 0 h = 0 l = 0 ðm −sÞ!ðm −t Þ! i i

2s

2t

× ∑∑

2s

f = 0g = 0

f

½f = 2 ½g = 2

h

!

2t g

0

!

!2 #)

( exp −

)   k2 2 2 4 2 p1 + q1 4Mij B

1     × mj −h ! mj −l !ð2sÞ!ð2t Þ!ð2hÞ!ð2lÞ! 1

0

1 1 − 2 Mij σi2

!s + t

Mij Nij

1

ikp1 ikq1 B C B C ffi AH2t−g @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi A H2s−f @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Mij2 σi2 −2Mij B4 Mij2 σi2 −2Mij B4

l

u + v + r + d

× ∑ ∑ ∑ ∑ ð−1Þ u=0 v=0 r=0 d=0

0

p1 p − 2 4Nij B4 α2ij 2B

1f

ð15Þ

f !g!ð2hÞ!ð2lÞ! 8 u!ðf −2uÞ!v!ðg−2vÞ!r!ð2h−2rÞ!d!ð2l−2dÞ! σj2

!h

+ l−r−d

0

+ g−2u−2v 2r + 2u−2h−f + 2d + 2v−2l−g

ð2iÞ qffiffiffiffiffiffi2h + Nij

2 B C × @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A 2 2 Mij σi −2Mij α2ij

f −2r−2u + 2l + g−2d−2r

H2h

1 k kp1 p2 − 4 2 B2B 4Mij B αij C B C qffiffiffiffiffiffi C + f −2r−2u B @ A Nij

0

1 k kq1 q − B2B 2 4M B4 α2 C ij ij C B qffiffiffiffiffiffi × H2l + g−2d−2v B C; @ A N ij

where the factor Nij in Eq. (15) is represented by

Nij = −

ik 1 1 1 A+ 2 + − ; 2B σj 2α2ij 4Mij α4ij

ð16Þ

Eq. (15) is the generalized analytical expression for the cross-spectral density matrix of PCFPG beams on propagation through ABCD optical systems. Subsequently we discuss some special cases of Eq. (15). It is assumed in this paper that the PCFPG beam is isotropic in the initial plane

J. Li et al. / Optics Communications 283 (2010) 3105–3114

3109

z = 0, so σi = σj = σ0, mi = mj = m should be satisfied both in Eqs. (5) and (15). When αij → ∞, which corresponds to the coherent case, the crossspectral density of scalar coherent four-petal Gaussian beams is obtained  Wcoh ðp1 ;q1 ;p2 ;q2 ;zÞ =

k 2jBj

2

   ½ð2mÞ!4 ik 4  2 ik  2 2 2 D p + exp − D p + q + q 1 1 2 2 2B 2B4 212m "

# " # 2 2    m m m m k  2 k 1 2 2 2 × exp − p2 + q2 exp − × ∑ ∑ ∑ ∑ p1 + q1 4NB2 s = 0 t = 0 h = 0 l = 0 ðm−sÞ!ðm−t Þ!ðm−hÞ!ðm−lÞ! 4M B4 2

1 × ð2sÞ!ð2t Þ!ð2hÞ!ð2lÞ! 0

1 1 − 2 Mσ02

!s + t 2s

2t

∑∑

MN

f = 0g = 0

2s

×

2t

f

!

g

0

1

ikp1 B ffi C H2s−f @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A 2 2 M σ0 −2MB4

ð17Þ

1

½f = 2 ½g = 2 h l ikq1 u B C ffi A ∑ ∑ ∑ ∑ ð−1Þ × H2t−g @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 4 u=0 v=0 r=0 d=0 M σ0 −2M B

8 σ02

!

!h +

l−r−d

+ v + r + d

ð2iÞ2r + 2u−2h−f + 2d + 2v−2l−g pffiffiffiffi2h + f −2r−2u + 2l + g−2d−2r H2h + N

f !g!ð2hÞ!ð2lÞ! u!ðf −2uÞ!v!ðg−2vÞ!r!ð2h−2rÞ!d!ð2l−2dÞ!

 f −2r−2u

 kp2 pffiffiffiffi H2l 2B N

 + g−2d−2v

 kq2 pffiffiffiffi ; 2B N

with factors M, and N in Eq. (17) given by ik 4 1 ik 1 A + 2 ;N = − A + 2 ; 2B 2B4 σ0 σ0

