Beam propagation of a higher-order cosh-Gaussian beam

ARTICLE IN PRESS Optics & Laser Technology 41 (2009) 202– 208

Contents lists available at ScienceDirect

Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec

Beam propagation of a higher-order cosh-Gaussian beam Guoquan Zhou , Jun Zheng School of Sciences, Zhejiang Forestry University, Lin’an 311300, Zhejiang Province, China

a r t i c l e in f o

a b s t r a c t

Article history: Received 22 February 2008 Received in revised form 4 May 2008 Accepted 6 May 2008 Available online 24 June 2008

On the basis of the second-order moment, the beam propagation of a higher-order cosh-Gaussian beam has been investigated. Two analytical expressions of the M2 factor have been derived, which corresponds to the parity of the beam order n. The M2 factor is determined by the beam order n and the parameter d. The influences of the beam order n and the parameter d on the M2 factor are illustrated with numerical examples. The analytical kurtosis parameter has also been presented. The analytical formula is further simplified in the source plane and the far field, respectively. With given appropriate values of n and d, the kurtosis parameter in the far field is smaller than 3. This research is helpful to the practical applications of a higher-order cosh-Gaussian beam. & 2008 Elsevier Ltd. All rights reserved.

PACS: 42.25.Bs 41.85.Ew Keywords: Beam propagation High-order cosh-Gaussian beam Second-order moment

1. Introduction The propagation of laser beams has received considerable interest. The propagation of coherent laser beams has been extensively studied in free space and through optical systems [1,2], and that of partially coherent Gaussian Schell-model beams have been widely investigated in turbulent atmosphere [3–5]. The average intensity and spreading of an elliptical Gaussian beam have been examined in a turbulent atmosphere [6]. The fractional Fourier transform of a partially coherent off-axis Gaussian Schell-model beam has been presented [7]. On the other hand, Hermite–sinusoidal-Gaussian (HSG) beams have been introduced as exact solutions of the paraxial wave equation [8,9], Gaussian beams and cosh-Gaussian beams can be regarded as special cases of HSG beams. A comprehensive investigation of the beam propagation of these beams has been reported [10–19]. The propagation properties of cosh-Gaussian beams and apertured cosh-Gaussian beams have been examined, and the corresponding beam propagation factor has been given [10–12]. The propagation characteristics of Hermite-cosh-Gaussian beams and elegant Hermite-cosh-Gaussian beams have been revealed, and the beam propagation factor has been expressed in a concise form [13,14]. The coherent superposition of off-axial Gaussian beams has been proved to be a nearly flattened beam with an axial shadow [15]. Similarly, the flattened beams with an axial shadow can be produced by coherent combination of certain cosh-Gaussian beams [16]. Recently, the research shows that the outcome of incoherent combination of cosh-Gaussian beams with different parameters is also a flattened Gaussian beam [17]. In the present paper, we investigate the propagation of a higher-order cosh-Gaussian beam. The higher-order cosh-Gaussian beam, which is defined as a higher-order cosh function multiplied by a Gaussian function, can be produced by coherent superposition of coshGaussian beams. Under the condition of appropriate parameters, the distribution of light intensity of a higher-order cosh-Gaussian beam is flattened. Moreover, the distribution of light intensity in the near field is nearly same as that in the source plane, while the distribution of light intensity in the far field no longer keeps flattened and emerges as a single apex [20]. Accordingly, a higher-order cosh-Gaussian beam is an appropriate model to describe the flattened laser beam [21,22]. To our best knowledge, however, the research in the beam propagation factor and the kurtosis parameter of a higher-order cosh-Gaussian beam has not been reported elsewhere. Therefore, analytical expressions of the beam propagation factor and the kurtosis parameter for a higher-order cosh-Gaussian beam will be presented in this paper.

 Corresponding author.

