- Email: [email protected]
Paraxial propagation of partially coherent flat-topped beam
Optics Communications 260 (2006) 687–690 www.elsevier.com/locate/optcom
Paraxial propagation of partially coherent flat-topped beam Guohua Wu
a,b,1
, Hong Guo
a,*
, Dongmei Deng
b
a
b
CREAM Group, Key Laboratory for Quantum Information and Measurements of Ministry of Education, School of Electronics Engineering and Computer Science, Peking University, Beijing 100871, PR China CREAM Group, Laboratory of Light Transmission Optics, South China Normal University, Guangzhou 510631, PR China Received 24 August 2005; received in revised form 16 November 2005; accepted 17 November 2005
Abstract The propagation of flat-topped beam passing through a paraxial optical ABCD system is extended to partially coherent case and has been studied in detail. Based on Wigner distribution function (WDF), the irradiance distribution of partially coherent flat-topped beam is derived. Furthermore, two propagation parameters, such as M2-factor and Power in Bucket (PIB) are illustrated analytically and numerically. Ó 2005 Elsevier B.V. All rights reserved. Keywords: Partially coherent flat-topped beam; Wigner distribution function; Beam quality
1. Introduction In the past several decades, light beams with flat-topped profiles [1–5] have been investigated with growing interest for their special application in high-power laser technology. Several models have been proposed to describe the flattopped beam, one of which is Super-Gaussian model [1]. However, some mathematical difficulties are met with this model; requiring numerical techniques even when evaluating the propagation in free space. Another model [2] avoided this difficulty, for which, instead, the flat-topped beam based on this model is used and it can be easily rewritten as a superposition of Laguerre–Gauss beams [2] in a cylindrical coordinate system or Hermite–Gauss beams [3] in a rectangular coordinate system whose propagation characteristics are well studied. So far, extensive study based on GoriÕs model is made by several authors [3–5]: the propagation factor and the kurtosis parameter are studied in detail [3], the general propagation of flattened Gaussian beam in *
Corresponding author. Tel.: +86 10 6275 7035; fax: +86 10 6275 3208. E-mail addresses: [email protected] (G. Wu), hongguo@pku. edu.cn (H. Guo). 1 Tel.: +86 20 8521 1603; fax: +86 20 8521 1920. 0030-4018/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2005.11.031
amplifying and absorbing media is studied [4]. More recently, another model of flat-topped beam expressed in terms of a finite series of lowest-order Gaussian modes with different parameters is also proposed [5]. The model based on fundamental Gaussian modes with different parameters is more satisfactory for propagation and focusing analysis and no orthogonal polynomials are needed in the study of beam propagation [6]. Furthermore, LiÕs model is able to describe beams with other profiles, e.g., the triangle and spike shape profiles [7]. Based on this model, we give the expression of the cross spectral density function for a partially coherent flat-topped beam. By means of WDF, the irradiance distribution is then given, in Section 2. Furthermore, two propagation parameters, i.e., M2-factor and PIB, are studied. 2. Propagation through a first-order optical ABCD system Let us consider a one-dimensional light beam with flattopped profile whose field distribution at the z = 0 plane is characterized by [5] n1 N X N ð1Þ nx2 EN ðx; 0Þ ¼ ð1Þ exp 2 ; N 4x0 n n¼1
688
G. Wu et al. / Optics Communications 260 (2006) 687–690
N represents a binomial n coefficient and x0 is the beam waist size of Gaussian beam. Assume that the complex degree of the spatial coherence of the partially coherent flat-topped beam depends on position coordinates of two points x1, x2 at z = 0 plane only through the difference x1 x2 and can be expressed as a finite sum of Gaussian functions with different parameters, as follows [8]: PN nx2 n¼1 exp 2r2 0 ; ð2Þ gðxÞ ¼ N where r0 is the correlation length of the Gaussian Schell model (GSM) beam. Recalling that Schell-model source is characterized by a cross-spectral density of the form [9] where N is the beam order,
Cðx1 ; x2 Þ ¼ EN ðx1 ; 0ÞEN ðx2 ; 0Þgðx1 x2 Þ.
