- Email: [email protected]
Propagation of Hermite-Gaussian beams in apertured fractional Fourier transforming systems
Optik 114, No. 11 (2003) 504 Daomu Zhao504–508 et al., Propagation of Hermite-Gaussian beams in apertured fractional Fourier transforming systems http://www.elsevier-deutschland.de/ijleo International Journal for Light and Electron Optics
Propagation of Hermite-Gaussian beams in apertured fractional Fourier transforming systems Daomu Zhao1, Haidan Mao1, Hongjie Liu1, 2, Feng Jing2, Qihua Zhu2, Xiaofeng Wei2 1 2
Department of Physics, Zhejiang University, Hangzhou 310027, China Research Center of Laser Fusion, CAEP, Mianyang 621900, China
Abstract: The apertured fractional Fourier transform (FRFT) system is introduced and applied to treat the propagation of Hermite-Gaussian beams. Based on the treatment that a rectangular function can be expanded into a approximate sum of complex Gaussian functions with finite numbers, the analytical expressions for the output field distribution of a Hermite-Gaussian beam through an apertured FRFT system are obtained and compared with those obtained from numerically integral calculation. The results show that our method can significantly improve the numerical calculation efficiency. Key words: Hermite-Gaussian beams – fractional Fourier transform – aperture – diffraction integral
1. Introduction The fractional Fourier transform (FRFT) is regarded as a generalization of the conventional Fourier transform. In the 1980s Namias [1], McBride and Kerr [2] took up the FRFT for the benefit of quantum mechanics, but did not gain much attention until its reintroduction to optics in 1993 by Mendlovic and Ozaktas [3, 4] and Lohmann [5]. Since then a lot of works have been done on its properties, optical implementations, and applications [6–20]. For instance, the FRFT has been used as a new method in signal processing, physical optics, laser optics, etc. From the optical point of view there exist two different approaches to obtain definitions of a FRFT. One definition is based on Hermite-Gaussian-modes and can be implemented optically by means of GRIN media [3, 4]. The other approach is based on rotating the Wigner distribution function of the input signal by a certain angle and can be implemented by means of bulk optical setups [5]. Both definitions are essentially equivalent [9]. The two kinds of optical setups for im-
plementing p order FRFT suggested by Lohmann are shown in fig. 1, respectively. Where fs is the standard focal length and f ¼ pp=2. Generally, the FRFT system without aperture had been extensively studied and used. The apertured FRFT system, to the best of our knowledge, has been paid little attention. Practically, aperture always exists in FRFT systems, for example, the finite size of lens. Therefore study on the behavior of light propagation through the apertured FRFT systems would be of practical interest. The paper is organized as follows. The theoretical analyses of a Hermite-Gaussian beam through the two types of apertured FRFT systems are given in section 2. Some numerical simulation comparisons by using the derived analytical formulae and the diffraction integral formulae are given in section 3. Finally, a simple conclusion is outlined in section 4.
2. Approximate analytical formulae for propagation of Hermite-Gaussian beams through apertured FRFT systems In this paper, we only consider the case of the finite size of lens in FRFT systems. For the sake of simplicity, we just consider the one-dimensional case, the extension to the two-dimensional one is straightforward. In case of the type I shown in fig. 1a, the whole system can be separated into two parts, the first is the free propagation from the input plane to z1 plane, and the second is the apertured propagation from z1 plane to the output plane because of the finite size of lens. For the first one we have Eðx; z1 Þ ¼
Received 18 August 2003; accepted 5 November 2003. Correspondence to: D. Zhao Fax: ++86-571-88863887 E-mail: [email protected]
1 ð i 1=2 Eðx1 ; 0Þ ld 1
ip 2 ðx1 2x1 x þ x2 Þ dx1 exp ld
ð1Þ
0030-4026/03/114/11-504 $ 15.00/0
505
Daomu Zhao et al., Propagation of Hermite-Gaussian beams in apertured fractional Fourier transforming systems
f = f s / sin φ
d = f s tan
Fig. 1. The two optical setups for implementing FRFT. a) type I, b) type II.