E-mail address: [email protected] (G. Zhou). 0030-3992/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2008.05.002

ARTICLE IN PRESS G. Zhou, J. Zheng / Optics & Laser Technology 41 (2009) 202–208

203

2. The beam propagation factor In the Cartesian coordinates system, a higher-order cosh-Gaussian beam propagates toward the half free space zX0, and z-axis is taken to be the propagation axis. The higher-order cosh-Gaussian beam at the source plane z ¼ 0 takes the form as En ðx; y; 0Þ ¼ En ðx; 0ÞEn ðy; 0Þ,

(1)

with En(x,0) and En(y,0) given by ! j2 n En ðj; 0Þ ¼ exp  2 cosh ðOjÞ; w0

n ¼ 1; 2; 3 . . .

(2)

where j ¼ x or y (hereafter), n is the beam order, w0 is the Gaussian waist width and O is the parameter associated with the cosh part. If n ¼ 0, Eq. (1) reduces to be the well-known Gaussian beam. En (j,0) can also be written in the form as follows: ( Intðn=2Þ coshð2mOjÞ; n is an even integer n! expðj2 =w20 Þ 1 X En ðj; 0Þ ¼ n1 ; (3) cosh½ð2m þ 1ÞOj; n is a odd integer fInt½ðn þ 1Þ=2 þ mg!½Intðn=2Þ  m! 2 m¼0 or En ðj; 0Þ ¼

"    2 # n 1 X n! n2 1 2m  n exp m  Ow20 d exp  2 j  , n 2 2 2 m¼0 m!ðn  mÞ! w0

(4)

where Int(  ) denotes taking the integral part of the expression, and d ¼ w20O2. En(j,0) can be produced by superposition of Int(n/2)+1 coshGaussian beams with appropriate weight coefficients. The case of n being an even integer corresponds to a higher-order even coshGaussian beam. Otherwise, it is a higher-order odd cosh-Gaussian beam. En (j,0) can also be produced by superposition of n+1 decentered Gaussian beams with the same waist width. The Mj2 factor is defined as follows [23]: M 2j ¼ 4psoj s1j ,

(5)

where s0j and sNj are the second-order moments of the intensity distribution in the waist plane and the far field, respectively. The 2s0j is just the beam waist, and 2psNj the far field divergence angle. The first-order moment of a higher-order cosh-Gaussian beam is given by R1 R1 R1 2 jjEn ðj; 0Þj2 dj 1 jjEn ðx; y; 0Þj dx dy ¯j ¼ R1 R1 ¼ ¼ 0. (6) R 1 1 1 2 2 1 1 jEn ðx; y; 0Þj dx dy 1 jjEn ðj; 0Þj dj Owing to the symmetry of the intensity distribution in the source plane, the beam center keeps located at the propagation axis. According to the second-order moment definition of s0j, one has s2oj ¼

R1 R1

2 ¯2 1 ðj  jÞ jEn ðx; y; 0Þj dx dy R1 R1 2 1 1 jEn ðx; y; 0Þj dx dy

1

R1 ¼ R1 1

j2 jEn ðj; 0Þj2 dj

1

jEn ðj; 0Þj2 dj

.

(7)

As Eq. (3) is convenient to calculation, it is used in the following derivation. To obtain s0j, Eq. (3) is substituted into Eq. (7). After length calculation, s0j of a higher-order even cosh-Gaussian beam yields 0

11=2 ðm þ m0 Þ2 exp½ðm þ m0 Þ2 d=2 þ ðm  m0 Þ2 exp½ðm  m0 Þ2 d=2 0 0 C w0 B B m¼0 m0 ¼0 fInt½ðn þ 1Þ=2 þ mg!½Intðn=2Þ  m!fInt½ðn þ 1Þ=2 þ m g!½Intðn=2Þ  m !C soj ¼ B1 þ dIntðn=2Þ Intðn=2Þ C . 0 2 0 2 A 2@ P P exp½ðm þ m Þ d=2 þ exp½ðm  m Þ d=2 Intðn=2Þ P Intðn=2Þ P

m¼0

m0 ¼0

(8)

fInt½ðn þ 1Þ=2 þ mg!½Intðn=2Þ  m!fInt½ðn þ 1Þ=2 þ m0 g!½Intðn=2Þ  m0 !

