- Email: [email protected]
Propagation and the kurtosis parameter of Gaussian flat-topped beams in uniaxial crystals orthogonal to the optical axis
ARTICLE IN PRESS Optics and Lasers in Engineering 48 (2010) 58–63
Contents lists available at ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Propagation and the kurtosis parameter of Gaussian flat-topped beams in uniaxial crystals orthogonal to the optical axis Dajun Liu , Zhongxiang Zhou Department of Physics, Harbin Institute of Technology, Harbin 150001, China
a r t i c l e in f o
a b s t r a c t
Article history: Received 17 April 2009 Received in revised form 9 June 2009 Accepted 18 July 2009 Available online 28 August 2009
The analytical formulae for the Gaussian flat-topped beams propagating in uniaxial crystals orthogonal to the optical axis are derived. The numerical results show that the Gaussian flat-topped beams spread at different rates in the directions parallel and orthogonal to the optical axis due to anisotropic crystals. An analytical expression for the kurtosis parameter of the Gaussian flat-topped beams propagating in uniaxial crystals is derived and illustrated with numerical examples. It is shown that the evolution of the kurtosis parameters Kx and Ky depend on the ratio of extraordinary to ordinary refractive indices. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Gaussian flat-topped beam Propagation properties Kurtosis parameter Uniaxial crystals
1. Introduction It is well known that anisotropic media are the subject of both theoretical and experimental study because of the increasing number of applications and devices that utilize these materials. In the past years, the propagation of various laser beams in uniaxial crystals has been widely studied [1–12]. Laser beams, such as Gaussian beams, Hermite–Gauss and Laguerre–Gauss beams, Bessel–Gauss beams, three-dimensional flattened Gaussian beams, hollow Gaussian beams and beams generated by Gaussian mirror resonator in uniaxial crystals [3–9], propagation in uniaxial crystals along the optical axis have been studied. And the propagation of Bessel-like beams [10], various hollow beams [11] and partially coherent flat-topped beams [12] in uniaxial crystals orthogonal to the optical axis has also been studied. Recently, much attention has been paid to the parameter characterization of laser beams based on the irradiance moments. The fourth-order moment has been used to define the so-called kurtosis parameter K, a shape parameter describing the degree of flatness of the beam intensity distribution [13,14]. Up to now, the kurtosis parameters of unapertured Gaussian beams, Hermit–Gaussian beams, Hermite-cosh-Gaussian beams, and super-Gaussian beams have been studied [15–19]. And the kurtosis parameters of flattened Gaussian beams, truncated standard and elegant Laguerre–Gaussian beams propagating through ABCD optical systems are also studied [20–22]. However,
Corresponding author.
E-mail address: [email protected] (D. Liu). 0143-8166/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2009.07.012
to our knowledge, there have been no reports on kurtosis parameters of laser beams in uniaxial crystals. In this paper, we have studied the propagation of Gaussian flattopped beams in uniaxial crystals orthogonal to the optical axis. Analytical formulae for Gaussian flat-topped beams propagating in uniaxial crystals and the kurtosis parameters of Gaussian flattopped beams propagating in uniaxial crystals are derived. The influences of the ratio of extraordinary to ordinary refractive indices on the spreading properties and kurtosis parameters Kx and Ky of Gaussian flat-topped beams are investigated by using the numerical illustrations.
2. Propagation of Gaussian flat-topped beams through uniaxial crystal When the electromagnetic beam propagates through uniaxial crystals orthogonal to the optical axis, the optical axis of uniaxial crystals coincides with the x-axis, and the beam propagates along the z-axis. Within the framework of paraxial propagation, the Cartesian components of the extraordinary and ordinary electromagnetic beams inside uniaxial crystals can be expressed as [12] ! Z n2e k2x þ n2o k2y 2 z E~ x ðkÞ Eex ðr; zÞ ¼ expðik0 ne zÞ d kexp ik r i 2k0 n2o ne ¼ expðik0 ne zÞAex ðr; zÞ
Eey ðr; zÞ ¼ expðik0 ne zÞ
Z
d2 kexp ik r i
ð1Þ n2e k2x þ n2o k2y 2k0 n2o ne
! z
ARTICLE IN PRESS D. Liu, Z. Zhou / Optics and Lasers in Engineering 48 (2010) 58–63
! kx ky ~ 2 2 E x ðkÞ ¼ expðik0 ne zÞAey ðr; zÞ k0 no Eez ðr; zÞ ¼ expðik0 ne zÞ
Z
2
d k exp ik r i
ð2Þ
n2e k2x þ n2o k2y 2k0 n2o ne
!
ð3Þ
Eox ðr; zÞ ¼ 0
ð4Þ
Eoy ðr; zÞ ¼ expðik0 no zÞ
k2x þ k2y d k exp ik r i z 2k0 no 2
!
kx ky ~ E x ðkÞ þ E~ y ðkÞ k20 n2o
¼ expðik0 no zÞAoy ðr; zÞ
! Z k2x þ k2y ky ~ z Eoz ðr; zÞ ¼ expðik0 no zÞ d2 k exp ik r i E y ðkÞ 2k0 no k0 no
expððnx2 =w20 qx Þ ðny2 =w20 qy ÞÞ
ð12Þ
ðqx qy Þ1=2
And using Eqs. (3) and (10), we can also obtain Aez ðr; zÞ¼
!