ð3Þ
Therefore, partially coherent flat-topped beam has crossspectral density function at the z = 0 plane of the form given by nþm2 N X N X N N ð1Þ Cðx1 ; x2 ; 0Þ ¼ 4 n m N n¼1 m¼1 ( " #) 2 nx21 þ mx22 ðn þ mÞðx1 x2 Þ exp þ ; 4x20 4r20 ð4Þ where x0 is the waist width of the GSM beam. When N = 1, Eq. (4) reduces to the cross-spectral density function of the GSM beam [10]. So the WDF of the partially coherent flat-topped beam at the z = 0 plane can be given by [11] Z þ1 s s W 0 ðx; u; 0Þ ¼ C x þ ; x ; 0 expðiksuÞ ds 2 2 1 pffiffiffi N X N X N N 4 px0 ð1Þnþm2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 4 2 m ðn þ mÞð1 þ 4g Þ n n¼1 m¼1 N 2 2 2 2 Ex þ 4k x0 u þ 2ikðn mÞxu exp ; ðn þ mÞð1 þ 4g2 Þx20 ð5Þ where 2
E ¼ nm þ ðn þ mÞ g2 ;
ð6Þ
and k is the wave number, g ¼ xr00 is the inverse of global coherence parameter. The two limiting cases of the fully coherent beam and the incoherent beam correspond to g ! 0 and g ! +1, respectively. As is well known, the propagation of WDF through a paraxial optical system A B parameterized by transfer matrix obeys [11–13] C D W ðx; u; zÞ ¼ W 0 ðDx Bu; Au CxÞ; ð7Þ where A, B, C, D are the elements of the wave matrix that characterizes the system with the condition that AD BC = 1. The substitution from Eq. (5) into Eq. (7), we obtain the WDF of the partially coherent flat-topped beam at the z plane
W ðx; u; zÞ ¼
N X N X n¼1 m¼1
4px0 ð1Þnþm2 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N 4 ðn þ mÞð1 þ 4g2 Þ
N n
!
N
!
m
D2 E 2iCDkðn mÞx20 þ 4C 2 k 2 x40 2 x exp ðn þ mÞð1 þ 4g2 Þx20 B2 E 2iABkðn mÞx20 þ 4A2 k 2 x40 2 exp u ðn þ mÞð1 þ 4g2 Þx20 2BDE 2ikx20 ðBC þ ADÞðn mÞ þ 8ACk 2 x40 exp xu . ðn þ mÞð1 þ 4g2 Þx20 ð8Þ
To illustrate the application of Eq. (8), we now consider in detail the important special case of beam propagation in free space. The corresponding transfer matrix reads A B 1 z ¼ . ð9Þ C D 0 1 With the help of Eqs. (8) and (9), the irradiance distribution of partially coherent flat-topped beam propagation in the free space is expressed as Z þ1 I f ðx; zÞ ¼ W ðx; u; zÞdu 1
¼
N X N X n¼1
4px20 ð1Þnþm2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi m¼1 N 4 Ez2 2ikzðn mÞx20 þ 4k 2 x40 ! ! ( ) N N k 2 x20 ðn þ mÞx2 exp 2 . Ez 2ikx20 ðn mÞz þ 4k 2 x40 n m ð10Þ
To characterize the behavior of irradiance distribution of partially coherent flat-topped beam propagation in free space, numerical illustrative examples are given. Fig. 1 displays the relative irradiance distribution of partially coherent flat-topped beam propagation in free space at different propagation distances: z = 0, 500, 5000 mm. The used parameters are x0 = 1 mm, g = 10, k = 850 lm, N = 15. As shown in Fig. 1, the partially coherent flat-topped beam can not keep invariant its shape as it propagates in free space. As can be known from Eq. (1), the partially coherent flat-topped beam can be considered as a sum of GSM beams with different parameters, i.e., a partially coherent flattopped beam can be seen as a multi-mode beam with different parameters. Different modes have different evolving velocities which lead to beam shape changing with propagation. Furthermore, in the far field (z = 5000 mm), the irradiance distribution behaves like the gaussian beam except with some lobes in two sides. This is because a partially coherent flat-topped beam with a bigger beam order N evolves slowly with propagation distance [14]. The lobes result from the interference superposition between different modes.