0
φ
d = f s tan
2
f = f s / tan
φ
d = f s sin φ
Z1
Z
Z
0
i ld
1=2 ða
(b)
ing eqs. (1) and (5) into (4) and using the following integral formula
and for the second one we have
Eðx; z1 Þ
1 ð
a
ip exp ld
d 2 2 1 dx ; x 2xx2 þ x2 f ð2Þ
where a means the half-width of the lens aperture and the constant phase terms in eqs. (1) and (2) have been omitted which have no influence on the output intensity distribution. Introducing the aperture function 1; jxj a Ap ðxÞ ¼ ð3Þ 0; jxj > a
"
# ðx yÞ2 exp Hm ðxÞ dx 2u
1
¼ ð2puÞ1=2 ð1 2uÞm=2 Hm ½yð1 2uÞ1=2 ;
i ld
1=2 1 ð
Eðx2 ; zÞ ¼
exp
ip ld
1
d 2 dx : x 2xx2 þ x22 f ð4Þ
Generally, the aperture function can be expanded as the sum of complex Gaussian functions with finite numbers and is given by N P Bn 2 An exp 2 x ; ð5Þ Ap ðxÞ ¼ a n¼1 where An and Bn are the expansion and Gaussian coefficients, respectively, which could be obtained by optimization-computation directly [21]. The Hermite-Gaussian beam is an important type of laser beams. Let us consider a Hermite-Gaussian beam pffiffiffi represented by exp ðx2 =W02 Þ Hm ð 2x=W0 Þ is incident on the input plane of the apertured FRFT system, where W0 is beam waist size of Gaussian part. Insert-
N P
An
n¼1
1 Cn
1=2 1
B2 Cn
m=2
ip p2 þ 2 exp x2 ld l d2 Cn 2 " 1=2 # iBpx2 B2 1 ; Hm ldCn Cn
Eðx; z1 Þ Ap ðxÞ
1
ð6Þ
after performing tedious integration, an analytical expression of the output field distribution in FRFT plane is obtained
then eq. (2) becomes Eðx2 ; zÞ ¼
2
2
(a)
Eðx2 ; zÞ ¼
φ
ð7Þ
where B¼
pffiffiffi 1=2 i 2p 2ld 1 ; 1 ip ld þ ipW02 þ ldW0 W02 ld
Cn ¼
Bn ip þ þ a2 ld
l2 d 2
p2 ip þ 1 ip ld þ W02 ld
1
ð8Þ
d : f (9)
In case of the type II shown in fig. 1b, there exist two apertures on the respective positions of the lenses, in fact the second aperture only truncates the output field distribution. According to Collins’ diffraction integral formula [22] and using the similar way as the type I, the analytical expression of the output field distribution of a Hermite-Gaussian beam in FRFT plane is ob-
506
Daomu Zhao et al., Propagation of Hermite-Gaussian beams in apertured fractional Fourier transforming systems
tained
Table 1. The coefficients An and Bn with N = 10.