Under the integral process, the following integral formulae is satisfied [24]: rffiffiffi Z 1 ð2n  1Þ!! p ; nX1; a40 x2n expðax2 Þ dx ¼ nþ1 n a 2 a 0

(9)

and Z

1

expðax2 Þ dx ¼

0

1 2

rffiffiffi p ; a

a40.

Similarly, s0j of a higher-order odd cosh-Gaussian beam turns out to be 0 1 Intðn=2Þ P Intðn=2Þ P ðm þ m0 þ 1Þ2 exp½ðm þ m0 þ 1Þ2 d=2 þ ðm  m0 Þ2 exp½ðm  m0 Þ2 d=2 B C 0 0 w0 B m¼0 m0 ¼0 fInt½ðn þ 1Þ=2 þ mg!½Intðn=2Þ  m!fInt½ðn þ 1Þ=2 þ m g!½Intðn=2Þ  m !C soj ¼ B1 þ dIntðn=2Þ Intðn=2Þ C. 2 2 0 0 A 2@ P P exp½ðm þ m þ 1Þ d=2 þ exp½ðm  m Þ d=2 m¼0

m0 ¼0

(10)

(11)

fInt½ðn þ 1Þ=2 þ mg!½Intðn=2Þ  m!fInt½ðn þ 1Þ=2 þ m0 g!½Intðn=2Þ  m0 !

The optical field in the spatial-frequency domain En(fx,fy,0) reads as En ðf x ; f y ; 0Þ ¼ En ðf x ; 0ÞEn ðf y ; 0Þ,

(12)

with En(fx,0) and En(fy,0) given by [25] Z 1 En ðj; 0Þ expði2pf j jÞ dj, En ðf j ; 0Þ ¼

(13)

1

ARTICLE IN PRESS 204

G. Zhou, J. Zheng / Optics & Laser Technology 41 (2009) 202–208

where fx and fy are the transverse spatial frequency, respectively. Inserting Eq. (3) into Eq. (13), one obtains 8 1=2 Intðn=2Þ expðm2 dÞ cosð2mpOw20 f j Þ P p w0 n! > 2 > > ; n is an even integer expðp2 w20 f j Þ > n2 < 2 m¼0 fInt½ðn þ 1Þ=2 þ mg!½Intðn=2Þ  m! En ðf j ; 0Þ ¼ Intðn=2Þ > P exp½ðm þ 1=2Þ2 d cos½ð2m þ 1ÞpOw20 f j  > p1=2 w0 n! 2 > ; n is a odd integer expðp2 w20 f j Þ > : n2 fInt½ðn þ 1Þ=2 þ mg!½Intðn=2Þ  m! 2 m¼0

(14)

Therefore, the first-order moment in the spatial-frequency domain yields R1 R1 R1 2 f j jEn ðf j ; 0Þj2 df j 1 f j jEn ðf x ; f y ; 0Þj df x df y f¯ j ¼ R1 ¼ R1 ¼ 0. R 1 2 2 1 1 1 jEn ðf j ; 0Þj df j 1 1 jEn ðf x ; f y ; 0Þj df x df y

(15)

The second-order moment in the spatial-frequency domain sNj is given by s21j ¼

R1 R1 1

2 ¯ 2 1 ðf j  f j Þ jEn ðf x ; f y Þj df x R1 R1 2 1 1 jEn ðf x ; f y Þj df x df y

df y

R1 ¼ R1 1

2

f j jEn ðf j ; 0Þj2 df j

1

jEn ðf j ; 0Þj2 df j

.

