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Propagation of flattened Gaussian beams passing through a misaligned optical system with finite aperture
Optik 115, No. 5 (2004) 193–196 http://www.elsevier.de/ijleo
International Journal for Light and Electron Optics
Propagation of flattened Gaussian beams passing through a misaligned optical system with finite aperture Meixiao Shen, Shaomin Wang, Daomu Zhao Department of Physics, Zhejiang University, Hangzhou 310027, P. R. China
Abstract: The propagation of flattened Gaussian beams (FGBs) passing though misaligned optical systems with aperture has been studied in detail. The corresponding approximate equation has been derived, where the rectangular aperture function is expanded into a finite sum of complex Gaussian functions. Numerical examples are given to illustrate the application of the propagation equation and the properties of flattened Gaussian beams passing through a misaligned optical system with aperture are discussed. Key words: Flattened Gaussian beams – misaligned optical system – finite aperture
1. Introduction Since Gari introduced the flattened Gaussian beams into optical field in 1994 [1], many studies in both theoretical and experimental respects for such beam have been carried out [2–5], due to their advantage in the simulation of beams with a flat-top spatial profile. Based on the well-known diffraction integral formula, the analytical propagation formula of FGBs passing through aligned optical systems has been derived [5]. However, this formula is only for the unapertured and aligned case. However, the propagation of flattened Gaussian beams passing through the misaligned optical systems with finite aperture, up to now, has not been considered elsewhere. While the misaligned optical systems with aperture always exists in practice, so the study of propagation for beams through misaligned optical systems with aperture is of practical significance. The purpose of this paper is to study the propagation of FGBs passing through misaligned optical systems with aperture. First, based on the fact that the rectangular function can be expanded into a finite sum of complex functions, the approximate propagation expressions of FGBs passing through a misaligned optical system with hard aperture are derived, respectively, where the FGB Received 19 December 2003; accepted 24 March 2004. Correspondence to: S. Wang Fax: ++86-571-8798-5970 E-mail: [email protected]
is regarded as a whole beam and a finite series expansion. Then, Numerical examples are given to illustrate the propagation properties of FGBs passing through the misaligned optical systems with finite aperture. Finally, the conclusion for this paper is summarized.
2. Propagation of flattened Gaussian Beams through misaligned optical systems with finite aperture The field distribution Eðx; zÞ of FGBs at the plane of z ¼ 0 in the rectangular coordinate system is expressed as follows [2]: Eðx1 ; 0Þ N ðN þ 1Þ x21 P 1 Nþ1 2 n x1 ; ð1Þ ¼ E0 exp w20 w20 n¼0 n! where N ðN ¼ 0; 1; . . .