M=

ð18Þ

further it is assumed that mx = my = 0, then Eq. (15) reduces to the expression for cross-spectral density matrix of electromagnetic Gaussian Schell-model beams in optical systems [23,25]

GSM

Wij

ðp1 ;q1 ;p2 ;q2 ;zÞ =

k2 2jBj

!2

   1 ik 4  2 ik  2 2 2 D p2 + q2 exp − 4 D p1 + q1 + Mij Nij 2B 2B (

× exp −

2

k Nij

"

q1 q − 2 4Nij B4 α2ij 2B

!2 +

p1 p − 2 4Nij B4 α2ij 2B

!2 #)

(

)   k 2 2 exp − 4 2 p1 + q1 ; 4Mij B

ð19Þ

2

Considering cases for free space diffraction of PCFPG beams, matrix elements of ABCD optical systems in Eq. (6) takes the form [26] 

A C

B D



 =

1 0

 z ; 1

ð20Þ

Recalling the formulae for intensity distributions and degree of polarization in the propagation field (z N 0), which are given by Iðp;q;zÞ = Tr ½W ðp;q;zÞ;

P ðp;q;zÞ =

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4Det ½W ðp;q;zÞ ; 1− fTr½W ðp;q;zÞg2

ð21Þ

ð22Þ

with W(p, q; z) is the cross-spectral density matrix which is defined as following  W ðp;q;zÞ =

 Wxx ðp;q;zÞ Wxy ðp;q;zÞ ; Wyx ðp;q;zÞ Wyy ðp;q;zÞ

ð23Þ

Where the elements of Eq. (23) have already been derived in Eq. (15). In this paper another assumption is made that off-diagonal element of cross-spectral density matrix for PCFPG beams being zero value in the initial plane [27] Wxy ðx;y;0Þ = Wyx ðx;y;0Þ = 0;

ð24Þ

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J. Li et al. / Optics Communications 283 (2010) 3105–3114

Substituting Eq. (15) into Eqs. (21) and (22), and using Eqs. (20) and (24), intensity distributions and degree of polarization of PCFPG beams in free space can be expressed as þ

Iðp;q;zÞ = A ðp;q;zÞ;

ð25Þ

   −  þ P ðp;q;zÞ = A ðp;q;zÞ = A ðp;q;zÞ;

ð26Þ

with A+, A− being represented by

 AFðp;q;zÞ =

k 2z

2

2s

½ð2mÞ!4 m m m m 1 ∑ ∑ ∑ ∑ 212m s = 0 t = 0 h = 0 l = 0 ðm−sÞ!ðm−t Þ!ðm−lÞ!ðm−hÞ!ð2sÞ!ð2t Þ!ð2hÞ!ð2lÞ!

2s

2t

× ∑∑

!

2t

f

f = 0g = 0

g

!

½f = 2 ½g = 2

h

l

h + l−r−d

∑ ∑ ∑ ∑ ð−1Þ

u=0 v=0 r=0 d=0

( 2r + 2u−2h−f + 2d + 2v−2l−g

× ð2iÞ

×

1 k − ˜ xx z0 2 2M

 h f !g!ð2hÞ!ð2lÞ! 4k u!ðf −2uÞ!v!ðg−2vÞ!r!ð2h−2rÞ!d!ð2l−2dÞ! z0

!s + t "

# 2   k 2 2 exp − p +q ˜ xx z2 4M

˜ xx N˜ xx M

0 "

2

× exp −

k N˜ xx

1 4N˜ xx α2xx z

−

1 2z

!2

1f

qffiffiffiffiffiffiffi−2h−f N˜ xx

+ g−2u−2v

#

pffiffiffi B C   B C 2 2 2 C ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s p +q ×B B C 2 @ M A ˜ xx z0 ˜ xx α2xx −M k 0

×

+ 2r + 2u−2l−g + 2d + 2r

0

1

1

B C B C B C B C ikp ikq sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC ffi C H2s−f B H2t−g B B C Bsffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C 2 2 @ A @ 2M A ˜ xx z0 ˜ xx z0 2M ˜ xx z ˜ xx z −2M −2M k k

! 3 1 1 kp7 6 2z − ˜ 6 7 4M xx α2xx z 7H2l q ffiffiffiffiffiffiffi × H2h + f −2r−2u 6 6 7 4 5 ˜ N