G. Wu et al. / Optics Communications 260 (2006) 687–690
689
8 > nþm2 : n¼1 m¼1 ðn þ mÞ3
1 Z=0 Z=500 z=5000
I / Imax
nþm2 N X N X ð1Þ E qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 n¼1 m¼1 ðn þ mÞ
N
!
n
m !
N n
N
"
nþm2 N X N X ð1Þ pffiffiffiffiffiffiffiffiffiffiffiffi nþm n¼1 m¼1
0 -20
-10
0
10
20
x Fig. 1. Relative irradiance distribution of partially coherent flat-topped beam propagation in free space at different propagation distances.
3. M2-factor and PIB In this section, we will study two propagation parameters of partially coherent flat-topped beam, i.e., M2-factor and PIB. 3.1. M2-factor The M2-factor in the rectangular coordinate system is defined as [15] 2 1=2
M 2 ¼ 2k½hx2 ihu2 i hxui
ð11Þ
;
m
n
hxm un io ¼ hðAx þ BuÞ ðCx þ DuÞ ii
3 91=2 ! 2> = 7 5 > ; m
!
N
n N
!#1 .
n
ð14Þ
m
20
N=1 N=6 N=15
M2
1
With the help of Eq. (8), we can present the M2-factor of partially coherent flat-topped beam. As known, the M2factor defined in the sense of second-order moments through paraxial first-order ABCD optical systems remains invariant upon propagation [16], which also can be derived from the well known propagation law of the irradiance moment of the order m + n [11]
!
N
Eq. (14) indicates that M2-factor of the partially coherent flat-topped beam depends on two parameters, i.e., the beam order N and the inverse of global coherence parameter g. When N = 1 Eq. (14) can be reduced to Eq. (36) in [10], which is the M2-factor of GSM beam. Fig. 2 shows M2-factor increases with the inverse of global coherence parameter g for a fixed beam order N. That is to say, the beam quality become worse and worse with the coherence of the beam decreasing. Also, the fully coherent beam has better beam quality than the partially coherent beam. Furthermore, if g is big enough, M2-factor increases greatly, i.e., the beam quality diminishes greatly. M2-factor increases with the beam order N for a very small fixed g and decreases with increasing beam order N for a very big fixed g. This indicates that a partially coherent flat-topped beam with a small beam order N evolves more rapidly with the degree of coherence.
where the irradiance moment of the order m + n is given by [11] Z þ1 Z þ1 m n hx u i ¼ xm un W ðx; u; zÞ dx du 1 1 Z þ1 Z þ1 W ðx; u; zÞ dx du. ð12Þ 1
N
!
m
2 nþm2 N X N 1 6X ð1Þ ðn mÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 4 n¼1 m¼1 ðn þ mÞ3
0.5
!
N
10
ð13Þ
where the subscript ÔoÕ and ÔiÕ refer to the output and input planes of the optical system, respectively. So it is convenient to use Eq. (5), i.e., the WDF at the plane of z = 0, to calculate the M2 factor of partially coherent flat-topped beam. On substituting from Eqs. (5) and (12) into Eq. (11), the final result can be arranged as
0
0
5
η
10
Fig. 2. The variation of M2-factor of partially coherent flat-topped beam versus g with the beam order N = 1, 6 and 15.