ð5Þ
N 1 X ð1Þn1 N 4n2 xy N n w40 qx qy
k20 n2o n¼1
z
ne kx ~ E x ðkÞ ¼ expðik0 ne zÞAez ðr; zÞ 2 k0 no
Z
¼
59
N ine @Aex ðr; zÞ ine X ð1Þn1 ¼ @x N k0 n2o k0 n2o n¼1
N n
expððnx2 =w20 qx Þ ðny2 =w20 qy ÞÞ ðqx qy Þ1=2
2nx w20 qx
ð13Þ
Similarly, by substituting Eq. (9) into (5) and performing some tedious but straightforward integration, we obtain Aoy ðr; zÞ¼
¼ expðik0 no zÞAoz ðr; zÞ
N 1 X ð1Þn1 N 4n2 xy 2 2 N n w40 q2o k0 no n¼1
expððnx2 =w20 qo Þ ðny2 =w20 qo ÞÞqo
ð6Þ
ð14Þ
where no and ne are the ordinary and extraordinary refractive indices, respectively; and Z 1 ~ ð7Þ d2 k expðik rÞEðr; 0Þ EðkÞ ¼ ð2pÞ2
with
is the two-dimensional Fourier transform of the transverse part of the electric field at the plane z ¼ 0, k ¼ kxe~x+kye~y, r ¼ xe~x+ye~y and k0 ¼ 2p/l. The boundary electric field of the Gaussian flat-topped beams polarized along x-axis at the plane z ¼ 0 can be expressed as [23,24] ! N X ð1Þn1 N x2 þ y2 e~ x EN ðrÞ ¼ exp n ð8Þ N n w20 n¼1
By using Eqs. (12) and (14), the Ey of Gaussian flat-topped beam inside uniaxial crystal can be expressed
where w0 is the waist size of the Gaussian beam,
N
denotes a n binomial coefficient, and N is the order of the Gaussian flat-topped beam. When N ¼ 1, Eq. (8) represents a pure Gaussian beam. On substituting Eq. (8) into (7), we can obtain the twodimensional Fourier transform of Gaussian flat-topped beam ! N k2x w20 þ k2y w20 w2 X ð1Þn1 N exp e~ x ð9Þ E~ N ðkÞ ¼ 0 4p n¼1 Nn 4n n
Substituting Eq. (9) into (1), we can obtain the following formula for the Gaussian flat-topped beam propagating in uniaxial crystal orthogonal to the optical axis N X ð1Þn1 N expððnx2 =w20 qx Þ ðny2 =w20 qy ÞÞ Aex ðr; zÞ ¼ ð10Þ N n ðqx qy Þ1=2 n¼1 with qx ¼ 1 þ
2inne z k0 n2o w20
ð11aÞ
qy ¼ 1 þ
2inz k0 ne w20
ð11bÞ
Then by using Eqs. (2) and (10), we can obtain Aey ðr; zÞ ¼
1 @2 Aex ðr; zÞ @x@y
k20 n2o
qo ¼ 1 þ
2inz k0 no w20
Ey ¼ expðik0 ne zÞAey ðr; zÞ þ expðik0 no zÞAoy ðr; zÞ
ð15Þ
ð16Þ
Eqs. (10)–(16) are the analytical expressions of Gaussian flattopped beam propagating in uniaxial crystals orthogonal to the optical axis.
3. Numerical examples In this section, the propagation properties of Gaussian flattopped beams in uniaxial crystal orthogonal to the optical axis are firstly illustrated by using the numerical calculations. The calculation parameters are w0 ¼ 20 mm, N ¼ 8, no ¼ 2, ne/no ¼ 1.5 and l ¼ 1.064 mm, and the beam parameters chosen in our calculations satisfy the paraxial constraint because of w0bl. Fig. 1 shows the |Aex| of the Gaussian flat-topped beam at several propagation distances in uniaxial crystal. It can be seen that |Aex| loses its flattened beam profile and its symmetry with the increase of propagation distance, and |Aex| spreads rapidly in x direction than y direction due to (ne/no)41 of the anisotropic crystal. And the evolution properties of Gaussian flat-topped beam in uniaxial crystals are accordance with the results of the Gaussian beam of Ref. [9]. Fig. 2 gives the |Ey| of the Gaussian flat-topped beam at several propagation distances in uniaxial crystal. It can be seen that the |Ey| spreads in xy plane with the propagation distance increasing, and |Ey| vanishes along the x and y axes. Fig. 3 shows the |Aez| of the Gaussian flat-topped beam at several propagation distances in uniaxial crystals. It can be seen from Fig. 3 that the |Aez| increases and spreads in xy plane with the increase of propagation distance. We can also see from Figs. 1–3 that |Ey|, |Aez| and |Aex| have the relations |Ey|5|Aex| and |Aez|5|Aex|; so the intensity of the Gaussian flat-topped beams in uniaxial crystals at the plane z can be written as I(x,y,z) ¼ |Aex|2. To learn the spreading properties of Gaussian flat-topped beams, we study the evolution properties of the effective beam
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Fig. 1. Profiles of transverse fields |Aex| originated by an incidence Gaussian flat-topped beams propagation in uniaxial crystal orthogonal to the optical axis at different plane (a) z ¼ 0, (b) z ¼ 1000 mm, (c) z ¼ 2000 mm, and (d) z ¼ 4000 mm.