Þ, the beam order and w0, the waist width, are two characteristic parameters of FGBs. E0 is a constant, which will be set to unity in the following. Assume there is a rectangular aperture, which is a in radius, at the plane of z ¼ 0. Just as the circ function can be expanded into a finite sum of complex Gaussian functions [6, 7], the rectangular function can be expressed as follows: m P Bm Am exp 2 x21 ; ð2Þ Hðx1 Þ ¼ a m¼1 where Am and Bm are expansion and Gaussian coefficients as shown in table 1. The propagation of FGBs passing through a misaligned optical system with aperture is governed by the generalized diffraction integral formulae for misaligned optical systems, which can be written as follows [8]: Eðx; zÞ rffiffiffiffiffiffi 1 ð i exp ðikl0 Þ Eðx1 ; 0Þ Hðx1 Þ ¼ lB 1 ik ðAx1 2x1 x þ Dx2 þ Ex1 þ GxÞ dx1 ; exp 2B (3) 0030-4026/04/115/05-193 $ 30.00/0
194
M. Shen et al., Propagation of flattened Gaussian beams passing through a misaligned optical system
has been used. Eq. (6) is the analytical equation of flattened Gaussian beams through misaligned optical systems with aperture, where the FGB is regarded as a whole beam. In the other hand, FGBs can be expanded in a finite sum of Hermite-Gaussian modes in the rectangular coordinate system, which is written as follows [8]:
Table 1. Expansion and Gaussian coefficients Am and Bm. Bm
Am 11.428 þ 0.95175i 0.06002 0.08013i 4:2734 8:5562i 1.6576 þ 2.7015i 5:0418 þ 3:2488i 1.1227 0.68854i 1:0106 0:26955i 2:5974 þ 3:2202i 0:14840 0:31193i 0:20850 0:23851i
4.0697 þ 0.22726i 1.1531 20.933i 4.4608 þ 5.1268i 4.3521 þ 14.997i 4.5443 þ 10.003i 3.8478 þ 20.078i 2.5280 10.310i 3.3197 4.8008i 1.9002 15.820i 2.6340 þ 25.009i
Eðx1 ; 0Þ
where k and l are the wave numer and wavelength, respectively. The parameters E and G take the following form: E ¼ 2ðaT ex þ bT e0x Þ ; G ¼ 2ðbgT daT Þex þ 2ðbdT dbT Þe0x ;
ð4Þ
e0x
denote the one-dimensional misawhere ex and ligned parameters, ex is the displacement, e0x is the tilting angle of the element. aT , bT , gT and dT represent the misaligned matrix elements determined by aT ¼ 1 A ; bT ¼ l B ; gT ¼ C ; dT ¼ 1 D :
N 1 1 P ð2lÞ! 3l l!ðl nÞ! n¼0 ð2nÞ! l¼n 2 "pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # 2ðN þ 1Þ ðN þ 1Þ 2 H2n x1 exp x1 : w0 w20
¼ E0
ð5Þ
Sign “þ” is for the case of the forward-going optical elements, and sign “” is for the case of the backwardgoing optical elements. Substituting eqs. (1) and (2) into (3), we obtain
N P
Substituting eqs. (8) and (2) into eq. (3), and using the integral formula
rffiffiffiffiffiffi ik ik exp ðDx2 þ GxÞ 2B 2B 9 8 > kð2x EÞ 2 > > > > > = < M P 4B Am exp N þ 1 ikA Bm > > m¼1 > > > þ 2> þ : a ; W02 2B 3nþ12 2 Nþ1 N 1 7 P w20 1 n 6 7 6 4N þ 1 ikA Bm 5 n! 4 n¼0 þ 2 þ 2B a w20 3 2 kð2x EÞ 7 6 7 6 4B 6 ffi7 H2n 6sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7: 4 N þ 1 ikA Bm 5 þ 2 þ a 2B w20
# ðx yÞ2 exp Hm ðxÞ dx 2u 1 pffiffiffiffiffiffiffiffi m y 2 p ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ¼ 2pu ð1 2uÞ Hm 1 2u we also can get
ikx ikDE2 ðG þ 2DEÞ 2B 2B N N 1 P 1 P ð2lÞ! E2n ðx; zÞ ; 3l l!ðl nÞ! n¼0 ð2nÞ! l¼n 2
Eðx; zÞ ¼ exp
n
1=2 ð2nþ1Þ
¼ ð1Þ p
2
y2 exp 2 4a
ð10Þ
E2n ðx; zÞ
ð6Þ
8 9 > E 2 D w20 Bm > > > > > Cþ 1þ < ik x = M P 2 q0 ðN þ 1Þ a2 ¼ Am exp > 2 B w20 Bm > m¼1 > > > > 1þ Aþ : ; q0 ðN þ 1Þ a2 n B w20 Bm A 1 q0 ðN þ 1Þa2 nþ1=2 B w20 Bm Aþ 1þ q0 ðN þ 1Þ a2 3 2 pffiffiffi 7 6 E 7 6 2 x 7 6 2 7 6 H2n 6vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ( ) 7; u 2 2 7 6u x2 i2BBm BlðN þ1Þ 5 4t 0 A2 þ N þ1 ka2 pw20 (11)
x2n exp ða2 x2 Þ cos ðxyÞ dx
0
ð9Þ
where
In the derivation of eq. (6), the integral formula 1 ð
"
1 ð
Eðx; zÞ
w0 ¼ pffiffiffiffiffiffiffiffiffiffiffi nþ1
ð8Þ
H2n
y 2a
where q0 ¼ ð7Þ
ipw20
lðN þ 1Þ plane of z ¼ 0.