690
G. Wu et al. / Optics Communications 260 (2006) 687–690
3.2. Power in bucket (PIB) Power in Bucket (PIB), another useful parameter for characterizing the beam quality, is defined as the fractional power within a given bucketÕs size [17], i.e., Ra Iðx; 0Þ dx a PIB ¼ R þ1 ; ð15Þ Iðx; 0Þ dx 1 where a is the radius of the given bucket, I(x, 0) is the irradiance of the partially coherent flat-topped beams at the z = 0 plane and can be evaluated by Eq. (4). By letting x01 ¼ x02 ¼ x, we obtained the irradiance of the partially coherent flat-topped beams at the z = 0 plane from Eq. (4). The substitution from Eq. (4) (letting x01 ¼ x02 ) into Eq. (15) and recalling the two integral equations [18] pffiffiffi Z a p ErfðaÞ; ð16Þ expðx2 Þ dx ¼ 2 0 pffiffiffi Z 1 p ½q > 0; ð17Þ expðq2 x2 Þ dx ¼ 2q 0 after tedious integral calculations, the final result can be written as h pffiffiffiffiffiffiffii PN PN ð1Þnþm2 N N pffiffiffiffiffiffiffi Erf a 2xnþm n¼1 m¼1 nþm 0 n m PIB ¼ ; ð18Þ PN PN ð1Þnþm2 N N pffiffiffiffiffiffiffi n¼1 m¼1 nþm n m where Erf [x] is the error function. Fig. 3 gives several typical curves of partially coherent flat-topped beam with different beam order N = 1, 6, 15. As shown in Fig. 3, for the same PIB, the bigger the beam order N the bigger the radius of the bucket a/x0. For a fixed radius of the bucket not too big, the bigger the beam order N the smaller the PIB. This
is because the flatness degree of a flat-topped beam increases with the beam order N increasing. So the fractional power increases with the decreasing beam order N when the radius of the given bucket is not too big. 4. Conclusions In conclusion, an extension of FGB characterizing for both from fully coherent to partially coherent case is presented. The GSM beam can be considered as special case of it. The irradiance distribution of partially coherent flattopped beam in free space is illustrated, which cannot keep invariant its shape during propagation. Also, the M2-factor of partially coherent flat-topped beam is studied. The results indicate that the M2-factor depends on the beam order N and the inverse of the global coherence parameter g. Fully coherent beam has better beam quality than partially coherent beam. The PIB of partially coherent flat-topped beam at the z = 0 plane is also studied. Although the calculation is performed in a rectangular coordinate system, the extension to a cylindrical coordinate system is straightforward. Acknowledgements This work is partially supported by the National Natural Science Foundation of China (Grant No. 10474004), National Fundamental Research Programme of China (Grant No. 2001CB309308), the Major Program of National Natural Science Foundation of China (Grant No. 60490280), DAAD exchange program: ‘‘D/0212785 personenaustausch VR China’’ and the National High Technology Research and Development Program of China (863 Program) international cooperation program (Grant No. 2004AA1Z1220). References
1
PIB
N=1 N=6 N=15 0.5
0
0
2
4
6
8
10
a/ω0 Fig. 3. The PIB curve versus the normalized bucket sized a/x0 with the beam order N = 1, 6 and 15 at the z = 0 plane.