Eðx2 ; zÞ ¼
N P
1=2
m=2
1 2 1 2 Cn W0 Cn n¼1 ip f p2 1 exp x22 þ 2 ld d l d2 Cn " pffiffiffi 1=2 # i 2 px2 2 1 2 Hm l dW0 Cn W0 Cn An
ð9Þ where Cn ¼
Bn 1 ip þ 2þ 2 a W0 ld
d 1 : f
ð11Þ
3. Numerical simulation comparisons For comparing the results obtained by using the analytical expression with those by using the diffraction integral formula, some numerical simulations are done. In calculation, we choose the coefficients An and Bn of a rectangular function [21] as the table 1. Fig. 2 and fig. 3 show the relative intensity distributions in FRFT plane of a Hermite-Gaussian beam (m ¼ 1) passing through the two types of apertured FRFT systems, respectively, where we choose a=W0 ¼ 1. From the figures we find
N 1 2 3 4 5 6 7 8 9 10
An 11.428 þ 0.95175i 0.06002 0.08013i 4.2743 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 0.1484 0.31193i 0.2085 0.23851i
Bn 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 20.087i 2.528 10.31i 3.3197 4.8008i 1.9002 15.82i 2.634 þ 25.009i
that the simulation results obtained by using the analytical expression are in keeping with those by using the numerical integral calculation quite well except for some slight decinations outside the center part, and the two types of apertured FRFT systems are no longer equivalent. When p 0:5, the simulation results have enough high consistency, especially on the Fourier transform plane. It should be pointed out that the numerical calculation efficiency by using the analytical expression is over 100 times than that by using the numerical integral calculation. Meanwhile, it should also be pointed out that the calculation accuracy could be improved if the N is increased. Furthermore, after some numerical simulations we find that the diffraction effect of the lens aperture can be neglected when a=W0 > 4.
Fig. 2. Relative intensity distribution in the FRFT output plane of the type I with the incidence of a HermiteGaussian beam (m ¼ 1) by using the two methods. The solid lines mean the case by using the analytical formula, and the dotted lines denote the case by using the diffraction integral formula. a) p ¼ 1; b) p ¼ 0.5; c) p ¼ 0.2; d) p ¼ 0.1.
Daomu Zhao et al., Propagation of Hermite-Gaussian beams in apertured fractional Fourier transforming systems
507
Fig. 3. Relative intensity distribution in the FRFT output plane of the type II with the incidence of a Hermite-Gaussian beam (m ¼ 1) by using the two methods. The others are the same as fig. 2.
4. Conclusions In conclusion, the apertured FRFT system is introduced and applied to treat the propagation of Hermite-Gaussian beams, and the analytical expressions for a Hermite-Gaussian beam through two types of apertured FRFT systems are obtained and compared with those by using the numerical integral calculation directly. It is shown that our method can significantly improve the numerical calculation efficiency. The results can be extended to the two-dimensional case, and the propagation of a Gaussian beam or a plane wave passing through the apertured FRFT systems can be regarded as the special cases. Acknowledgements. This work is supported by the National Natural Science Foundation of China (NSAF United Foundation), Grant No. 10276034.
References [1] Namias V: The fractional order Fourier transform and its applications to quantum mechanics. J. Inst. Math. Its Appl. 25 (1980) 241–265 [2] McBride AC, Kerr FH: On Namias fractional Fourier transform. IMA J. Appl. Math. 39 (1987) 159–175 [3] Mendlovic D, Ozaktas HM: Fractional Fourier transforms and their optical implementation: I. J. Opt. Soc. Am. A 10 (1993) 1875–1881
[4] Ozaktas HM, Mendlovic D: Fractional Fourier transforms and their optical implementation:II. J. Opt. Soc. Am. A 10 (1993) 2522–2531 [5] Lohmann AW: Image rotation, Wigner rotation, and the fractional Fourier transform. J. Opt. Soc. Am. A 10 (1993) 2181–2186 [6] Lohmann AW, Mendlovic D, Zalevsky Z: Fractional transformations in optics. In Wolf E (ed.): Progress in optics. Elsevier, Amsterdam, 1998 [7] Ozaktas HM, Zalevsky Z, Kutay MA: The Fractional Fourier Transform with applications in optics and signal processing. Wiley, New York 2000 [8] Torre A: The fractional Fourier transfomr and some of its applications to optics. In Wolf E (ed.): Progress in optics. Elsevier, Amsterdam 2002 [9] Mendlovic D, Ozaktas HM, Lohmann AW: Graded index fibers, Wigner-distribution functions, and the fractional Fourier transforms. Appl. Opt. 33 (1994) 6188–6193 [10] Pallat-Finet P: Fresnel diffraction and the fractionalorder Fourier transforms. Opt. Lett. 19 (1994) 1388– 1390 [11] Ozaktas HM, Mendlovic D: Fractional Fourier transforms as a tool for analyzing beam propagation and spherical mirror resonators. Opt. Lett. 19 (1994) 1678–1680 [12] Shih C: Optical interpretation of a complex-order Fourier transform. Opt. Lett. 20 (1995) 1178–1180 [13] Zalevsky Z, Mendlovic D, Dorsch RG: Gerchberg-Saxton algorithm applied in the fractional Fourier or the Fresnel domain. Opt. Lett. 21 (1996) 842–844 [14] Sahin A, Ozaktas HM, Mendlovic D: Optical implenations of two-dimensional fractional Fourier transform and linear canonical transform with arbitrary parameters. Appl. Opt. 37 (1998) 2130–2141 [15] Dragoman D, Dragoman M: Temporal implementation of Fourier-related transforms. Opt. Commun. 145 (1998) 33– 37
508
Daomu Zhao et al., Propagation of Hermite-Gaussian beams in apertured fractional Fourier transforming systems
[16] Zhao D: Multielement resonators and scaled fractional Fourier transforms. Opt. Commun. 168 (1999) 85–88 [17] Zhao D, Wang S: Effect of misalignment on optical fractional Fourier transforming systems. Opt. Commun. 198 (2001) 281–286 [18] Zhao D: A thick lens as a fractional Fourier transformer. Optik 111 (2000) 35–37 [19] Zhao D: Collins formula in frequency-domain described by fractional Fourier transform or fractional Hankel transform. Optik 111 (2000) 9–12
[20] Wang S, Zhao D: Matrix optics. Chapt. 2. CHEP-Springer, Beijing 2000 [21] Wen JJ, Breazeale MA: A diffraction beam field expressed as the superposition of Gaussian beams. J. Acoust. Soc. Am. 83 (1988) 1752–1756 [22] Collins SA: Lens-system diffraction integral written in terms of matrix optics. J. Opt. Soc. Am. 60 (1970) 1168– 1177
Propagation of Hermite-Gaussian beams in apertured fractional Fourier transforming systems Daomu Zhao1, Haidan Mao1, Hongjie Liu1, 2, Feng Jing2, Qihua Zhu2, Xiaofeng Wei2 1 2
Department of Physics, Zhejiang University, Hangzhou 310027, China Research Center of Laser Fusion, CAEP, Mianyang 621900, China
Abstract: The apertured fractional Fourier transform (FRFT) system is introduced and applied to treat the propagation of Hermite-Gaussian beams. Based on the treatment that a rectangular function can be expanded into a approximate sum of complex Gaussian functions with finite numbers, the analytical expressions for the output field distribution of a Hermite-Gaussian beam through an apertured FRFT system are obtained and compared with those obtained from numerically integral calculation. The results show that our method can significantly improve the numerical calculation efficiency. Key words: Hermite-Gaussian beams – fractional Fourier transform – aperture – diffraction integral
1. Introduction The fractional Fourier transform (FRFT) is regarded as a generalization of the conventional Fourier transform. In the 1980s Namias [1], McBride and Kerr [2] took up the FRFT for the benefit of quantum mechanics, but did not gain much attention until its reintroduction to optics in 1993 by Mendlovic and Ozaktas [3, 4] and Lohmann [5]. Since then a lot of works have been done on its properties, optical implementations, and applications [6–20]. For instance, the FRFT has been used as a new method in signal processing, physical optics, laser optics, etc. From the optical point of view there exist two different approaches to obtain definitions of a FRFT. One definition is based on Hermite-Gaussian-modes and can be implemented optically by means of GRIN media [3, 4]. The other approach is based on rotating the Wigner distribution function of the input signal by a certain angle and can be implemented by means of bulk optical setups [5]. Both definitions are essentially equivalent [9]. The two kinds of optical setups for im-
plementing p order FRFT suggested by Lohmann are shown in fig. 1, respectively. Where fs is the standard focal length and f ¼ pp=2. Generally, the FRFT system without aperture had been extensively studied and used. The apertured FRFT system, to the best of our knowledge, has been paid little attention. Practically, aperture always exists in FRFT systems, for example, the finite size of lens. Therefore study on the behavior of light propagation through the apertured FRFT systems would be of practical interest. The paper is organized as follows. The theoretical analyses of a Hermite-Gaussian beam through the two types of apertured FRFT systems are given in section 2. Some numerical simulation comparisons by using the derived analytical formulae and the diffraction integral formulae are given in section 3. Finally, a simple conclusion is outlined in section 4.