2

xx

  k2 k 2 2 × exp − p +q − 2 ˜ yy z 4M N˜ yy

!2

1 4N˜ yy α2yy z

−

1 2z

 2 2 p +q

0 ×

qffiffiffiffiffiffiffiffi−2h−f N˜ yy

2 × H2h

6 6 6 4

+ f −2r−2u 6

t

xx

0 2

ð27Þ

! 3 !s + 1 1 1 k − kq 6 2z − ˜ 7 ˜ yy z0 2 2M 6 7 4M xx α2xx z 6 7F qffiffiffiffiffiffiffi + g−2d−2v 6 7 ˜ yy N˜ yy M 4 5 N˜

2

"

+ l−r−d

+ 2r + 2u−2l−g + 2d + 2r

# 

1f

+ g−2u−2v

pffiffiffi B C B C 2 C ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s ×B B C 2 @ M A ˜ yy z0 2 ˜ −M yy αyy k 1

0

1

B C B C B C B C ikp ikq ffi C ffi C sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H2s−f B ; H2t−g B Bsffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C B C 2 2 @ 2M A @ A ˜ yy z0 ˜ yy z0 2M ˜ ˜ −2Myy z −2Myy z k k

! 3 1 1 − kp7 ˜ yy α2yy z 2z 4M 7 7H2l qffiffiffiffiffiffiffiffi 7 5 N˜ yy

2 6 6 6 4

+ g−2d−2v 6

! 3 1 1 − kq7 ˜ yy α2yy z 2z 4M 7 7 qffiffiffiffiffiffiffiffi 7 5 N˜ yy

g

J. Li et al. / Optics Communications 283 (2010) 3105–3114

3111

where factors M̃ ii, Ñii (i = x, y) in Eq. (26) being expressed as ˜ ii = ik + k + 1 ;N˜ ii = − ik + k + 1 − 1 ;ði = x;yÞ; M ˜ ii α4 2z 2z0 2z 2z0 2α2ii 2α2ii 4M ii

ð28Þ

In Eqs. (27) and (28), z0 =kσ20 /2 is defined as the Rayleigh range of fundamental Gaussian beams [12,14,17,28]. Eqs. (25)–(28) are the main analytical results of this paper, which allow one to investigate intensity distributions and polarization of PCFPG beams in free space. Although these derived expressions seem complicated in their forms, which involve exponent functions, sums of binomial coefficients and Hermite polynomial etc., it only takes several minutes to run the calculation by using MatLab. On the contrary it would cost much more time performing numerical integrations of Eq. (8) due to the fact that it involves four integrals. 3. Numerical examples Based on the derived analytical expressions above, here we study intensity distributions and degree of polarization of PCFPG beams in free space. Their behaviors are shown in Figs. 1–4. Effects of transversal coherence lengths on intensity distributions and polarization are also discussed in detail. For Figs. 1–4, uniform beam parameters are chosen λ = 632.8 nm, σ0 = 0.01 m, unless otherwise stated. The slant axis R is set in a condition that the angle between the slanted axis and the p direction or q direction is 45, which is also equivalent with the relation by setting p = q in Eqs. (25)–(27). Fig. 1(c) shows normalized intensity distributions of PCFPG beams versus the slanted axis R at different propagation distances. For comparisons, Fig. 1(a) and (b) represent profiles of partially coherent electromagnetic Gaussian Schell-model beams and coherent electromagnetic four-petal Gaussian beams, respectively. From Fig. 1 it is found that intensity distributions of PCFPG beams (see Fig. 1(c)) are slightly different from their coherent counterparts (Fig. 1(b)). It also can be seen that, the PCFPB beam tends to a Gaussian profile when the propagation distance is large enough (see the dash lines in Fig. 1(b) and (c)), and this phenomenon has been explained in Appendix. Fig. 2 reflects effects of transversal coherence lengths αxx and αyy on intensity distributions of PCFPG beams with mx = my = 1. From Fig. 2(a) it can be concluded that transversal coherence lengths have no effect on intensity distributions in the source plane z = 0 due to Eqs. (3) and (5). However, when propagation distance z increases, intensity distributions start to depend on coherence lengths αxx and αyy (see Fig. 2(b)). Another phenomenon that can be seen in Fig. 2 is that, when coherence lengths αxx or αyy reduce, intensity distributions of PCFPG beams would convert into the Gauss profile much more rapidly (see dash–dot lines in Fig. 2(c) and (d)), and this phenomenon has also been explained in Appendix. Fig. 3 shows the degree of polarization of PCFPG beams versus slanted axis R at different propagation distances with mx = my = 2. From Fig. 3(a) it is found that when αxx = αyy, the degree of polarization of PCFPG beams is invariant upon propagation and equivalent to the initial value in the plane z = 0 (see Eqs. (5) and (22)). Detailed criterion of this phenomenon has been discussed in [29–31]. Fig. 3(b)–(c) represent that when αxx ≠ αyy, the degree of polarization changes upon propagation and the variations of αxx and αyy have strong modulations on the degree of polarization. Fig. 4 further show effects of transversal coherence lengths on the degree of polarization at propagation distance z = 0, 3z0, 6z0, 9z0, respectively, other beam parameters are the same as Fig. 3. Similar to Fig. 2(a), in the initial plane z = 0, coherence lengths αxx, and αyy have no effect on the degree of polarization (see Eqs. (5) and (22)). However, when propagation distance increases, degree of polarization strongly depends on coherence lengths. It also can be found that when z is small, polarization shows a more linearly characteristic (see Fig. 4(b)). However, as z subsequently increases, polarization represents more oscillating distributions, and this oscillation would enhance with the increase of coherence length αxx or αyy (see Fig. 4(c) and (d)). These phenomenons can be correspondingly explained by analytical forms shown in Appendix A.