[1] A. Parent, M. Marin, P. Lavigne, Opt. Quantum Electron. 24 (1992) 1071. [2] F. Gori, Opt. Commun. 107 (1994) 335. [3] S.A. Amarande, Opt. Commun. 129 (1996) 311. [4] B.D. Lu¨, J. Opt. Soc. Am. A 17 (2000) 2001. [5] Y.J. Li, Opt. Lett. 27 (2002) 1007. [6] Y.J. Li, Opt. Commun. 206 (2002) 225. [7] Y.J. Li, J. Mod. Opt. 50 (2003) 1957. [8] G.H. Wu, H. Guo, D.M. Deng, Appl. Opt. (accepted). [9] L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, Cambridge U. Press, Cambridge, UK, 1995. [10] M. Santarsiero, F. Gori, R. Borghi, G. Cincotti, P. Vahimaa, J. Opt. Soc. Am. A 16 (1996) 106. [11] R. Martı´nez-Herrero, G. Piquero, P.M. Mejı´as, Opt. Commun. 115 (1995) 225. [12] M.J. Bastiaans, J. Opt. Soc. Am. A 3 (1986) 1227. [13] S. Lavi, R. Prochaska, E. Keren, Appl. Opt. 27 (1988) 3696. [14] Di Ge, Y.J. Cai, Q. Lin, Appl. Opt. 43 (2004) 4732. [15] J. Serna, R. Martı´nez-Herrero, P.M. Mejı´as, J. Opt. Soc. Am. A 8 (1991) 1094. [16] B. Zhang, X.L. Chu, Q. Li, J. Opt. Soc. Am. A 19 (2002) 1370. [17] A.E. Siegman (Ed.), Trends in Optics and Photonics Series, 17, Optical Society of America, Washington, DC, 1998, 184. [18] I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products, sixth ed. Academic Press, New York, p. 333.
Paraxial propagation of partially coherent flat-topped beam Guohua Wu
a,b,1
, Hong Guo
a,*
, Dongmei Deng
b
a
b
CREAM Group, Key Laboratory for Quantum Information and Measurements of Ministry of Education, School of Electronics Engineering and Computer Science, Peking University, Beijing 100871, PR China CREAM Group, Laboratory of Light Transmission Optics, South China Normal University, Guangzhou 510631, PR China Received 24 August 2005; received in revised form 16 November 2005; accepted 17 November 2005
Abstract The propagation of flat-topped beam passing through a paraxial optical ABCD system is extended to partially coherent case and has been studied in detail. Based on Wigner distribution function (WDF), the irradiance distribution of partially coherent flat-topped beam is derived. Furthermore, two propagation parameters, such as M2-factor and Power in Bucket (PIB) are illustrated analytically and numerically. Ó 2005 Elsevier B.V. All rights reserved. Keywords: Partially coherent flat-topped beam; Wigner distribution function; Beam quality
1. Introduction In the past several decades, light beams with flat-topped profiles [1–5] have been investigated with growing interest for their special application in high-power laser technology. Several models have been proposed to describe the flattopped beam, one of which is Super-Gaussian model [1]. However, some mathematical difficulties are met with this model; requiring numerical techniques even when evaluating the propagation in free space. Another model [2] avoided this difficulty, for which, instead, the flat-topped beam based on this model is used and it can be easily rewritten as a superposition of Laguerre–Gauss beams [2] in a cylindrical coordinate system or Hermite–Gauss beams [3] in a rectangular coordinate system whose propagation characteristics are well studied. So far, extensive study based on GoriÕs model is made by several authors [3–5]: the propagation factor and the kurtosis parameter are studied in detail [3], the general propagation of flattened Gaussian beam in *
Corresponding author. Tel.: +86 10 6275 7035; fax: +86 10 6275 3208. E-mail addresses: [email protected] (G. Wu), hongguo@pku. edu.cn (H. Guo). 1 Tel.: +86 20 8521 1603; fax: +86 20 8521 1920. 0030-4018/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2005.11.031
amplifying and absorbing media is studied [4]. More recently, another model of flat-topped beam expressed in terms of a finite series of lowest-order Gaussian modes with different parameters is also proposed [5]. The model based on fundamental Gaussian modes with different parameters is more satisfactory for propagation and focusing analysis and no orthogonal polynomials are needed in the study of beam propagation [6]. Furthermore, LiÕs model is able to describe beams with other profiles, e.g., the triangle and spike shape profiles [7]. Based on this model, we give the expression of the cross spectral density function for a partially coherent flat-topped beam. By means of WDF, the irradiance distribution is then given, in Section 2. Furthermore, two propagation parameters, i.e., M2-factor and PIB, are studied. 2. Propagation through a first-order optical ABCD system Let us consider a one-dimensional light beam with flattopped profile whose field distribution at the z = 0 plane is characterized by [5] n1 N X N ð1Þ nx2 EN ðx; 0Þ ¼ ð1Þ exp 2 ; N 4x0 n n¼1
688
G. Wu et al. / Optics Communications 260 (2006) 687–690
N represents a binomial n coefficient and x0 is the beam waist size of Gaussian beam. Assume that the complex degree of the spatial coherence of the partially coherent flat-topped beam depends on position coordinates of two points x1, x2 at z = 0 plane only through the difference x1 x2 and can be expressed as a finite sum of Gaussian functions with different parameters, as follows [8]: PN nx2 n¼1 exp 2r2 0 ; ð2Þ gðxÞ ¼ N where r0 is the correlation length of the Gaussian Schell model (GSM) beam. Recalling that Schell-model source is characterized by a cross-spectral density of the form [9] where N is the beam order,
Cðx1 ; x2 Þ ¼ EN ðx1 ; 0ÞEN ðx2 ; 0Þgðx1 x2 Þ.