2. Approximate analytical formulae for propagation of Hermite-Gaussian beams through apertured FRFT systems In this paper, we only consider the case of the finite size of lens in FRFT systems. For the sake of simplicity, we just consider the one-dimensional case, the extension to the two-dimensional one is straightforward. In case of the type I shown in fig. 1a, the whole system can be separated into two parts, the first is the free propagation from the input plane to z1 plane, and the second is the apertured propagation from z1 plane to the output plane because of the finite size of lens. For the first one we have Eðx; z1 Þ ¼
Received 18 August 2003; accepted 5 November 2003. Correspondence to: D. Zhao Fax: ++86-571-88863887 E-mail: [email protected]
1 ð i 1=2 Eðx1 ; 0Þ ld 1
ip 2 ðx1 2x1 x þ x2 Þ dx1 exp ld
ð1Þ
0030-4026/03/114/11-504 $ 15.00/0
505
Daomu Zhao et al., Propagation of Hermite-Gaussian beams in apertured fractional Fourier transforming systems
f = f s / sin φ
d = f s tan
Fig. 1. The two optical setups for implementing FRFT. a) type I, b) type II.
0
φ
d = f s tan
2
f = f s / tan
φ
d = f s sin φ
Z1
Z
Z
0
i ld
1=2 ða
(b)
ing eqs. (1) and (5) into (4) and using the following integral formula
and for the second one we have
Eðx; z1 Þ
1 ð
a
ip exp ld
d 2 2 1 dx ; x 2xx2 þ x2 f ð2Þ
where a means the half-width of the lens aperture and the constant phase terms in eqs. (1) and (2) have been omitted which have no influence on the output intensity distribution. Introducing the aperture function 1; jxj a Ap ðxÞ ¼ ð3Þ 0; jxj > a
"
# ðx yÞ2 exp Hm ðxÞ dx 2u
1
¼ ð2puÞ1=2 ð1 2uÞm=2 Hm ½yð1 2uÞ1=2 ;
i ld
1=2 1 ð
Eðx2 ; zÞ ¼
exp
ip ld
1
d 2 dx : x 2xx2 þ x22 f ð4Þ
Generally, the aperture function can be expanded as the sum of complex Gaussian functions with finite numbers and is given by N P Bn 2 An exp 2 x ; ð5Þ Ap ðxÞ ¼ a n¼1 where An and Bn are the expansion and Gaussian coefficients, respectively, which could be obtained by optimization-computation directly [21]. The Hermite-Gaussian beam is an important type of laser beams. Let us consider a Hermite-Gaussian beam pffiffiffi represented by exp ðx2 =W02 Þ Hm ð 2x=W0 Þ is incident on the input plane of the apertured FRFT system, where W0 is beam waist size of Gaussian part. Insert-
N P
An
n¼1
1 Cn
1=2 1
B2 Cn
m=2
ip p2 þ 2 exp x2 ld l d2 Cn 2 " 1=2 # iBpx2 B2 1 ; Hm ldCn Cn
Eðx; z1 Þ Ap ðxÞ
1
ð6Þ
after performing tedious integration, an analytical expression of the output field distribution in FRFT plane is obtained
then eq. (2) becomes Eðx2 ; zÞ ¼
2
2
(a)
Eðx2 ; zÞ ¼
φ
ð7Þ
where B¼
pffiffiffi 1=2 i 2p 2ld 1 ; 1 ip ld þ ipW02 þ ldW0 W02 ld
Cn ¼
Bn ip þ þ a2 ld
l2 d 2
p2 ip þ 1 ip ld þ W02 ld
1
ð8Þ
d : f (9)
In case of the type II shown in fig. 1b, there exist two apertures on the respective positions of the lenses, in fact the second aperture only truncates the output field distribution. According to Collins’ diffraction integral formula [22] and using the similar way as the type I, the analytical expression of the output field distribution of a Hermite-Gaussian beam in FRFT plane is ob-
506
Daomu Zhao et al., Propagation of Hermite-Gaussian beams in apertured fractional Fourier transforming systems
tained
Table 1. The coefficients An and Bn with N = 10.