(16)

sNj of a higher-order even cosh-Gaussian beam is found to be 0

s1j

11=2 Intðn=2Þ P Intðn=2Þ P ðm þ m0 Þ2 exp½ðm þ m0 Þ2 d=2 þ ðm  m0 Þ2 exp½ðm  m0 Þ2 d=2 B 0 0 C 1 B m¼0 m0 ¼0 fInt½ðn þ 1Þ=2 þ mg!½Intðn=2Þ  m!fInt½ðn þ 1Þ=2 þ m g!½Intðn=2Þ  m ! C ¼ B1  dIntðn=2Þ Intðn=2Þ C . 2 2 A 2pw0 @ P P exp½ðm þ m0 Þ d=2 þ exp½ðm  m0 Þ d=2 m¼0

m0 ¼0

fInt½ðn þ 1Þ=2 þ mg!½Intðn=2Þ  m!fInt½ðn þ 1Þ=2 þ

m0 g!½Intðn=2Þ



(17)

m0 !

sNj of a higher-order odd cosh-Gaussian beam yields 0

s1j

11=2 þ m0 þ 1Þ2 exp½ðm þ m0 þ 1Þ2 d=2 þ ðm  m0 Þ2 exp½ðm  m0 Þ2 d=2 0 0 C 1 B B m¼0 m0 ¼0 fInt½ðn þ 1Þ=2 þ mg!½Intðn=2Þ  m!fInt½ðn þ 1Þ=2 þ m g!½Intðn=2Þ  m ! C ¼ B1  d Intðn=2Þ Intðn=2Þ C . 2 2 0 0 @ A 2pw0 P P exp½ðm þ m þ 1Þ d=2 þ exp½ðm  m Þ d=2 Intðn=2Þ P Intðn=2Þ P ðm

m¼0

m0 ¼0

(18)

fInt½ðn þ 1Þ=2 þ mg!½Intðn=2Þ  m!fInt½ðn þ 1Þ=2 þ m0 g!½Intðn=2Þ  m0 !

The M2 factor of a higher-order cosh-Gaussian beam can be readily obtained [26] M2 ¼ ðM 2x M 2y Þ1=2 ¼ M 2x ¼ M2y .

(19) 2

2

The parity of the beam order n results in two different calculated formulae of the M factor. The M factor of a higher-order coshGaussian beam is determined by the beam order n and the parameter d. If n ¼ 0, M2 ¼ 1, which corresponds to a Gaussian beam. If n ¼ 1, it reduces to be a cosh-Gaussian beam. In this case, the M2 factor is simplified to be M2 ¼

½ð1  dÞ expðdÞ þ ð2  d2 Þ expðd=2Þ þ d þ 11=2 . 1 þ expðd=2Þ

(20)

The above result is consistent with Eq. (8) of Ref. [10]. Now, we compare our result with that of Ref. [16], which is also the case of superposition of cosh-Gaussian beams. The two-dimensional optical distribution at the source plane z ¼ 0 is expressed as [16] En ðx; 0Þ ¼ exp 

x2 w20

!

N X

Zm expðm2 Þ coshð2mx=w0 Þ,

(21)

m¼L

where Zm ¼ 1dm0/2. dm0 is the Kronecker symbol. N and L (N ¼ 0,1,2,y; NXL) are two constant integers. When L ¼ 0, n ¼ 2N, and O ¼ 1/w0, Eq. (21) is similar to a higher-order even cosh-Gaussian beam. The only difference is the weight coefficient. In this case, the beam propagation factor of the flattened light beam with an axial shadow is found to be [16] 2

M0 ¼ 4ps0 0x s0 1x ,

(22)

with s0 0x and s0 Nx given by 0

s0 0x

11=2 N N P P Zm Zm0 exp½ðm2 þ m0 2 Þ=2½ðm2 þ m0 2 Þ coshðmm0 Þ þ 2mm0 sinhðmm0 ÞC B w0 B C m¼0 m0 ¼0 ¼ B1 þ C , N N A 2 @ P P Zm Zm0 exp½ðm2 þ m0 2 Þ=2 coshðmm0 Þ