ð3Þ
Therefore, partially coherent flat-topped beam has crossspectral density function at the z = 0 plane of the form given by nþm2 N X N X N N ð1Þ Cðx1 ; x2 ; 0Þ ¼ 4 n m N n¼1 m¼1 ( " #) 2 nx21 þ mx22 ðn þ mÞðx1 x2 Þ exp þ ; 4x20 4r20 ð4Þ where x0 is the waist width of the GSM beam. When N = 1, Eq. (4) reduces to the cross-spectral density function of the GSM beam [10]. So the WDF of the partially coherent flat-topped beam at the z = 0 plane can be given by [11] Z þ1 s s W 0 ðx; u; 0Þ ¼ C x þ ; x ; 0 expðiksuÞ ds 2 2 1 pffiffiffi N X N X N N 4 px0 ð1Þnþm2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 4 2 m ðn þ mÞð1 þ 4g Þ n n¼1 m¼1 N 2 2 2 2 Ex þ 4k x0 u þ 2ikðn mÞxu exp ; ðn þ mÞð1 þ 4g2 Þx20 ð5Þ where 2
E ¼ nm þ ðn þ mÞ g2 ;
ð6Þ
and k is the wave number, g ¼ xr00 is the inverse of global coherence parameter. The two limiting cases of the fully coherent beam and the incoherent beam correspond to g ! 0 and g ! +1, respectively. As is well known, the propagation of WDF through a paraxial optical system A B parameterized by transfer matrix obeys [11–13] C D W ðx; u; zÞ ¼ W 0 ðDx Bu; Au CxÞ; ð7Þ where A, B, C, D are the elements of the wave matrix that characterizes the system with the condition that AD BC = 1. The substitution from Eq. (5) into Eq. (7), we obtain the WDF of the partially coherent flat-topped beam at the z plane
W ðx; u; zÞ ¼
N X N X n¼1 m¼1
4px0 ð1Þnþm2 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N 4 ðn þ mÞð1 þ 4g2 Þ
N n
!
N
!