is the beam q-parameter at the
M. Shen et al., Propagation of flattened Gaussian beams passing through a misaligned optical system
195
3. Numerical examples and discussion Let us consider a one-dimensional flattened Gaussian beam propagating through a system with a transverse shifted thin lens with focal length f, whose aperture effect is considered, followed by a free space. The displacements and angle misalignment of the lens with respect to the optical axis of the system are ex, e0x ¼ 0. The thin lens is located at the plane of z ¼ 0. The ABCD transfer matrix for such system reads A B Dz f ð1 þ DzÞ ¼ ; ð12Þ C D 1=f 1 where f denotes the focal length of the thin lens, z is the propagation distance after the lens, and Dz ¼ ðz f Þ=f :
ð13Þ
The misalignment parameters aT , bT , gT and dT take the following form: aT ¼ 1 þ Dz ;
bT ¼ 0 ;
gT ¼ 1=f ;
dT ¼ 0 ð14Þ
and the corresponding parameters E and G are given by E ¼ 2ð1 þ DzÞ ex ;
G ¼ 2ð1 þ DzÞex :
ð15Þ
Substituting eqs. (2) and (15) into eq. (6) and eq. (10), we can obtain the one-dimensional relative intensity distribution of the flattened Gaussian beam at several propagation distances after a misaligned thin lens with aperture as shown in fig. 1, where the FGBs are regarded as the whole beam and a finite series expansion, respectively. The parameters used in the calculation are l ¼ 632:8 nm, ex ¼ 1:0 mm, f ¼ 150 mm. The solid line denotes the intensity derived by using eq. (6) and the dotted line denotes that derived by using eq. (10). As can be seen, although eqs. (6) and (10) have different mathematical formulations, many numerical calculations have shown the same results.
a)
b)
Fig. 2. Relative intensity distribution of flattened Gaussian beam through a thin lens, followed by a free space, where the thin lens is aligned and the aperture of that is neglected. a) Dz ¼ 0:1; b) Dz ¼ 1.
In order to compare the propagation properties of flattened Gaussian beam through misaligned thin lens with aperture and through aligned thin lens, plot of 1-D intensity distribution of flattened Gaussian beam passing though an aligned thin lens at several propagation distances are also shown in fig. 2. From fig. 1 and fig. 2, we can find that the flattened Gaussian beam becomes a decectered flattened Gaussian beam after passing through the misaligned thin lens and the aperture diffraction effect results in the larger lobes.
4. Conclusions In conclusions, the propagation of FGBs passing through a misaligned optical system with aperture has been studied. The approximate equation for both forms of FGBs of such case have been derived, where the rectangular function is expanded into a sum of complex Gaussian functions, and numerical examples have been shown that the two results are equivalent. Numerical calculations are also given to illustrate the propagation properties of FGBs passing through misaligned optical systems with aperture. Actually, the method used here has a general applicable advantage and the approximate formulae derived in this paper provide a convenient tool to treat the propagation of FGBs through a misaligned optical system, where the aperture effect is been considered. Acknowledgments. The authors are grateful to the financial support of Natural Science Foundation of China, Grant No. 60276035.
a)
b)
Fig. 1. Relative intensity distribution of flattened Gaussian beam through a transverse shifted thin lens, followed by a free space, where the aperture of the thin lens is considered. a) Dz ¼ 0; b) Dz ¼ 0:1; c) Dz ¼ 1. c)
References [1] Gori F: Flattened Gaussian beams. Opt. Commun. 107 (1994) 335–341 [2] Bagini V, Borghi R, Gori F: Propagation of axially symmetric flattened Gaussian beams. J. Opt. Soc. Am. A 13 (1996) 1385–1394 [3] Santarsiero M, Aiello D, Borghi R: Focusing of axially symmetric flattened Gaussian beams. J. Mod. Opt. 44 (1997) 633–650 [4] Borghi R, Santarsiero M: Modal decomposition of partially coherent flat-topped beams produced by multimode lasers. Opt. Lett. 23 (1998) 313–315
