- Email: [email protected]
Collins formula in frequency-domain and fractional Fourier transforms
1 October 1998
Optics Communications 155 Ž1998. 7–11
Collins formula in frequency-domain and fractional Fourier transforms Zhongyong Liu, Xiuying Wu, Dianyuan Fan Shanghai Institute of Optics and Fine Mechanics, Academia Sinica, P.O. Box 800216, Shanghai 201800, China Received 15 April 1998; accepted 22 June 1998
Abstract Collins formula in frequency-domain has been derived. This formula gives the direct relationship between input and output spatial frequency spectra of a light field. Furthermore, by applying this formula the fractional Fourier transform ŽFRT. in frequency-domain can be investigated. It has been found that an arbitrary reciprocally symmetric ABCD optical system ŽRSOS. can implement the FRT in both space-domain and frequency-domain, with the same order. In other words, the FRT is a symmetric transform that performs the same operation in both space-domain and frequency-domain. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Collins formula in frequency-domain; Fractional Fourier transform; Reciprocally symmetric optical system; Symmetric transform
1. Introduction When investigating the light propagation through an ABCD optical system, one usually uses the diffraction integral of such a system, i.e. the so-called Collins formula w1x, which gives the relationship between the output and input complex amplitude distributions of the light field, so we may call this formula Collins formula in space-domain. On the other hand, the spatial frequency transfer properties through this kind of optical system are also important. The transfer function is very useful for describing an image-forming optical system in the frequency-domain. However, an arbitrary ABCD system is not, generally, space-invariant for the complex amplitude of the light field, as a result there does not exist a corresponding coherent transfer function ŽCTF. for this system, only an effective one can be found w2x. It is worthwhile to investigate the transfer law of the spatial Fourier frequency spectrum which is also called plane-wave angle spectrum, directly in the frequency-domain. In this paper, the Collins diffraction integral formula in frequency-domain, which establishes the input–output relationship of the angle spectrum, has been derived. In addition, fractional Fourier transforms ŽFRTs. have been examined extensively in recent years w3–11x. We concluded in this paper that an arbitrary reciprocally symmetric optical system ŽRSOS. can implement a FRT of the complex amplitude of the light field. Applying the Collins formula in frequency-domain derived in this paper, we discuss the FRT in frequency-domain, i.e. the FRT of the angle spectrum. It is found that an arbitrary RSOS can implement the FRT not only in space-domain but also in frequency-domain at the same time, with the same order. In other words, FRT is a symmetric transform that performs simultaneously the same operation in both space-domain and frequency-domain.
2. Collins formula in the frequency-domain Collins formula in space-domain which gives the relationship between the input complex amplitude u1Ž x 1, y 1 . and the output one u 2 Ž x 2 , y 2 ., provided that the optical system is lossless and rotationally symmetric and the input and output 0030-4018r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 8 . 0 0 3 4 3 - 5
Z. Liu et al.r Optics Communications 155 (1998) 7–11
8
medium is the same, is known as w1x u2 Ž x 2 , y2 . s y
i
HHu Ž x , y . exp lB 1
1
1
½
ip
A Ž x 12 q y 12 . q D Ž x 22 q y 22 . y 2 Ž x 1 x 2 q y 1 y 2 .
lB
5
d x 1 d y1 ,
Ž1.
where, A, B, C and D are the elements of the ray transfer matrix of the optical system, l the wavelength. Applying a Fourier transform on both sides of Eq. Ž1. yields
HHu Ž x 2
2 , y2
=
HHexp
. exp yi2p Ž nx 2 x 2 q n y 2 y 2 . d x 2 d y 2 s y ip
½
D Ž x 22 q y 22 . y 2 Ž x 1 x 2 q y1 y 2 .
lB
5
i
HHu Ž x , y . exp lB 1
1
1
ip
lB
A Ž x 12 q y 12 .
exp yi2p Ž nx 2 x 2 q n y 2 y 2 . d x 2 d y 2 d x 1 d y 1 ,
the term on the left-hand side of the above equation is just the output angle spectrum, then we have A 2 Ž nx 2 , n y 2 . s
1 D
exp y
=exp
ipl B D
ip
½ ž lB
Ž nx22 q ny22 . HHu1Ž x 1 , y1 .
Ay
1 D
/Ž
x 12 q y 12 . y
2lB D
Ž nx 2 x 1 q n y 2 y1 .