m
D2 E 2iCDkðn mÞx20 þ 4C 2 k 2 x40 2 x exp ðn þ mÞð1 þ 4g2 Þx20 B2 E 2iABkðn mÞx20 þ 4A2 k 2 x40 2 exp u ðn þ mÞð1 þ 4g2 Þx20 2BDE 2ikx20 ðBC þ ADÞðn mÞ þ 8ACk 2 x40 exp xu . ðn þ mÞð1 þ 4g2 Þx20 ð8Þ
To illustrate the application of Eq. (8), we now consider in detail the important special case of beam propagation in free space. The corresponding transfer matrix reads A B 1 z ¼ . ð9Þ C D 0 1 With the help of Eqs. (8) and (9), the irradiance distribution of partially coherent flat-topped beam propagation in the free space is expressed as Z þ1 I f ðx; zÞ ¼ W ðx; u; zÞdu 1
¼
N X N X n¼1
4px20 ð1Þnþm2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi m¼1 N 4 Ez2 2ikzðn mÞx20 þ 4k 2 x40 ! ! ( ) N N k 2 x20 ðn þ mÞx2 exp 2 . Ez 2ikx20 ðn mÞz þ 4k 2 x40 n m ð10Þ
To characterize the behavior of irradiance distribution of partially coherent flat-topped beam propagation in free space, numerical illustrative examples are given. Fig. 1 displays the relative irradiance distribution of partially coherent flat-topped beam propagation in free space at different propagation distances: z = 0, 500, 5000 mm. The used parameters are x0 = 1 mm, g = 10, k = 850 lm, N = 15. As shown in Fig. 1, the partially coherent flat-topped beam can not keep invariant its shape as it propagates in free space. As can be known from Eq. (1), the partially coherent flat-topped beam can be considered as a sum of GSM beams with different parameters, i.e., a partially coherent flattopped beam can be seen as a multi-mode beam with different parameters. Different modes have different evolving velocities which lead to beam shape changing with propagation. Furthermore, in the far field (z = 5000 mm), the irradiance distribution behaves like the gaussian beam except with some lobes in two sides. This is because a partially coherent flat-topped beam with a bigger beam order N evolves slowly with propagation distance [14]. The lobes result from the interference superposition between different modes.
G. Wu et al. / Optics Communications 260 (2006) 687–690
689
8 > nþm2
1 Z=0 Z=500 z=5000
I / Imax
nþm2 N X N X ð1Þ E qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 n¼1 m¼1 ðn þ mÞ
N
!
n
m !
N n
N
"
nþm2 N X N X ð1Þ pffiffiffiffiffiffiffiffiffiffiffiffi nþm n¼1 m¼1
0 -20
-10
0
10
20
x Fig. 1. Relative irradiance distribution of partially coherent flat-topped beam propagation in free space at different propagation distances.
3. M2-factor and PIB In this section, we will study two propagation parameters of partially coherent flat-topped beam, i.e., M2-factor and PIB. 3.1. M2-factor The M2-factor in the rectangular coordinate system is defined as [15] 2 1=2
M 2 ¼ 2k½hx2 ihu2 i hxui
ð11Þ
;
m
n
hxm un io ¼ hðAx þ BuÞ ðCx þ DuÞ ii
3 91=2 ! 2> = 7 5 > ; m
!
N
n N
!#1 .
n
ð14Þ
m
20
N=1 N=6 N=15
M2
1
With the help of Eq. (8), we can present the M2-factor of partially coherent flat-topped beam. As known, the M2factor defined in the sense of second-order moments through paraxial first-order ABCD optical systems remains invariant upon propagation [16], which also can be derived from the well known propagation law of the irradiance moment of the order m + n [11]
!
N
Eq. (14) indicates that M2-factor of the partially coherent flat-topped beam depends on two parameters, i.e., the beam order N and the inverse of global coherence parameter g. When N = 1 Eq. (14) can be reduced to Eq. (36) in [10], which is the M2-factor of GSM beam. Fig. 2 shows M2-factor increases with the inverse of global coherence parameter g for a fixed beam order N. That is to say, the beam quality become worse and worse with the coherence of the beam decreasing. Also, the fully coherent beam has better beam quality than the partially coherent beam. Furthermore, if g is big enough, M2-factor increases greatly, i.e., the beam quality diminishes greatly. M2-factor increases with the beam order N for a very small fixed g and decreases with increasing beam order N for a very big fixed g. This indicates that a partially coherent flat-topped beam with a small beam order N evolves more rapidly with the degree of coherence.
where the irradiance moment of the order m + n is given by [11] Z þ1 Z þ1 m n hx u i ¼ xm un W ðx; u; zÞ dx du 1 1 Z þ1 Z þ1 W ðx; u; zÞ dx du. ð12Þ 1
N
!
m
2 nþm2 N X N 1 6X ð1Þ ðn mÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 4 n¼1 m¼1 ðn þ mÞ3
0.5
!