196
M. Shen et al., Propagation of flattened Gaussian beams passing through a misaligned optical system
[5] Lu¨ B, Luo SR, Zhang B: Propagation of flattened Gaussian beams with rectangular symmetry passing through a paraxial optical ABCD system with and without aperture. Opt. Commun. 164 (1999) 1–6 [6] Wen JJ, Breazeale MA: A diffraction beam field expressed as the superposition of Gaussian beams. J. Acoust. Soc. Am. 83 (1988) 1752–1756
[7] Ding DS, Liu XJ: Approximate description of Bessel, Bessel-Gauss, and Gaussian beams with finite aperture. J. Opt. Soc. Am. A 16 (1999) 1286–1293 [8] Wang SM, Ronchi L: Principles and design of optical array. In: Wolf E (ed): Progress in optics, Vol. 25, pp. 279. Elsevier Science BV, Amsterdam 1998
International Journal for Light and Electron Optics
Propagation of flattened Gaussian beams passing through a misaligned optical system with finite aperture Meixiao Shen, Shaomin Wang, Daomu Zhao Department of Physics, Zhejiang University, Hangzhou 310027, P. R. China
Abstract: The propagation of flattened Gaussian beams (FGBs) passing though misaligned optical systems with aperture has been studied in detail. The corresponding approximate equation has been derived, where the rectangular aperture function is expanded into a finite sum of complex Gaussian functions. Numerical examples are given to illustrate the application of the propagation equation and the properties of flattened Gaussian beams passing through a misaligned optical system with aperture are discussed. Key words: Flattened Gaussian beams – misaligned optical system – finite aperture
1. Introduction Since Gari introduced the flattened Gaussian beams into optical field in 1994 [1], many studies in both theoretical and experimental respects for such beam have been carried out [2–5], due to their advantage in the simulation of beams with a flat-top spatial profile. Based on the well-known diffraction integral formula, the analytical propagation formula of FGBs passing through aligned optical systems has been derived [5]. However, this formula is only for the unapertured and aligned case. However, the propagation of flattened Gaussian beams passing through the misaligned optical systems with finite aperture, up to now, has not been considered elsewhere. While the misaligned optical systems with aperture always exists in practice, so the study of propagation for beams through misaligned optical systems with aperture is of practical significance. The purpose of this paper is to study the propagation of FGBs passing through misaligned optical systems with aperture. First, based on the fact that the rectangular function can be expanded into a finite sum of complex functions, the approximate propagation expressions of FGBs passing through a misaligned optical system with hard aperture are derived, respectively, where the FGB Received 19 December 2003; accepted 24 March 2004. Correspondence to: S. Wang Fax: ++86-571-8798-5970 E-mail: [email protected]
is regarded as a whole beam and a finite series expansion. Then, Numerical examples are given to illustrate the propagation properties of FGBs passing through the misaligned optical systems with finite aperture. Finally, the conclusion for this paper is summarized.