spot of Gaussian flat-topped beams propagating in uniaxial crystals orthogonal to the optical axis. The effective beam width of beams at the plane z is defined as [25] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R1 R1 2 1 1 s2 Iðx; y; zÞdxdy R1 R1 Ws ¼ ðs ¼ x; yÞ ð17Þ 1 1 Iðx; y; zÞdxdy where Wx and Wy are the effective beam widths of beams in the x and y directions, respectively. By using Eq. (10), the intensity of the Gaussian flat-topped beams can be expressed as
Making use of the integral transform technique, after some tedious but straightforward integration, we can obtain Z 1 Z 1 Z 1 Z 1 F0 ¼ Iðx; y; zÞdxdy ¼ Iðx; y; z ¼ 0Þdxdy 1
1
¼
Iðx; y; zÞ ¼ jAex ðx; y; zÞj2 ¼ Aex ðx; y; zÞAex ðx; y; zÞ N X N X N ð1Þmþn N ¼ 2 N n m m¼1 n¼1 ðqx1 qy1 qx2 qy2 Þ1=2
with
qx1
2inne z 2inz ¼1þ qy1 ¼ 1 þ k0 n2o w20 k0 ne w20
qx2 ¼ 1 þ
2imne z 2imz qy2 ¼ 1 þ k0 n2o w20 k0 ne w20
Fy2 ¼ ¼
ð1Þmþn
2 m¼1 n¼1 N2 ðqx1 qy1 qx2 qy2 Þ1=2 n m þ w20 qx1 w20 qx2
Z
1 1
ð18Þ
1
1
ð19Þ
1
N X N pX
expððnx2 =w20 qx1 Þ ðmx2 =w20 qx2 Þ ðny2 =w20 qy1 Þ ðmy2 =w20 qy2 ÞÞ
1
2 N X N X N w0 p ð1Þmþn N ¼ 2 mþn N n m m n Z 1 Z 1 x2 Iðx; y; zÞdxdy Fx2 ¼
Z
1
!3=2
N n
N m
n m þ w20 qy1 w20 qy2
!1=2 ð20Þ
y2 Iðx; y; zÞdxdy
1
N X N pX
ð1Þmþn
N
N
2 m¼1 n¼1 N2 ðqx1 qy1 qx2 qy2 Þ1=2 n m !1=2 !3=2 n m n m þ þ w20 qx1 w20 qx2 w20 qy1 w20 qy2
ð21Þ
Thus by using Eqs. (17)–(21), we can easily calculate the effective beam widths Wx and Wy of Gaussian flat-topped beams in uniaxial crystals.
ARTICLE IN PRESS D. Liu, Z. Zhou / Optics and Lasers in Engineering 48 (2010) 58–63
Fig. 2. Profiles of transverse fields |Ey| originated by an incidence Gaussian flattopped beam propagation in uniaxial crystal orthogonal to the optical axis at different plane (a) z ¼ 1000 mm and (b) z ¼ 2000 mm.
Fig. 4 gives the dependence of the effective beam spot sizes Wx and Wy of Gaussian flat-topped beam on the propagation distance z in uniaxial crystals for the different ratio of extraordinary to ordinary refractive indices. One can see from Fig. 4 that the effective beam spot size Wx of Gaussian flat-topped beam increases with the increase of the ratio of extraordinary to ordinary refractive indices at a given propagation distance, while the effective beam spot size Wy of Gaussian flat-topped beam decreases with the increase of the ratio of extraordinary to ordinary refractive indices due to (ne/no)41 of the anisotropic crystal. In order to characterize the behavior of the electromagnetic beam passing through uniaxial crystals orthogonal to the optical axis, the kurtosis parameter, which characterizes the sharpness (or flatness) of the transverse intensity profile, is defined by Ks ¼
/s4 S /s2 S2
ðs ¼ x; yÞ
ð22Þ
where R1 R1 n 1 s Iðx; y; zÞdxdy ðs ¼ x; yÞ /sn S ¼ R1 1 R1 1 1 Iðx; y; zÞdxdy
ð23Þ
61
Fig. 3. Profiles of transverse fields |Aez| originated by an incidence Gaussian flattopped beams propagation in uniaxial crystal orthogonal to the optical axis with at different plane (a) z ¼ 1000 mm and (b) z ¼ 2000 mm.