(23)

m¼0 m0 ¼0

0

s0 0x

11=2 N N P P Zm Zm0 exp½ðm2 þ m0 2 Þ=2½ðm2 þ m0 2 Þ coshðmm0 Þ  2mm0 sinhðmm0 ÞC B 1 B C m¼0 m0 ¼0 ¼ B1  C . N N A 2pw0 @ P P 2 2 0 0 Zm Zm0 exp½ðm þ m Þ=2 coshðmm Þ m¼0 m0 ¼0

(24)

ARTICLE IN PRESS G. Zhou, J. Zheng / Optics & Laser Technology 41 (2009) 202–208

205

The superscript is adopted to distinguish it from that of a higher-order cosh-Gaussian beam. s0x of a higher-order even cosh-Gaussian beam can be rewritten as 0

s0x

11=2 N N P P cm cm0 exp½ðm2 þ m02 Þ=2½ðm2 þ m0 2 Þ coshðmm0 Þ þ 2mm0 sinhðmm0 ÞC B w0 B C m¼0 m0 ¼0 ¼ B1 þ C , N N A 2 @ P P 2 2 0 0 cm cm0 exp½ðm þ m Þ=2 coshðmm Þ

(25)

m¼0 m0 ¼0

where cm ¼ (1/2N1)(N!/(N+1+m)!(Nm)!). sNx of a higher-order even cosh-Gaussian beam reduces to be 0

11=2 2 02 2 02 0 0 0 0 c c exp½ðm þ m Þ=2½ðm þ m Þ coshðmm Þ  2mm sinhðmm Þ m m C 1 B B C m¼0 m0 ¼0 ¼ B1  C . N N @ A 2pw0 P P 2 2 0 0 cm cm0 exp½ðm þ m Þ=2 coshðmm Þ

s1x

N N P P

(26)

m¼0 m0 ¼0

Apparently, the beam propagation factor of these two beams is different. To show the quantitative difference, we consider the case of N ¼ 1. The beam propagation factor of the flattened light beam with an axial shadow turns out to be 2

M0 ¼

ð3  6e2 Þ1=2 ð11 þ 8e1=2 þ 2e2 Þ1=2 ¼ 1:04 3 þ 4e1=2 þ 2e2

(27)

and the beam propagation factor of a second-order cosh-Gaussian beam yields M2 ¼



1þ

12e1=2 þ 4e2 19 þ 12e1=2 þ e2

1=2  1

12e1=2 þ 4e2 19 þ 12e1=2 þ e2

1=2

¼ 1:21.

(28)

The beam waist of the flattened light beam with an axial shadow is far larger than that of the second-order cosh-Gaussian beam, which is caused by the existence of an axial shadow. However, the divergence of the second-order cosh-Gaussian beam is far larger than that of the flattened light beam with an axial shadow. The difference of the beam propagation factor results from their respective weight coefficients. For the sake of intuition, numerical calculations of the M2 factor obtained here are performed. First, we analyze the influence of the beam order n on the M2 factor. The M2 factor is plotted as a function of n in Fig. 1. d ¼ 0.1 in Fig. 1(a), and d ¼ 1 in Fig. 1(b). d ¼ 3 in Fig. 1(c), and d ¼ 10 in Fig. 1(d). The circle denotes that n is even, and the square means that n is odd. When d is very small, the two curves of the M2 factor versus n partially overlap. The two curves overlap at the position of small n. As n increases and gets large enough, the two curves separate from each other. Moreover, the slope of the curve of the M2 factor versus the odd n is smaller than that versus the even n. When d is an intermediate value, the two curves only overlap at the beginning of the abscissa. When d is large enough, the two curves completely overlap. The M2 factor augments as n increases. Except the case of d being large enough, the parity of the beam order n affects the variational rule of the M2 factor. Then, the influence of the parameterd on the M2 factor is also investigated, which is shown in Fig. 2. n ¼ 2 in Fig. 2(a), and n ¼ 9 in Fig. 2(b). n ¼ 50 in Fig. 2(c), and n ¼ 99 in Fig. 2(d). With a given n, the M2 factor augments with d increasing.