2. Propagation of flattened Gaussian Beams through misaligned optical systems with finite aperture The field distribution Eðx; zÞ of FGBs at the plane of z ¼ 0 in the rectangular coordinate system is expressed as follows [2]: Eðx1 ; 0Þ N ðN þ 1Þ x21 P 1 Nþ1 2 n x1 ; ð1Þ ¼ E0 exp w20 w20 n¼0 n! where N ðN ¼ 0; 1; . . .Þ, the beam order and w0, the waist width, are two characteristic parameters of FGBs. E0 is a constant, which will be set to unity in the following. Assume there is a rectangular aperture, which is a in radius, at the plane of z ¼ 0. Just as the circ function can be expanded into a finite sum of complex Gaussian functions [6, 7], the rectangular function can be expressed as follows: m P Bm Am exp 2 x21 ; ð2Þ Hðx1 Þ ¼ a m¼1 where Am and Bm are expansion and Gaussian coefficients as shown in table 1. The propagation of FGBs passing through a misaligned optical system with aperture is governed by the generalized diffraction integral formulae for misaligned optical systems, which can be written as follows [8]: Eðx; zÞ rffiffiffiffiffiffi 1 ð i exp ðikl0 Þ Eðx1 ; 0Þ Hðx1 Þ ¼ lB 1 ik ðAx1 2x1 x þ Dx2 þ Ex1 þ GxÞ dx1 ; exp 2B (3) 0030-4026/04/115/05-193 $ 30.00/0
194
M. Shen et al., Propagation of flattened Gaussian beams passing through a misaligned optical system
has been used. Eq. (6) is the analytical equation of flattened Gaussian beams through misaligned optical systems with aperture, where the FGB is regarded as a whole beam. In the other hand, FGBs can be expanded in a finite sum of Hermite-Gaussian modes in the rectangular coordinate system, which is written as follows [8]:
Table 1. Expansion and Gaussian coefficients Am and Bm. Bm
Am 11.428 þ 0.95175i 0.06002 0.08013i 4:2734 8:5562i 1.6576 þ 2.7015i 5:0418 þ 3:2488i 1.1227 0.68854i 1:0106 0:26955i 2:5974 þ 3:2202i 0:14840 0:31193i 0:20850 0:23851i
4.0697 þ 0.22726i 1.1531 20.933i 4.4608 þ 5.1268i 4.3521 þ 14.997i 4.5443 þ 10.003i 3.8478 þ 20.078i 2.5280 10.310i 3.3197 4.8008i 1.9002 15.820i 2.6340 þ 25.009i
Eðx1 ; 0Þ
where k and l are the wave numer and wavelength, respectively. The parameters E and G take the following form: E ¼ 2ðaT ex þ bT e0x Þ ; G ¼ 2ðbgT daT Þex þ 2ðbdT dbT Þe0x ;
ð4Þ
e0x
denote the one-dimensional misawhere ex and ligned parameters, ex is the displacement, e0x is the tilting angle of the element. aT , bT , gT and dT represent the misaligned matrix elements determined by aT ¼ 1 A ; bT ¼ l B ; gT ¼ C ; dT ¼ 1 D :
N 1 1 P ð2lÞ! 3l l!ðl nÞ! n¼0 ð2nÞ! l¼n 2 "pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # 2ðN þ 1Þ ðN þ 1Þ 2 H2n x1 exp x1 : w0 w20
¼ E0
ð5Þ
Sign “þ” is for the case of the forward-going optical elements, and sign “” is for the case of the backwardgoing optical elements. Substituting eqs. (1) and (2) into (3), we obtain
N P
Substituting eqs. (8) and (2) into eq. (3), and using the integral formula
rffiffiffiffiffiffi ik ik exp ðDx2 þ GxÞ 2B 2B 9 8 > kð2x EÞ 2 > > > > > = < M P 4B Am exp N þ 1 ikA Bm > > m¼1 > > > þ 2> þ : a ; W02 2B 3nþ12 2 Nþ1 N 1 7 P w20 1 n 6 7 6 4N þ 1 ikA Bm 5 n! 4 n¼0 þ 2 þ 2B a w20 3 2 kð2x EÞ 7 6 7 6 4B 6 ffi7 H2n 6sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7: 4 N þ 1 ikA Bm 5 þ 2 þ a 2B w20
# ðx yÞ2 exp Hm ðxÞ dx 2u 1 pffiffiffiffiffiffiffiffi m y 2 p ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ¼ 2pu ð1 2uÞ Hm 1 2u we also can get
ikx ikDE2 ðG þ 2DEÞ 2B 2B N N 1 P 1 P ð2lÞ! E2n ðx; zÞ ; 3l l!ðl nÞ! n¼0 ð2nÞ! l¼n 2
Eðx; zÞ ¼ exp
n
1=2 ð2nþ1Þ
¼ ð1Þ p
2
y2 exp 2 4a
ð10Þ
E2n ðx; zÞ
ð6Þ
8 9 > E 2 D w20 Bm > > > > > Cþ 1þ < ik x = M P 2 q0 ðN þ 1Þ a2 ¼ Am exp > 2 B w20 Bm > m¼1 > > > > 1þ Aþ : ; q0 ðN þ 1Þ a2 n B w20 Bm A 1 q0 ðN þ 1Þa2 nþ1=2 B w20 Bm Aþ 1þ q0 ðN þ 1Þ a2 3 2 pffiffiffi 7 6 E 7 6 2 x 7 6 2 7 6 H2n 6vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ( ) 7; u 2 2 7 6u x2 i2BBm BlðN þ1Þ 5 4t 0 A2 þ N þ1 ka2 pw20 (11)
x2n exp ða2 x2 Þ cos ðxyÞ dx
0
ð9Þ
where
In the derivation of eq. (6), the integral formula 1 ð
"
1 ð
Eðx; zÞ
w0 ¼ pffiffiffiffiffiffiffiffiffiffiffi nþ1
ð8Þ
H2n
y 2a
where q0 ¼ ð7Þ
ipw20
lðN þ 1Þ plane of z ¼ 0.