By using the integral transform technique, we can obtain Z 1 Z 1 Fx4 ¼ x4 Iðx; y; zÞdxdy 1
1
N X N N N 3p X ð1Þmþn ¼ 4 m¼1 n¼1 N2 ðqx1 qy1 qx2 qy2 Þ1=2 n m !5=2 !1=2 n m n m þ þ w20 qx1 w20 qx2 w20 qy1 w20 qy2
Fy4 ¼
Z
1 1
¼
Z
1
ð24Þ
y4 Iðx; y; zÞdxdy
1
N X N N N 3p X ð1Þmþn 4 m¼1 n¼1 N2 ðqx1 qy1 qx2 qy2 Þ1=2 n m !1=2 !5=2 n m n m þ þ w20 qx1 w20 qx2 w20 qy1 w20 qy2
ð25Þ
Thus by using Eqs. (20)–(25), we can easily obtain the kurtosis parameters Kx and Ky of Gaussian flat-topped beams in uniaxial crystals orthogonal to the optical axis. At the incident plane z ¼ 0, the kurtosis parameters Kx and Ky of Gaussian flat-topped beams have the same value. Fig. 5 gives the kurtosis parameter of Gaussian flat-topped beam at the incident plane for the different beam order N. We can find that the kurtosis parameter of Gaussian flat-topped beam has the
ARTICLE IN PRESS 62
D. Liu, Z. Zhou / Optics and Lasers in Engineering 48 (2010) 58–63
maximum value equal to 3 when it is pure Gaussian beam (N ¼ 1), and the kurtosis parameter of Gaussian flat-topped beam decreases with increasing N. Fig. 6 gives the kurtosis parameter of Gaussian flat-topped beam in uniaxial crystals on the propagation distance for the different ratio of extraordinary to ordinary refractive indices. One can see that the kurtosis parameters Kx and Ky increase, as the propagation distance increases. So the Gaussian flat-topped beam will lose its initial profile and become more-Gaussian-like with the propagation distance increasing. And with the propagation distance increasing, the Gaussian flat-topped beam propagating in uniaxial crystals become more Gaussian-like in the direction parallel and orthogonal to the optical axis, then the kurtosis parameters Kx and Ky become almost the same for the different ratios of extraordinary to ordinary refractive indices. We can also see that kurtosis parameter Kx of Gaussian flat-topped beam increases with the increase of the ratio of extraordinary to ordinary refractive indices at a certain propagation distance, while the kurtosis parameter Ky of Gaussian flat-topped beam decreases with the increase of the ratio of extraordinary to ordinary refractive indices due to (ne/no)41 of the anisotropic crystal.
Fig. 4. Dependence of the effective beam spot sizes Wx and Wy of Gaussian flattopped beam in uniaxial crystals on the propagation distance for the different ratios of extraordinary to ordinary refractive indices (a) Wx and (b) Wy.
Fig. 5. The kurtosis parameter of Gaussian flat-topped beam at the incidenc plane vs. beam order N.
Fig. 6. The kurtosis parameter of Gaussian flat-topped beam in uniaxial crystals on the propagation distance for the different ratios of extraordinary to ordinary refractive indices (a) Kx and (b) Ky.
ARTICLE IN PRESS D. Liu, Z. Zhou / Optics and Lasers in Engineering 48 (2010) 58–63
4. Conclusions In conclusion, we have derived the analytical formulae of Gaussian flat-topped beams propagating in uniaxial crystals orthogonal to the optical axis, and studied the propagation and spreading properties of Gaussian flat-topped beams in uniaxial crystals in detail by using the derived formulae. The numerical results show that the effective beam spot size Wx of Gaussian flattopped beam increases with the increase of the ratio of extraordinary to ordinary refractive indices at a given propagation distance, while the effective beam spot size Wy of Gaussian flattopped beam decreases with the increase of the ratio of extraordinary to ordinary refractive indices due to (ne/no)41 of the anisotropic crystal. With the help of numerical calculations, we find that the evolution properties of kurtosis parameters Kx and Kx depend on the ratio of extraordinary to ordinary refractive indices.
Acknowledgement The research has been financially supported by the Science Fund for Distinguished Young Scholars of Heilongjiang Province (No. JC200710). References [1] Stamnes J, Sherman G. Radiation of electromagnetic fields in uniaxially anisotropic medium. J Opt Soc Am 1976;66:780–8. [2] Ciattoni A, Crosignani B, Di Porto P. Vectorial theory of propagation in uniaxially anisotropic media. J Opt Soc Am A 2001;18:1656–61. [3] Ciattoni A, Cincotti G, Palma C. Ordinary and extraordinary beams characterization in uniaxially anisotropic crystals. Opt Commun 2001;195: 55–61. [4] Cincotti G, Ciattoni A, Palma C. Laguerre–Gauss and Bessel–Gauss beams in uniaxial crystals. J Opt Soc Am A 2002;19:1680–8.