3. The kurtosis parameter The higher-order cosh-Gaussian beam propagating towards the half free space zX0 obeys the well-known Collins integral formula   Z 1Z 1 i ik En ðx0 ; y0 ; 0Þ exp  ðx20 þ y20  2x0 x  2y0 y þ x2 þ y2 Þ dx0 dy0 , (29) En ðx; y; zÞ ¼ lz 1 1 2z where k ¼ 2p/l is the wave number with l the wavelength. Substituting Eq. (4) into Eq. (29), one can obtain the propagating higher-order cosh-Gaussian beam En ðx; y; zÞ ¼ En ðx; zÞEn ðy; zÞ with En(x,z) and En(y,z) given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! " "

2 # pffiffiffi# 2 n X m  n=2 d 1 ipN w ipNw j n! ipNw ðn  2mÞj d exp exp  En ðj; zÞ ¼ n exp , 1 þ ipNw ð1 þ ipNw Þw0 1 þ ipN w 2 ð1 þ ipNw Þw20 m¼0 m!ðn  mÞ!

(30)

(31)

where Nw ¼ w20/lz is the Fresnel number. As the kurtosis parameter is employed to describe the flatness degree of the beams, it is an important parameter to valuate the beam propagation. The kurtosis parameter in one transversal dimension, e.g. the x-direction, is defined as [27,28] K¼

hx4 i , hx2 i2

(32)

ARTICLE IN PRESS 206

G. Zhou, J. Zheng / Optics & Laser Technology 41 (2009) 202–208

Fig. 1. The M2 factor as a function of n: (a) d ¼ 0.1, (b) d ¼ 1, (c) d ¼ 3, and (d) d ¼ 10. The circle denotes that n is even, and the square means that n is odd.

Fig. 2. The M2 factor as a function of d: (a) n ¼ 2, (b) n ¼ 9, (c) n ¼ 50, and (d) n ¼ 99.

where /x4S and /x2S are given by R 1 n x jEðx; y; zÞj2 dx R 1 ; n ¼ 2 or 4. hxn i ¼ 1 2 1 jEðx; y; zÞj dx

(33)

ARTICLE IN PRESS G. Zhou, J. Zheng / Optics & Laser Technology 41 (2009) 202–208

207

Instead of Eqs. (3) and (4) is used in the derivation of the propagating higher-order cosh-Gaussian beam, which simplifies the expression of the kurtosis parameter. After lengthy calculation, the kurtosis parameter of a higher-order cosh-Gaussian beam finally turns out to be K¼

b0 b1

(34)

2

b2

with b0, b1 and b2 given by b0 ¼

b1 ¼

" # n X n 1X 1 ð2m  nÞ2 þ ð2m0  nÞ2  2ðm0  mÞ2 þ 2p2 N2w ðn  m  m0 Þ2 exp d , 2 m¼0 m0 ¼0 m!m0 !ðn  mÞ!ðn  m0 Þ! 4ð1 þ p2 N2w Þ n X n  X m¼0 m0 ¼0



b2 ¼

" # 1 ð2m  nÞ2 þ ð2m0  nÞ2  2ðm0  mÞ2 þ 2p2 N2w ðn  m  m0 Þ2 exp d m!m0 !ðn  mÞ!ðn  m0 Þ! 4ð1 þ p2 N2w Þ

3ð1 þ p2 N2w Þ2 32p4 N 4w

þ

3½n  m  m0 þ iðm0  mÞ=pN w 2 ð1 þ p2 N 2w dÞ 16p2 N 2w

þ

½n  m  m0 þ iðm0  mÞ=pN w 4 d2 32

(35)

!)