N
10
ð13Þ
where the subscript ÔoÕ and ÔiÕ refer to the output and input planes of the optical system, respectively. So it is convenient to use Eq. (5), i.e., the WDF at the plane of z = 0, to calculate the M2 factor of partially coherent flat-topped beam. On substituting from Eqs. (5) and (12) into Eq. (11), the final result can be arranged as
0
0
5
η
10
Fig. 2. The variation of M2-factor of partially coherent flat-topped beam versus g with the beam order N = 1, 6 and 15.
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3.2. Power in bucket (PIB) Power in Bucket (PIB), another useful parameter for characterizing the beam quality, is defined as the fractional power within a given bucketÕs size [17], i.e., Ra Iðx; 0Þ dx a PIB ¼ R þ1 ; ð15Þ Iðx; 0Þ dx 1 where a is the radius of the given bucket, I(x, 0) is the irradiance of the partially coherent flat-topped beams at the z = 0 plane and can be evaluated by Eq. (4). By letting x01 ¼ x02 ¼ x, we obtained the irradiance of the partially coherent flat-topped beams at the z = 0 plane from Eq. (4). The substitution from Eq. (4) (letting x01 ¼ x02 ) into Eq. (15) and recalling the two integral equations [18] pffiffiffi Z a p ErfðaÞ; ð16Þ expðx2 Þ dx ¼ 2 0 pffiffiffi Z 1 p ½q > 0; ð17Þ expðq2 x2 Þ dx ¼ 2q 0 after tedious integral calculations, the final result can be written as h pffiffiffiffiffiffiffii PN PN ð1Þnþm2 N N pffiffiffiffiffiffiffi Erf a 2xnþm n¼1 m¼1 nþm 0 n m PIB ¼ ; ð18Þ PN PN ð1Þnþm2 N N pffiffiffiffiffiffiffi n¼1 m¼1 nþm n m where Erf [x] is the error function. Fig. 3 gives several typical curves of partially coherent flat-topped beam with different beam order N = 1, 6, 15. As shown in Fig. 3, for the same PIB, the bigger the beam order N the bigger the radius of the bucket a/x0. For a fixed radius of the bucket not too big, the bigger the beam order N the smaller the PIB. This
is because the flatness degree of a flat-topped beam increases with the beam order N increasing. So the fractional power increases with the decreasing beam order N when the radius of the given bucket is not too big. 4. Conclusions In conclusion, an extension of FGB characterizing for both from fully coherent to partially coherent case is presented. The GSM beam can be considered as special case of it. The irradiance distribution of partially coherent flattopped beam in free space is illustrated, which cannot keep invariant its shape during propagation. Also, the M2-factor of partially coherent flat-topped beam is studied. The results indicate that the M2-factor depends on the beam order N and the inverse of the global coherence parameter g. Fully coherent beam has better beam quality than partially coherent beam. The PIB of partially coherent flat-topped beam at the z = 0 plane is also studied. Although the calculation is performed in a rectangular coordinate system, the extension to a cylindrical coordinate system is straightforward. Acknowledgements This work is partially supported by the National Natural Science Foundation of China (Grant No. 10474004), National Fundamental Research Programme of China (Grant No. 2001CB309308), the Major Program of National Natural Science Foundation of China (Grant No. 60490280), DAAD exchange program: ‘‘D/0212785 personenaustausch VR China’’ and the National High Technology Research and Development Program of China (863 Program) international cooperation program (Grant No. 2004AA1Z1220). References
1
PIB
N=1 N=6 N=15 0.5
0
0
2
4
6
8
10
a/ω0 Fig. 3. The PIB curve versus the normalized bucket sized a/x0 with the beam order N = 1, 6 and 15 at the z = 0 plane.
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