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[5] Ciattoni A, Cincotti G, Palma C. Propagation of cylindrically symmetric fields in uniaxial crystals. J Opt Soc Am A 2002;19:792–6. [6] Lu¨ B, Luo S. Propagation properties of three-dimensional flattened Gaussian beams in uniaxially anisotropic crystals. Opt Laser Technol 2004;36:51–6. [7] Deng D, Yu H, Xu S, Shao J, Fan Z. Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals. Opt Commun 2008;281:202–9. [8] Deng D, Shen J, Tian Y, Shao J, Fan Z. Propagation properties of beams generated by Gaussian mirror resonator in uniaxial crystals. Optik 2007;118:547–51. [9] Ciattoni A, Palma C. Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond. J Opt Soc Am A 2003;20:2163–71. [10] Ciattoni A, Palma C. Nondiffracting beams in uniaxial media propagating orthogonally to the optical axis. Opt Commun 2003;224:175–83. [11] Liu D, Zhou Z. Various dark hollow beams propagating in uniaxial crystals orthogonal to the optical axis. J Opt A: Pure Appl Opt 2008;10:095005. [12] Liu D, Zhou Z. Propagation of partially coherent flat-topped beams in uniaxial crystals orthogonal to the optical axis. J Opt Soc Am A 2009;26:924–30. [13] Weber H. Propagation of higher-order intensity moments in quadratic-index media. Opt Quantum Electron 1992;24:1027–49. [14] Martinez-Herrero R, Mejias P, Sanchez M, Neira J. Third- and fourth-order parametric characterization of partially coherent beams propagating through ABCD optical systems. Opt Quantum Electron 1992;24:1021–6. [15] Martinez-Herrero R, Piquero G, Mejias P. On the propagation of the kurtosis parameter of general beams. Opt Commun 1995;115:225–32. [16] Amarande S. Beam propagation factor and the kurtosis parameter of flattened Gaussian beams. Opt Commun 1996;19:311–17. [17] Saghafi S, Sheppard C, Piper J. Characterizing elegant and standard and stand Hermite–Gaussian beam modes. Opt Commun 2001;191:173–9. [18] Luo S, Lu¨ B. Propagation of the kurtosis parameter of Hermite-cosh-Gaussian beams. Optik 2002;113:329–32. [19] Ji X, Lu¨ B. Changes in the kurtosis parameter of super Gaussian beams passing through a spherically aberrated lens. Opt Quantum Electron 2003;35:749–55. [20] Lu¨ B, Luo S. Analytical expression for the kurtosis parameter of flattened Gaussian beams propagation through ABCD optical system. J Mod Opt 2002;49:1731–8. [21] Luo S, Lu¨ B. Propagation of the kurtosis parameter of elegant Hermite–Gaussian and Laguerre–Gaussian beams passing through ABCD systems. Optik 2002;113:227–31. [22] Mei Z, Zhao D. Kurtosis parameter of truncated standard and elegant Laguerre–Gaussian beams propagating through ABCD optical systems. Opt Laser Technol 2007;39:586–92. [23] Li Y. Light beam with flat-topped profiles. Opt Lett 2002;27:1007–9. [24] Li Y. New expressions for flat-topped beams. Opt Commun 2002;206:225–34. [25] Carter WH. Spot size and divergence for Hermite Gaussian beams of any order. Appl Opt 1980;19:1027–9.
Contents lists available at ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Propagation and the kurtosis parameter of Gaussian flat-topped beams in uniaxial crystals orthogonal to the optical axis Dajun Liu , Zhongxiang Zhou Department of Physics, Harbin Institute of Technology, Harbin 150001, China
a r t i c l e in f o
a b s t r a c t
Article history: Received 17 April 2009 Received in revised form 9 June 2009 Accepted 18 July 2009 Available online 28 August 2009
The analytical formulae for the Gaussian flat-topped beams propagating in uniaxial crystals orthogonal to the optical axis are derived. The numerical results show that the Gaussian flat-topped beams spread at different rates in the directions parallel and orthogonal to the optical axis due to anisotropic crystals. An analytical expression for the kurtosis parameter of the Gaussian flat-topped beams propagating in uniaxial crystals is derived and illustrated with numerical examples. It is shown that the evolution of the kurtosis parameters Kx and Ky depend on the ratio of extraordinary to ordinary refractive indices. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Gaussian flat-topped beam Propagation properties Kurtosis parameter Uniaxial crystals
1. Introduction It is well known that anisotropic media are the subject of both theoretical and experimental study because of the increasing number of applications and devices that utilize these materials. In the past years, the propagation of various laser beams in uniaxial crystals has been widely studied [1–12]. Laser beams, such as Gaussian beams, Hermite–Gauss and Laguerre–Gauss beams, Bessel–Gauss beams, three-dimensional flattened Gaussian beams, hollow Gaussian beams and beams generated by Gaussian mirror resonator in uniaxial crystals [3–9], propagation in uniaxial crystals along the optical axis have been studied. And the propagation of Bessel-like beams [10], various hollow beams [11] and partially coherent flat-topped beams [12] in uniaxial crystals orthogonal to the optical axis has also been studied. Recently, much attention has been paid to the parameter characterization of laser beams based on the irradiance moments. The fourth-order moment has been used to define the so-called kurtosis parameter K, a shape parameter describing the degree of flatness of the beam intensity distribution [13,14]. Up to now, the kurtosis parameters of unapertured Gaussian beams, Hermit–Gaussian beams, Hermite-cosh-Gaussian beams, and super-Gaussian beams have been studied [15–19]. And the kurtosis parameters of flattened Gaussian beams, truncated standard and elegant Laguerre–Gaussian beams propagating through ABCD optical systems are also studied [20–22]. However,
Corresponding author.
E-mail address: [email protected] (D. Liu). 0143-8166/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2009.07.012
to our knowledge, there have been no reports on kurtosis parameters of laser beams in uniaxial crystals. In this paper, we have studied the propagation of Gaussian flattopped beams in uniaxial crystals orthogonal to the optical axis. Analytical formulae for Gaussian flat-topped beams propagating in uniaxial crystals and the kurtosis parameters of Gaussian flattopped beams propagating in uniaxial crystals are derived. The influences of the ratio of extraordinary to ordinary refractive indices on the spreading properties and kurtosis parameters Kx and Ky of Gaussian flat-topped beams are investigated by using the numerical illustrations.