" # 1 ð2m  nÞ2 þ ð2m0  nÞ2  2ðm0  mÞ2 þ 2p2 N2w ðn  m  m0 Þ2 exp d m!m0 !ðn  mÞ!ðn  m0 Þ! 4ð1 þ p2 N2w Þ m¼0 m0 ¼0 !) ðn  m  m0 Þ2 d ðm0  mÞ2 d 1 þ p2 N2w ðn  m  m0 Þðm0  mÞd   þ þ i . 8 2pN w 8p2 N2w 8p2 N2w

,

(36)

n X n  X

(37)

Due to the symmetry, the kurtosis parameter in the y-direction is same as that in the x-direction. If n ¼ 0, the kurtosis parameter is 3, which is the well-known result of a Gaussian beam. As z is large enough in the far field, the Fresnel number Nw is approximately equal to be zero. The kurtosis parameter of a higher-order cosh-Gaussian beam in the far field reduces to be

K¼

n n expf½ð2m  nÞ2 þ ð2m0  nÞ2  2ðm0  mÞ2 d=4g P P m!m0 !ðn  mÞ!ðn  m0 Þ! 0 m¼0 m ¼0

!2 n n expf½ð2m  nÞ2 þ ð2m0  nÞ2  2ðm0  mÞ2 d=4g P P 0  mÞ2 d ½1  ðm m!m0 !ðn  mÞ!ðn  m0 Þ! m¼0 m0 ¼0 

n X n X expf½ð2m  nÞ2 þ ð2m0  nÞ2  2ðm0  mÞ2 d=4g ½3  6ðm0  mÞ2 d þ ðm0  mÞ4 d2 : m!m0 !ðn  mÞ!ðn  m0 Þ! m¼0 m0 ¼0

(38)

In the source plane, the kurtosis parameter of a higher-order cosh-Gaussian beam is simplified to be n n exp½dðn  m  m0 Þ2 =2 P n n exp½dðn  m  m0 Þ2 =2 P P P ½3 þ 6ðn  m  m0 Þ2 d þ ðn  m  m0 Þ4 d2  0 0 m!m !ðn  mÞ!ðn  m Þ! m¼0 m0 ¼0 m!m0 !ðn  mÞ!ðn  m0 Þ! 0 . K ¼ m¼0 m ¼0 !2 n n exp½dðn  m  m0 Þ2 =2 P P 2 ½ðn  m  m0 Þ d þ 1 0 0 m¼0 m0 ¼0 m!m !ðn  mÞ!ðn  m Þ!

(39)

The kurtosis parameter of the flattened light beams with an axial shadow has been presented in the source plane [16]. In order to compare our result with that of Ref. [16], the condition of L ¼ 0, O ¼ 1/w0, and n ¼ 2N ¼ 2 is still considered. In this case, the kurtosis parameter of a second-order cosh-Gaussian beam in the source plane is K ¼ 2.36. While, the kurtosis parameter of the flattened light

Fig. 3. The kurtosis parameter in the reference plane as a function of d. The solid curve denotes that n ¼ 1, and the dotted means that n ¼ 2: (a) in the source plane, (b) in the far field plane.

ARTICLE IN PRESS 208

G. Zhou, J. Zheng / Optics & Laser Technology 41 (2009) 202–208

beams with an axial shadow in the source plane is K ¼ 1.17. However, Ref. [16] points out that the kurtosis parameter may be insufficient to describe the flatness of laser beams with axial hole. The kurtosis parameters of some certain higher-order cosh-Gaussian beams are depicted as a function of the parameter d, which is shown in Fig. 3. The reference plane is just the source plane in Fig. 3(a) and the far field in Fig. 3(b). The solid curve denotes that n ¼ 1, and the dotted means that n ¼ 2. The kurtosis parameter in the source plane decreases as d increases. With a given value of d, the kurtosis parameter decreases as the beam order increases. When d is larger than a certain value, the kurtosis parameter in the far field maintains the value of 3. As the beam order augments, the peak value of the kurtosis parameter remarkably minishes. When 0odo1.5, the kurtosis parameter with n ¼ 1 is smaller than 3. However, this case will not happen to the kurtosis parameter with n ¼ 2. Under the condition of appropriate parameters, therefore, the light intensity distribution of a higher-order cosh-Gaussian beam is flattened.