5
d x 1 d y1 .
Ž2.
Writing u1Ž x 1, y 1 . in terms of the input angle spectrum as u1Ž x 1 , y 1 . s
HH A Ž n 1
x1 , n y1
. exp i2p Ž nx1 x 1 q n y1 y1 . d nx1 d n y1 ,
Ž3.
and substituting Eq. Ž3. into Ž2., we get A 2 Ž nx 2 , n y 2 . s
il
HH A Ž n C 1
x1 , n y1
½
. exp y
ipl
2 2 D Ž nx1 q n y1 . q A Ž nx22 q ny22 . y 2 Ž nx1nx 2 q ny1ny 2 .
C
5
d nx1 d n y1 .
Ž4. The above equation is just the wanted Collins formula in frequency-domain, which gives the relationship between the input and output angle spectra through a general ABCD optical system. One can know from Eq. Ž4. that for a general ABCD system, the angle spectrum propagation follows a law which is very similar to that of the complex amplitude propagation, but the changes in the position of the matrix elements should be especially noted. Eq. Ž4. can be more convenient for studies on light propagation and transform in frequency-domain. Now special attention should be given to those situations where the matrix elements A, B, C and D are equal to zero, respectively. For C s 0, using the property of the d-function, Eq. Ž4. becomes A 2 Ž nx 2 , n y 2 . s
1 D
exp y
ipl B D
Ž nx22 q ny22 .
A1
ž
nx 2 n y2 , . D D
/
Ž5.
It can be seen that there exists a simple input–output relationship of the angle spectrum. In view of the similarity between Eq. Ž5. and that in space-domain when B s 0 Žcorresponding to the image-forming system in space-domain., we can call the system where C s 0 the image-forming in frequency-domain. For D s 0, from Eq. Ž4. one can find A 2 Ž nx 2 , n y 2 . s y
i
lB
exp ipl AB Ž nx22 q n y22 .
HH A Ž n 1
x1 , n y1
. exp yi2pl B Ž nx 2 nx1 q n y 2 n y1 . d nx1 d n y1 .
Ž6.
Obviously, the above equation can be rewritten as A 2 Ž nx 2 , n y 2 . s y
i
lB
exp ipl AB Ž nx22 q n y22 . u1 Ž yl Bnx 2 ,y l Bn y 2 . ,
Ž7.
where u1Ž x 1, y 1 . is the complex amplitude distribution of the input field. In this case, the input plane is just the equivalent object focal plane of the optical system.
Z. Liu et al.r Optics Communications 155 (1998) 7–11
9
For B s 0, one can get il
A 2 Ž nx 2 , n y 2 . s
HH A Ž n C 1
x1 , n y1
½
. exp y
ipl C
2 2 D Ž nx1 q n y1 .q
1 D
Ž nx22 q ny22 . y 2 Ž nx1nx 2 q ny1ny 2 .
5
d nx1 d n y1 .
Ž8. It is known that the case when B s 0 is related to the image-forming in space-domain.
3. Fractional Fourier transforms in the frequency-domain It is well-known that the fractional Fourier transform ŽFRT. performs the following operation w5x, u2 Ž x 2 , y2 . s
HHu Ž x , y . exp 1
1
1
ip
l f1
Ž x 2 q y12 q x 22 q y 22 . tan f 1
exp y
i2p
l f 1 sin f
Ž x 1 x 2 q y1 y 2 . d x 1 d y1 ,
Ž9.
where f 1 is called the standard focal length which decides the scale factor of the coordinates, the order is given by P s 2 frp , and when P s 1 the transform goes back to the conventional Fourier transform. Now let us consider how to implement the FRT by using the ABCD optical system. We note that if we assume that the following relations hold A s D s cos f ,
yf 1C s Brf 1 s sin f ,
Ž 10 .
substituting into Eq. Ž1., we can rewritte Eq. Ž1. as u2 Ž x 2 , y2 . s y
ip
i
l f1
HHu Ž x , y . exp sin f 1
=exp y
i2p
l f 1 sin f
1
1
l f 1 tan f
Ž x 12 q y12 q x 22 q y 22 .
Ž x 1 x 2 q y1 y 2 . d x 1 d y1 .
Ž 11.