2. Propagation of Gaussian flat-topped beams through uniaxial crystal When the electromagnetic beam propagates through uniaxial crystals orthogonal to the optical axis, the optical axis of uniaxial crystals coincides with the x-axis, and the beam propagates along the z-axis. Within the framework of paraxial propagation, the Cartesian components of the extraordinary and ordinary electromagnetic beams inside uniaxial crystals can be expressed as [12] ! Z n2e k2x þ n2o k2y 2 z E~ x ðkÞ Eex ðr; zÞ ¼ expðik0 ne zÞ d kexp ik r i 2k0 n2o ne ¼ expðik0 ne zÞAex ðr; zÞ
Eey ðr; zÞ ¼ expðik0 ne zÞ
Z
d2 kexp ik r i
ð1Þ n2e k2x þ n2o k2y 2k0 n2o ne
! z
ARTICLE IN PRESS D. Liu, Z. Zhou / Optics and Lasers in Engineering 48 (2010) 58–63
! kx ky ~ 2 2 E x ðkÞ ¼ expðik0 ne zÞAey ðr; zÞ k0 no Eez ðr; zÞ ¼ expðik0 ne zÞ
Z
2
d k exp ik r i
ð2Þ
n2e k2x þ n2o k2y 2k0 n2o ne
!
ð3Þ
Eox ðr; zÞ ¼ 0
ð4Þ
Eoy ðr; zÞ ¼ expðik0 no zÞ
k2x þ k2y d k exp ik r i z 2k0 no 2
!
kx ky ~ E x ðkÞ þ E~ y ðkÞ k20 n2o
¼ expðik0 no zÞAoy ðr; zÞ
! Z k2x þ k2y ky ~ z Eoz ðr; zÞ ¼ expðik0 no zÞ d2 k exp ik r i E y ðkÞ 2k0 no k0 no
expððnx2 =w20 qx Þ ðny2 =w20 qy ÞÞ
ð12Þ
ðqx qy Þ1=2
And using Eqs. (3) and (10), we can also obtain Aez ðr; zÞ¼
!
ð5Þ
N 1 X ð1Þn1 N 4n2 xy N n w40 qx qy
k20 n2o n¼1
z
ne kx ~ E x ðkÞ ¼ expðik0 ne zÞAez ðr; zÞ 2 k0 no
Z
¼
59
N ine @Aex ðr; zÞ ine X ð1Þn1 ¼ @x N k0 n2o k0 n2o n¼1
N n
expððnx2 =w20 qx Þ ðny2 =w20 qy ÞÞ ðqx qy Þ1=2
2nx w20 qx
ð13Þ
Similarly, by substituting Eq. (9) into (5) and performing some tedious but straightforward integration, we obtain Aoy ðr; zÞ¼
¼ expðik0 no zÞAoz ðr; zÞ
N 1 X ð1Þn1 N 4n2 xy 2 2 N n w40 q2o k0 no n¼1
expððnx2 =w20 qo Þ ðny2 =w20 qo ÞÞqo
ð6Þ
ð14Þ
where no and ne are the ordinary and extraordinary refractive indices, respectively; and Z 1 ~ ð7Þ d2 k expðik rÞEðr; 0Þ EðkÞ ¼ ð2pÞ2
with
is the two-dimensional Fourier transform of the transverse part of the electric field at the plane z ¼ 0, k ¼ kxe~x+kye~y, r ¼ xe~x+ye~y and k0 ¼ 2p/l. The boundary electric field of the Gaussian flat-topped beams polarized along x-axis at the plane z ¼ 0 can be expressed as [23,24] ! N X ð1Þn1 N x2 þ y2 e~ x EN ðrÞ ¼ exp n ð8Þ N n w20 n¼1
By using Eqs. (12) and (14), the Ey of Gaussian flat-topped beam inside uniaxial crystal can be expressed
where w0 is the waist size of the Gaussian beam,
N
denotes a n binomial coefficient, and N is the order of the Gaussian flat-topped beam. When N ¼ 1, Eq. (8) represents a pure Gaussian beam. On substituting Eq. (8) into (7), we can obtain the twodimensional Fourier transform of Gaussian flat-topped beam ! N k2x w20 þ k2y w20 w2 X ð1Þn1 N exp e~ x ð9Þ E~ N ðkÞ ¼ 0 4p n¼1 Nn 4n n
Substituting Eq. (9) into (1), we can obtain the following formula for the Gaussian flat-topped beam propagating in uniaxial crystal orthogonal to the optical axis N X ð1Þn1 N expððnx2 =w20 qx Þ ðny2 =w20 qy ÞÞ Aex ðr; zÞ ¼ ð10Þ N n ðqx qy Þ1=2 n¼1 with qx ¼ 1 þ
2inne z k0 n2o w20
ð11aÞ
qy ¼ 1 þ
2inz k0 ne w20
ð11bÞ
Then by using Eqs. (2) and (10), we can obtain Aey ðr; zÞ ¼
1 @2 Aex ðr; zÞ @x@y
k20 n2o
qo ¼ 1 þ
2inz k0 no w20
Ey ¼ expðik0 ne zÞAey ðr; zÞ þ expðik0 no zÞAoy ðr; zÞ
ð15Þ
ð16Þ
Eqs. (10)–(16) are the analytical expressions of Gaussian flattopped beam propagating in uniaxial crystals orthogonal to the optical axis.