Eðx2 ; zÞ ¼
N P
1=2
m=2
1 2 1 2 Cn W0 Cn n¼1 ip f p2 1 exp x22 þ 2 ld d l d2 Cn " pffiffiffi 1=2 # i 2 px2 2 1 2 Hm l dW0 Cn W0 Cn An
ð9Þ where Cn ¼
Bn 1 ip þ 2þ 2 a W0 ld
d 1 : f
ð11Þ
3. Numerical simulation comparisons For comparing the results obtained by using the analytical expression with those by using the diffraction integral formula, some numerical simulations are done. In calculation, we choose the coefficients An and Bn of a rectangular function [21] as the table 1. Fig. 2 and fig. 3 show the relative intensity distributions in FRFT plane of a Hermite-Gaussian beam (m ¼ 1) passing through the two types of apertured FRFT systems, respectively, where we choose a=W0 ¼ 1. From the figures we find
N 1 2 3 4 5 6 7 8 9 10
An 11.428 þ 0.95175i 0.06002 0.08013i 4.2743 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 0.1484 0.31193i 0.2085 0.23851i
Bn 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 20.087i 2.528 10.31i 3.3197 4.8008i 1.9002 15.82i 2.634 þ 25.009i
that the simulation results obtained by using the analytical expression are in keeping with those by using the numerical integral calculation quite well except for some slight decinations outside the center part, and the two types of apertured FRFT systems are no longer equivalent. When p 0:5, the simulation results have enough high consistency, especially on the Fourier transform plane. It should be pointed out that the numerical calculation efficiency by using the analytical expression is over 100 times than that by using the numerical integral calculation. Meanwhile, it should also be pointed out that the calculation accuracy could be improved if the N is increased. Furthermore, after some numerical simulations we find that the diffraction effect of the lens aperture can be neglected when a=W0 > 4.
Fig. 2. Relative intensity distribution in the FRFT output plane of the type I with the incidence of a HermiteGaussian beam (m ¼ 1) by using the two methods. The solid lines mean the case by using the analytical formula, and the dotted lines denote the case by using the diffraction integral formula. a) p ¼ 1; b) p ¼ 0.5; c) p ¼ 0.2; d) p ¼ 0.1.
Daomu Zhao et al., Propagation of Hermite-Gaussian beams in apertured fractional Fourier transforming systems
507
Fig. 3. Relative intensity distribution in the FRFT output plane of the type II with the incidence of a Hermite-Gaussian beam (m ¼ 1) by using the two methods. The others are the same as fig. 2.
4. Conclusions In conclusion, the apertured FRFT system is introduced and applied to treat the propagation of Hermite-Gaussian beams, and the analytical expressions for a Hermite-Gaussian beam through two types of apertured FRFT systems are obtained and compared with those by using the numerical integral calculation directly. It is shown that our method can significantly improve the numerical calculation efficiency. The results can be extended to the two-dimensional case, and the propagation of a Gaussian beam or a plane wave passing through the apertured FRFT systems can be regarded as the special cases. Acknowledgements. This work is supported by the National Natural Science Foundation of China (NSAF United Foundation), Grant No. 10276034.