4. Conclusions On the basis of the second-order moment, the beam propagation of a higher-order cosh-Gaussian beam has been investigated. Two analytical expressions of the M2 factor have been derived, which corresponds to the parity of the beam order n. The M2 factor is determined by the beam order n and the parameter d. The influences of the beam order n and the parameter d on the M2 factor are illustrated with numerical examples. An analytical formula of kurtosis parameter of a higher-order cosh-Gaussian beam has been also presented. The analytical formula is further simplified in the source plane and the far field, respectively. The kurtosis parameters of some certain higher-order cosh-Gaussian beams are depicted as a function of the parameter d. With given appropriate values of n and d, the kurtosis parameter in the far field is smaller than 3. Under the condition of appropriate parameters, therefore, the distribution of light intensity of a higher-order cosh-Gaussian beam is flattened. This research is helpful to the practical applications of a higher-order coshGaussian beam.

Acknowledgments This research was supported by Scientific Research Fund of Zhejiang Provincial Education Department under Grant 20060677. The authors are indebted to the reviewers for valuable comments. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]

Saghafi S, Sheppard CJR. Opt Commun 1998;153:207. Deng DM. Phys Lett A 2004;333:485. Cai YJ, He SL. Appl Phys Lett 2006;89:041117. Cai YJ, Hu L. Opt Lett 2006;31:685. Cai YJ, Lin Q, Baykal Y, Eyyuboglu HT. Opt Commun 2007;278:157. Cai YJ, He SL. Opt Lett 2006;31:568. Zheng CW. J Opt Soc Am A 2006;23:2161. Caperson LW, Tover AA. J Opt Soc Am A 1998;15:954. Tover AA, Caperson LW. J Opt Soc Am A 1998;15:2425. Lu¨ BD, Zhang B, Ma H. Opt Lett 1999;24:640. Lu¨ BD, Ma H, Zhang B. Opt Commun 1999;64:165. Lu¨ BD, Luo SR. Opt Commun 2000;178:275. Yu S, Guo H, Fu XQ, Hu W. Opt Commun 2002;204:59. Wang XQ, Lu¨ BD. J Mod Opt 2001;48:2097. Zhu KC, Tang HQ, Sun XM, Wang XW, Liu TN. Opt Commun 2002;207:29. Zhu KC, Tang HQ, Wang XW, Liu TN. Optik 2002;113:222. Tang QJ, Yu YG, Hu QQ. Chin Opt Lett 2007;5:S46. Mei ZR, Zhao DM, Sun D, Gu JG. Optik 2001;115:89. Lu¨ BD, Ma H. Appl Opt 2000;39:1279. Zhu KC, Tang HQ, Sun XM. Laser Tech 2002;26:364 [in Chinese]. Gori F. Opt Commun 1994;107:335. Sheppard CJR, Saghafi S. Opt Commun 1996;132:144. Siegman A. Proc SPIE 1990;1224:2. Gradshteyn IS, Ryzhik IM. Tables of integrals, series, and product. San Diego, CA: Academic Press; 1980. Goodman JW. Introduction to Fourier optics. 2nd ed. New York: McGraw-Hill; 1996. Nemes G, Serna J. OSA TOPS 1998;17:200. Martinez-Herrero R, Piquero G, Mejias PM. Opt Commun 1996;115:225. Amarande SA. Opt Commun 1996;129:311.