Fig. 1. Normalized intensity distributions of PCFPG beams versus slanted axis with different propagation distances. (a) αxx = αyy = 10− 2 m, mx = my = 0, (b) αxx = αyy = infinity, mx =my =1, and (c) αxx =αyy =10− 2 m, mx =my =1.

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Fig. 2. Normalized intensity distributions of PCFPG beams versus slanted axis with different αxx and αyy, mx = my = 1. (a) z = 0, (b) z = 5z0, (c) z = 10z0, and (d) z = 15z0.

4. Conclusions In this paper the existing scalar coherent four-petal Gaussian beam is extended into a more general partially coherent electromagnetic four-petal Gaussian beam. The concept of partially coherent electromagnetic four-petal Gaussian beam (PCFPG) is proposed and its crossspectral density matrix is derived when it propagates in free space. Based on the derived formulae, intensity distributions and degree of polarization are investigated by numerical examples. Results show that propagation properties of PCFPG beam are different from its scalar coherent or scalar partially coherent counterparts, and its diffraction properties strongly depend on transversal coherence lengths. When the propagation distance decreases to a large value, intensity distributions would convert into the Gauss profile. On the other hand, polarization should be influenced by transversal coherence lengths and shows oscillating characteristics when z increases, and this oscillation would enhance with the increase of coherence lengths. These results may provide a new approach in laser shaping and modulating partially coherent electromagnetic beams in free space optical communications. Acknowledgement This work is supported by the National High Tech Research and Development Program of China (2007AA04Z181), the High Tech Industrial Development Project of the University in Jiangsu Province (BG2005006) and the NUST Research Funding, No.

2010ZYTS031. The author is indebted to the reviewers for valuable comments. Appendix A. Analytical explanations to the phenomenon shown in Figs. 1–4 Firstly, Eq. (27) is derived in a so sophisticated form, in order to investigate effects of parameters on light intensity and polarization of PCFPG beam upon propagation; we extract the main body of Eq. (27) by using characteristics of Hermite polynomials [24]. After tedious operations, Eqs. (25)–(27) can be rewritten into the following forms   Iðp;q;zÞ = Dðαxx ;p;q;zÞ + D αyy ;p;q;z ;

ðA1Þ

        P ðp;q;zÞ = Dðαxx ;p;q;zÞ−D αyy ;p;q; z  = Dðαxx ;p;q;zÞ + D αyy ;p;q;z ;

ðA2Þ with D(αii, p, q; z) given by Dðαii ;p;q;zÞ = Sðm;u;vÞ⋅Oðαii ;p;q;zÞ⋅Lðαii ;p;q;zÞ;ði = x;yÞ

ðA3Þ

for the partially coherent scalar four-petal Gaussian beam, its intensity distribution can be represented by Iscalar ðp;q;zÞ = Dðα0 ;p;q;zÞ;

ðA4Þ

J. Li et al. / Optics Communications 283 (2010) 3105–3114

where 2m

2m

Sðm;u;vÞ = ∑ ∑

ð2mÞ!