3. Numerical examples In this section, the propagation properties of Gaussian flattopped beams in uniaxial crystal orthogonal to the optical axis are firstly illustrated by using the numerical calculations. The calculation parameters are w0 ¼ 20 mm, N ¼ 8, no ¼ 2, ne/no ¼ 1.5 and l ¼ 1.064 mm, and the beam parameters chosen in our calculations satisfy the paraxial constraint because of w0bl. Fig. 1 shows the |Aex| of the Gaussian flat-topped beam at several propagation distances in uniaxial crystal. It can be seen that |Aex| loses its flattened beam profile and its symmetry with the increase of propagation distance, and |Aex| spreads rapidly in x direction than y direction due to (ne/no)41 of the anisotropic crystal. And the evolution properties of Gaussian flat-topped beam in uniaxial crystals are accordance with the results of the Gaussian beam of Ref. [9]. Fig. 2 gives the |Ey| of the Gaussian flat-topped beam at several propagation distances in uniaxial crystal. It can be seen that the |Ey| spreads in xy plane with the propagation distance increasing, and |Ey| vanishes along the x and y axes. Fig. 3 shows the |Aez| of the Gaussian flat-topped beam at several propagation distances in uniaxial crystals. It can be seen from Fig. 3 that the |Aez| increases and spreads in xy plane with the increase of propagation distance. We can also see from Figs. 1–3 that |Ey|, |Aez| and |Aex| have the relations |Ey|5|Aex| and |Aez|5|Aex|; so the intensity of the Gaussian flat-topped beams in uniaxial crystals at the plane z can be written as I(x,y,z) ¼ |Aex|2. To learn the spreading properties of Gaussian flat-topped beams, we study the evolution properties of the effective beam
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Fig. 1. Profiles of transverse fields |Aex| originated by an incidence Gaussian flat-topped beams propagation in uniaxial crystal orthogonal to the optical axis at different plane (a) z ¼ 0, (b) z ¼ 1000 mm, (c) z ¼ 2000 mm, and (d) z ¼ 4000 mm.
spot of Gaussian flat-topped beams propagating in uniaxial crystals orthogonal to the optical axis. The effective beam width of beams at the plane z is defined as [25] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R1 R1 2 1 1 s2 Iðx; y; zÞdxdy R1 R1 Ws ¼ ðs ¼ x; yÞ ð17Þ 1 1 Iðx; y; zÞdxdy where Wx and Wy are the effective beam widths of beams in the x and y directions, respectively. By using Eq. (10), the intensity of the Gaussian flat-topped beams can be expressed as
Making use of the integral transform technique, after some tedious but straightforward integration, we can obtain Z 1 Z 1 Z 1 Z 1 F0 ¼ Iðx; y; zÞdxdy ¼ Iðx; y; z ¼ 0Þdxdy 1
1
¼
Iðx; y; zÞ ¼ jAex ðx; y; zÞj2 ¼ Aex ðx; y; zÞAex ðx; y; zÞ N X N X N ð1Þmþn N ¼ 2 N n m m¼1 n¼1 ðqx1 qy1 qx2 qy2 Þ1=2
with
qx1
2inne z 2inz ¼1þ qy1 ¼ 1 þ k0 n2o w20 k0 ne w20
qx2 ¼ 1 þ
2imne z 2imz qy2 ¼ 1 þ k0 n2o w20 k0 ne w20
Fy2 ¼ ¼
ð1Þmþn
2 m¼1 n¼1 N2 ðqx1 qy1 qx2 qy2 Þ1=2 n m þ w20 qx1 w20 qx2
Z
1 1
ð18Þ
1
1
ð19Þ
1
N X N pX
expððnx2 =w20 qx1 Þ ðmx2 =w20 qx2 Þ ðny2 =w20 qy1 Þ ðmy2 =w20 qy2 ÞÞ
1
2 N X N X N w0 p ð1Þmþn N ¼ 2 mþn N n m m n Z 1 Z 1 x2 Iðx; y; zÞdxdy Fx2 ¼
Z
1
!3=2
N n
N m
n m þ w20 qy1 w20 qy2
!1=2 ð20Þ
y2 Iðx; y; zÞdxdy
1
N X N pX
ð1Þmþn
N
N
2 m¼1 n¼1 N2 ðqx1 qy1 qx2 qy2 Þ1=2 n m !1=2 !3=2 n m n m þ þ w20 qx1 w20 qx2 w20 qy1 w20 qy2
ð21Þ
Thus by using Eqs. (17)–(21), we can easily calculate the effective beam widths Wx and Wy of Gaussian flat-topped beams in uniaxial crystals.
ARTICLE IN PRESS D. Liu, Z. Zhou / Optics and Lasers in Engineering 48 (2010) 58–63
Fig. 2. Profiles of transverse fields |Ey| originated by an incidence Gaussian flattopped beam propagation in uniaxial crystal orthogonal to the optical axis at different plane (a) z ¼ 1000 mm and (b) z ¼ 2000 mm.