is the beam q-parameter at the
M. Shen et al., Propagation of flattened Gaussian beams passing through a misaligned optical system
195
3. Numerical examples and discussion Let us consider a one-dimensional flattened Gaussian beam propagating through a system with a transverse shifted thin lens with focal length f, whose aperture effect is considered, followed by a free space. The displacements and angle misalignment of the lens with respect to the optical axis of the system are ex, e0x ¼ 0. The thin lens is located at the plane of z ¼ 0. The ABCD transfer matrix for such system reads A B Dz f ð1 þ DzÞ ¼ ; ð12Þ C D 1=f 1 where f denotes the focal length of the thin lens, z is the propagation distance after the lens, and Dz ¼ ðz f Þ=f :
ð13Þ
The misalignment parameters aT , bT , gT and dT take the following form: aT ¼ 1 þ Dz ;
bT ¼ 0 ;
gT ¼ 1=f ;
dT ¼ 0 ð14Þ
and the corresponding parameters E and G are given by E ¼ 2ð1 þ DzÞ ex ;
G ¼ 2ð1 þ DzÞex :
ð15Þ
Substituting eqs. (2) and (15) into eq. (6) and eq. (10), we can obtain the one-dimensional relative intensity distribution of the flattened Gaussian beam at several propagation distances after a misaligned thin lens with aperture as shown in fig. 1, where the FGBs are regarded as the whole beam and a finite series expansion, respectively. The parameters used in the calculation are l ¼ 632:8 nm, ex ¼ 1:0 mm, f ¼ 150 mm. The solid line denotes the intensity derived by using eq. (6) and the dotted line denotes that derived by using eq. (10). As can be seen, although eqs. (6) and (10) have different mathematical formulations, many numerical calculations have shown the same results.
a)
b)
Fig. 2. Relative intensity distribution of flattened Gaussian beam through a thin lens, followed by a free space, where the thin lens is aligned and the aperture of that is neglected. a) Dz ¼ 0:1; b) Dz ¼ 1.
In order to compare the propagation properties of flattened Gaussian beam through misaligned thin lens with aperture and through aligned thin lens, plot of 1-D intensity distribution of flattened Gaussian beam passing though an aligned thin lens at several propagation distances are also shown in fig. 2. From fig. 1 and fig. 2, we can find that the flattened Gaussian beam becomes a decectered flattened Gaussian beam after passing through the misaligned thin lens and the aperture diffraction effect results in the larger lobes.
4. Conclusions In conclusions, the propagation of FGBs passing through a misaligned optical system with aperture has been studied. The approximate equation for both forms of FGBs of such case have been derived, where the rectangular function is expanded into a sum of complex Gaussian functions, and numerical examples have been shown that the two results are equivalent. Numerical calculations are also given to illustrate the propagation properties of FGBs passing through misaligned optical systems with aperture. Actually, the method used here has a general applicable advantage and the approximate formulae derived in this paper provide a convenient tool to treat the propagation of FGBs through a misaligned optical system, where the aperture effect is been considered. Acknowledgments. The authors are grateful to the financial support of Natural Science Foundation of China, Grant No. 60276035.
a)
b)
Fig. 1. Relative intensity distribution of flattened Gaussian beam through a transverse shifted thin lens, followed by a free space, where the aperture of the thin lens is considered. a) Dz ¼ 0; b) Dz ¼ 0:1; c) Dz ¼ 1. c)
References [1] Gori F: Flattened Gaussian beams. Opt. Commun. 107 (1994) 335–341 [2] Bagini V, Borghi R, Gori F: Propagation of axially symmetric flattened Gaussian beams. J. Opt. Soc. Am. A 13 (1996) 1385–1394 [3] Santarsiero M, Aiello D, Borghi R: Focusing of axially symmetric flattened Gaussian beams. J. Mod. Opt. 44 (1997) 633–650 [4] Borghi R, Santarsiero M: Modal decomposition of partially coherent flat-topped beams produced by multimode lasers. Opt. Lett. 23 (1998) 313–315
196
M. Shen et al., Propagation of flattened Gaussian beams passing through a misaligned optical system
[5] Lu¨ B, Luo SR, Zhang B: Propagation of flattened Gaussian beams with rectangular symmetry passing through a paraxial optical ABCD system with and without aperture. Opt. Commun. 164 (1999) 1–6 [6] Wen JJ, Breazeale MA: A diffraction beam field expressed as the superposition of Gaussian beams. J. Acoust. Soc. Am. 83 (1988) 1752–1756
[7] Ding DS, Liu XJ: Approximate description of Bessel, Bessel-Gauss, and Gaussian beams with finite aperture. J. Opt. Soc. Am. A 16 (1999) 1286–1293 [8] Wang SM, Ronchi L: Principles and design of optical array. In: Wolf E (ed): Progress in optics, Vol. 25, pp. 279. Elsevier Science BV, Amsterdam 1998