Obviously, Eq. Ž11. is just a FRT except for an unimportant scaling constant. Then it can be concluded that any ABCD optical system satisfying the relation Ž10. can implement a FRT. On the other hand, let us recall the concept of reciprocally symmetric optical system ŽRSOS. in geometric optics, the RSOS is a system that performs the same operation when used either in positive direction or in its converse along the optical axis. If using the terms of matrix optics, we can find the following conditions for an ABCD system to be a RSOS A s D,
AD y BC s 1,
Ž 12.
where the latter relation requires that the input and output medium is the same, which is usually satisfied. Thus the former relation A s D becomes relatively more important. A simple comparison between Eqs. Ž10. and Ž12. allows us to find that conditions Ž10. must lead to conditions Ž12.. Consequently, we can say that an arbitrary RSOS can implement a FRT. It is worthwhile to point out that from relations Ž10. there exist FRTs with not only real but also complex orders according to the ray matrix elements A, D and the standard focal length f 1. Especially for FRTs with complex orders, one can find them in several references, such as Refs. w8x and w9x, etc. Now we focus our attention on the transform of the angle spectrum performed by a RSOS. Here in order to give a comparison between the operations in space-domain and frequency-domain, we assume that the RSOS satisfies the following relations, A s D s cos f ,
yCrl2 f 2 s l2 f 2 B s sin f .
Ž 13.
Using the above equations in Eq. Ž4. yields that A 2 Ž nx 2 , n y 2 . s y
i
l f 2 sin f
= exp y
HH A Ž n 1
i2p
l f 2 sin f
x1 , n y 2
. exp
ip
l f 2 tan f
Ž nx12 q ny12 q nx22 q ny22 .
Ž nx1 nx 2 q n y1 n y 2 . d nx1 d n y1 .
Ž 14.
Z. Liu et al.r Optics Communications 155 (1998) 7–11
10
Fig. 1. Setup for observing the FRT of the angle spectrum. The order is P s Ž2rp .arccosŽ1 y frZ ., where f is the focal length of the two lenses. By changing the distance Z, various orders can be obtained.
Obviously, this is a FRT describing the operation in frequency-domain, and it has the same order as that in space-domain by Eq. Ž11., although the scale factor may be different due to the difference between f 2 and f 1. We can conclude that an arbitrary RSOS can implement a FRT in both space-domain and frequency-domain with the same order. On the other hand, we introduced the concept of symmetric transform w12x, which is a kind of transform that implements the same operation at the same time in both space-domain and frequency-domain. Thus the FRT is, in other words, a symmetric transform. This feature can also be derived from the pure mathematical properties of the FRT as w6x F Ž a . F Žb . w u Ž x , y . x 4 s F Ž b . F Ža . w u Ž x , y . x 4 s F Ž a q b . w u Ž x , y . x ,
Ž 15 .
Ža .
where F represents the FRT with the order of a . Consider a FRT of function u1Ž x 1, y 1 . in space-domain with the order of P as follows, u 2 Ž x 2 , y 2 . s F Ž P . w u1Ž x 1 , y 1 . x . Then performing a conventional Fourier transform on both sides of the above equation, and taking into account relations Ž15., we can get F Ž1. w u 2 Ž x 2 , y 2 . x s F Ž1. F Ž P . w u1Ž x 1 , y 1 . x 4 s F Ž P . F Ž 1 . w u1Ž x 1 , y 1 . x 4 s F Ž P . A1Ž nx1 , n y1 . 4 , so we have A 2 Ž nx 2 , n y 2 . s F Ž P . A1 Ž nx1 , n y1 . , which is just the FRT in frequency-domain with the same order P. In order to observe the FRT of the angle spectrum, one can use the simple optical setup shown in Fig. 1. But it is worthwhile to note that the distributions on the Fourier plane and output plane are not directly the actual angle spectrum distributions, but after scaling the coordinates, one can get the actual angle spectra.
4. Conclusion The propagation and transfer of the plane-wave angle spectrum of the optical signal through an arbitrary ABCD optical system are found to be very similar to those of the complex amplitude, which can be described by the Collins diffraction integral formula in frequency-domain. In this paper, we have derived this frequency-domain Collins formula. In addition, by applying this formula we discussed the fractional Fourier transform in frequency-domain, it has been found that any arbitrary reciprocally symmetric ABCD optical system can implement simultaneously the FRT in both space-domain and frequencydomain. In other words, FRT belongs to such a kind of symmetric transform which can perform the same operation in both space-domain and frequency-domain.