Fig. 4 gives the dependence of the effective beam spot sizes Wx and Wy of Gaussian flat-topped beam on the propagation distance z in uniaxial crystals for the different ratio of extraordinary to ordinary refractive indices. One can see from Fig. 4 that the effective beam spot size Wx of Gaussian flat-topped beam increases with the increase of the ratio of extraordinary to ordinary refractive indices at a given propagation distance, while the effective beam spot size Wy of Gaussian flat-topped beam decreases with the increase of the ratio of extraordinary to ordinary refractive indices due to (ne/no)41 of the anisotropic crystal. In order to characterize the behavior of the electromagnetic beam passing through uniaxial crystals orthogonal to the optical axis, the kurtosis parameter, which characterizes the sharpness (or flatness) of the transverse intensity profile, is defined by Ks ¼
/s4 S /s2 S2
ðs ¼ x; yÞ
ð22Þ
where R1 R1 n 1 s Iðx; y; zÞdxdy ðs ¼ x; yÞ /sn S ¼ R1 1 R1 1 1 Iðx; y; zÞdxdy
ð23Þ
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Fig. 3. Profiles of transverse fields |Aez| originated by an incidence Gaussian flattopped beams propagation in uniaxial crystal orthogonal to the optical axis with at different plane (a) z ¼ 1000 mm and (b) z ¼ 2000 mm.
By using the integral transform technique, we can obtain Z 1 Z 1 Fx4 ¼ x4 Iðx; y; zÞdxdy 1
1
N X N N N 3p X ð1Þmþn ¼ 4 m¼1 n¼1 N2 ðqx1 qy1 qx2 qy2 Þ1=2 n m !5=2 !1=2 n m n m þ þ w20 qx1 w20 qx2 w20 qy1 w20 qy2
Fy4 ¼
Z
1 1
¼
Z
1
ð24Þ
y4 Iðx; y; zÞdxdy
1
N X N N N 3p X ð1Þmþn 4 m¼1 n¼1 N2 ðqx1 qy1 qx2 qy2 Þ1=2 n m !1=2 !5=2 n m n m þ þ w20 qx1 w20 qx2 w20 qy1 w20 qy2
ð25Þ
Thus by using Eqs. (20)–(25), we can easily obtain the kurtosis parameters Kx and Ky of Gaussian flat-topped beams in uniaxial crystals orthogonal to the optical axis. At the incident plane z ¼ 0, the kurtosis parameters Kx and Ky of Gaussian flat-topped beams have the same value. Fig. 5 gives the kurtosis parameter of Gaussian flat-topped beam at the incident plane for the different beam order N. We can find that the kurtosis parameter of Gaussian flat-topped beam has the
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maximum value equal to 3 when it is pure Gaussian beam (N ¼ 1), and the kurtosis parameter of Gaussian flat-topped beam decreases with increasing N. Fig. 6 gives the kurtosis parameter of Gaussian flat-topped beam in uniaxial crystals on the propagation distance for the different ratio of extraordinary to ordinary refractive indices. One can see that the kurtosis parameters Kx and Ky increase, as the propagation distance increases. So the Gaussian flat-topped beam will lose its initial profile and become more-Gaussian-like with the propagation distance increasing. And with the propagation distance increasing, the Gaussian flat-topped beam propagating in uniaxial crystals become more Gaussian-like in the direction parallel and orthogonal to the optical axis, then the kurtosis parameters Kx and Ky become almost the same for the different ratios of extraordinary to ordinary refractive indices. We can also see that kurtosis parameter Kx of Gaussian flat-topped beam increases with the increase of the ratio of extraordinary to ordinary refractive indices at a certain propagation distance, while the kurtosis parameter Ky of Gaussian flat-topped beam decreases with the increase of the ratio of extraordinary to ordinary refractive indices due to (ne/no)41 of the anisotropic crystal.
Fig. 4. Dependence of the effective beam spot sizes Wx and Wy of Gaussian flattopped beam in uniaxial crystals on the propagation distance for the different ratios of extraordinary to ordinary refractive indices (a) Wx and (b) Wy.
Fig. 5. The kurtosis parameter of Gaussian flat-topped beam at the incidenc plane vs. beam order N.
Fig. 6. The kurtosis parameter of Gaussian flat-topped beam in uniaxial crystals on the propagation distance for the different ratios of extraordinary to ordinary refractive indices (a) Kx and (b) Ky.
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4. Conclusions In conclusion, we have derived the analytical formulae of Gaussian flat-topped beams propagating in uniaxial crystals orthogonal to the optical axis, and studied the propagation and spreading properties of Gaussian flat-topped beams in uniaxial crystals in detail by using the derived formulae. The numerical results show that the effective beam spot size Wx of Gaussian flattopped beam increases with the increase of the ratio of extraordinary to ordinary refractive indices at a given propagation distance, while the effective beam spot size Wy of Gaussian flattopped beam decreases with the increase of the ratio of extraordinary to ordinary refractive indices due to (ne/no)41 of the anisotropic crystal. With the help of numerical calculations, we find that the evolution properties of kurtosis parameters Kx and Kx depend on the ratio of extraordinary to ordinary refractive indices.
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