References [1] Namias V: The fractional order Fourier transform and its applications to quantum mechanics. J. Inst. Math. Its Appl. 25 (1980) 241–265 [2] McBride AC, Kerr FH: On Namias fractional Fourier transform. IMA J. Appl. Math. 39 (1987) 159–175 [3] Mendlovic D, Ozaktas HM: Fractional Fourier transforms and their optical implementation: I. J. Opt. Soc. Am. A 10 (1993) 1875–1881
[4] Ozaktas HM, Mendlovic D: Fractional Fourier transforms and their optical implementation:II. J. Opt. Soc. Am. A 10 (1993) 2522–2531 [5] Lohmann AW: Image rotation, Wigner rotation, and the fractional Fourier transform. J. Opt. Soc. Am. A 10 (1993) 2181–2186 [6] Lohmann AW, Mendlovic D, Zalevsky Z: Fractional transformations in optics. In Wolf E (ed.): Progress in optics. Elsevier, Amsterdam, 1998 [7] Ozaktas HM, Zalevsky Z, Kutay MA: The Fractional Fourier Transform with applications in optics and signal processing. Wiley, New York 2000 [8] Torre A: The fractional Fourier transfomr and some of its applications to optics. In Wolf E (ed.): Progress in optics. Elsevier, Amsterdam 2002 [9] Mendlovic D, Ozaktas HM, Lohmann AW: Graded index fibers, Wigner-distribution functions, and the fractional Fourier transforms. Appl. Opt. 33 (1994) 6188–6193 [10] Pallat-Finet P: Fresnel diffraction and the fractionalorder Fourier transforms. Opt. Lett. 19 (1994) 1388– 1390 [11] Ozaktas HM, Mendlovic D: Fractional Fourier transforms as a tool for analyzing beam propagation and spherical mirror resonators. Opt. Lett. 19 (1994) 1678–1680 [12] Shih C: Optical interpretation of a complex-order Fourier transform. Opt. Lett. 20 (1995) 1178–1180 [13] Zalevsky Z, Mendlovic D, Dorsch RG: Gerchberg-Saxton algorithm applied in the fractional Fourier or the Fresnel domain. Opt. Lett. 21 (1996) 842–844 [14] Sahin A, Ozaktas HM, Mendlovic D: Optical implenations of two-dimensional fractional Fourier transform and linear canonical transform with arbitrary parameters. Appl. Opt. 37 (1998) 2130–2141 [15] Dragoman D, Dragoman M: Temporal implementation of Fourier-related transforms. Opt. Commun. 145 (1998) 33– 37
508
Daomu Zhao et al., Propagation of Hermite-Gaussian beams in apertured fractional Fourier transforming systems
[16] Zhao D: Multielement resonators and scaled fractional Fourier transforms. Opt. Commun. 168 (1999) 85–88 [17] Zhao D, Wang S: Effect of misalignment on optical fractional Fourier transforming systems. Opt. Commun. 198 (2001) 281–286 [18] Zhao D: A thick lens as a fractional Fourier transformer. Optik 111 (2000) 35–37 [19] Zhao D: Collins formula in frequency-domain described by fractional Fourier transform or fractional Hankel transform. Optik 111 (2000) 9–12
[20] Wang S, Zhao D: Matrix optics. Chapt. 2. CHEP-Springer, Beijing 2000 [21] Wen JJ, Breazeale MA: A diffraction beam field expressed as the superposition of Gaussian beams. J. Acoust. Soc. Am. 83 (1988) 1752–1756 [22] Collins SA: Lens-system diffraction integral written in terms of matrix optics. J. Opt. Soc. Am. 60 (1970) 1168– 1177