ð2mÞ!

u = 0 v = 0 u!ð2m−uÞ! v!ð2m−vÞ!

;

9 8 2  2 3 > > > > k k > − 2 = < 6 k2 7 > z ˜ z 2M 2 ii 6 7 2 ; Oðαii ;p;q;zÞ = exp −6 2 + + q p 7 2 > > 44M 5 ˜ ii z2 > > 4N˜ ii > > ; :

Lðαii ;p;q;zÞ =

1 2 N˜ ii z2

1+

1 ˜ 2ii N˜ 2ii α4 2M ii

× H2m−u −

!u = 2

ðA5Þ

3113

PCFPG beams depends on Eq. (A6) and converts into the Gaussian profile. Similarly, when αii increases, oscillations also play a more dominant role than the linear characteristic. This is why polarization changes from mainly linear distributions to oscillating characteristics (Figs. 3–4). These effects have provided plenty of information and shown constructive for applications related to the propagation of PCFPG beams in free space by modulating polarization.

ðA6Þ

+ v=2

pffiffiffi ! pffiffiffi ! 2k 2k p H2m−v − q ˜ ii z ˜ ii z 2M 2M

pffiffiffi 9 8 2 > > > > > > ! > > = < ˜ ii N˜ 2ii α2 4M k k ii − p × Hu − sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > ˜ 2ii α2 z z > 1 > > 2M > > ii > > 1+ ; : 2 2 4 ˜ ˜ 2Mii Nii αii

ðA7Þ

pffiffiffi 9 8 2 > > > > > > ! > > < ˜ ii N˜ 2ii α2 4M k k = ii × Hv − sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; − q > ˜ 2ii α2 z z > 1 > > 2M > > ii > > 1 + ; : ˜ 2ii N˜ 2ii α2 2M ii where M̃ ii and Ñii have been given in Eq. (28). It can be observed that Eq. (A6) is an exponential function and Eq. (A7) is a linear distributed function (due to the linear expansion of its Hermite polynomials). It shows that intensity distributions and polarization of PCFPG beams can be composed by linear distributed functions and exponential functions with different radius curvatures (see Eq. (A6)). This is why so many lobes appearing in intensity distributions (Figs. 1–2) and also oscillating characteristic shown in polarization behaviors (Figs. 3–4). Comparing Eqs. (A1) with (A4), we can observe that intensity distributions of partially coherent electromagnetic four-petal Gaussian beams are different from their partially coherent scalar counterparts. This is because intensity distributions of partially coherent scalar beams should be modulated by scalar coherence length α0 (Eq. (A4)). However, for the partially coherent electromagnetic beams, the electric optical components in x and y directions should be simultaneously considered, so their intensity distributions should be both modulated by components αxx and αyy. For example, when α0 = 0.01 m, αxx = αyy = 0.01 m, intensity profiles of partially coherent electromagnetic four-petal Gaussian beams are the same as their partially coherent scalar counterparts. However, when one of parameters αxx or αyy changes (αxx ≠ αyy), their intensity distributions are totally different (see Eqs. (A1), (A4), (A6) and (A7)). This difference is induced by different coherence length components αxx and αyy of electromagnetic beams. However, for partially coherent scalar beams, only the uniform coherence length α0 involves into modulations of intensity distributions. Polarization behaviors shown in Figs. 3–4 can also be explained by Eqs. (A2)–(A7). It can be observed that, when the propagation distance z increases, the value of Eq. (A6) increases and Eq. (A7) decreases. As a result, Eq. (A6) plays a dominant role in polarization modulations (see Eq. (A2)), so oscillating characteristic can be observed in Fig. 4(c)–(d). When z is large enough, the value of Eq. (A7) is so small compared to that of Eq. (A6), so the intensity of

Fig. 3. Degree of polarization of PCFPG beams versus slanted axis with different propagation distances, mx = m y = 2. (a) αxx = αyy = 10 − 2 m, (b) α xx = 0.01 m, αyy = 0.012 m, and (c) αxx = 0.01 m, αyy = 0.015 m.

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Fig. 4. Degree of polarization of PCFPG beams versus slanted axis with different αxx and αyy, mx = my = 2. (a) z = 0, (b) z = 3z0, (c) z = 6z0, and (d) z = 9z0.

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