Acknowledgements This work is supported by the key project of the National Natural Science Foundation and National Hi-Tech Foundation of China. The authors are grateful to the reviewers for useful comments.
Z. Liu et al.r Optics Communications 155 (1998) 7–11
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x
S.A. Collins, J. Opt. Soc. Am. 60 Ž1970. 1168. D. Fan, Acta Optica Sinica Žin Chinese. 1 Ž1981. 395. D. Mendlovic, H.M. Ozaktas, J. Opt. Soc. Am. A 10 Ž1993. 1875. H.M. Ozaktas, D. Mendlovic, J. Opt. Soc. Am. A 10 Ž1993. 2522. A.W. Lohmann, J. Opt. Soc. Am. A 10 Ž1993. 2181. V. Namias, J. Inst. Math. Appl. 25 Ž1980. 241. L.M. Bernardo, O.D.D. Soares, Optics Comm. 110 Ž1994. 517. L.M. Bernardo, O.D.D. Soares, Appl. Optics 35 Ž1996. 3163. C. Shih, Optics Lett. 20 Ž1995. 1178. T. Alieva, V. Lopez, F. Agullo-Lopez, L.B. Almeida, J. Mod. Optics 41 Ž1994. 1037. S. Abe, J.T. Sheridan, Optics Comm. 137 Ž1997. 214. Z. Liu, X. Wu, D. Fan, J. Opt. Soc. Am. A, submitted.
11
Optics Communications 155 Ž1998. 7–11
Collins formula in frequency-domain and fractional Fourier transforms Zhongyong Liu, Xiuying Wu, Dianyuan Fan Shanghai Institute of Optics and Fine Mechanics, Academia Sinica, P.O. Box 800216, Shanghai 201800, China Received 15 April 1998; accepted 22 June 1998
Abstract Collins formula in frequency-domain has been derived. This formula gives the direct relationship between input and output spatial frequency spectra of a light field. Furthermore, by applying this formula the fractional Fourier transform ŽFRT. in frequency-domain can be investigated. It has been found that an arbitrary reciprocally symmetric ABCD optical system ŽRSOS. can implement the FRT in both space-domain and frequency-domain, with the same order. In other words, the FRT is a symmetric transform that performs the same operation in both space-domain and frequency-domain. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Collins formula in frequency-domain; Fractional Fourier transform; Reciprocally symmetric optical system; Symmetric transform
1. Introduction When investigating the light propagation through an ABCD optical system, one usually uses the diffraction integral of such a system, i.e. the so-called Collins formula w1x, which gives the relationship between the output and input complex amplitude distributions of the light field, so we may call this formula Collins formula in space-domain. On the other hand, the spatial frequency transfer properties through this kind of optical system are also important. The transfer function is very useful for describing an image-forming optical system in the frequency-domain. However, an arbitrary ABCD system is not, generally, space-invariant for the complex amplitude of the light field, as a result there does not exist a corresponding coherent transfer function ŽCTF. for this system, only an effective one can be found w2x. It is worthwhile to investigate the transfer law of the spatial Fourier frequency spectrum which is also called plane-wave angle spectrum, directly in the frequency-domain. In this paper, the Collins diffraction integral formula in frequency-domain, which establishes the input–output relationship of the angle spectrum, has been derived. In addition, fractional Fourier transforms ŽFRTs. have been examined extensively in recent years w3–11x. We concluded in this paper that an arbitrary reciprocally symmetric optical system ŽRSOS. can implement a FRT of the complex amplitude of the light field. Applying the Collins formula in frequency-domain derived in this paper, we discuss the FRT in frequency-domain, i.e. the FRT of the angle spectrum. It is found that an arbitrary RSOS can implement the FRT not only in space-domain but also in frequency-domain at the same time, with the same order. In other words, FRT is a symmetric transform that performs simultaneously the same operation in both space-domain and frequency-domain.
2. Collins formula in the frequency-domain Collins formula in space-domain which gives the relationship between the input complex amplitude u1Ž x 1, y 1 . and the output one u 2 Ž x 2 , y 2 ., provided that the optical system is lossless and rotationally symmetric and the input and output 0030-4018r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 8 . 0 0 3 4 3 - 5
Z. Liu et al.r Optics Communications 155 (1998) 7–11
8
medium is the same, is known as w1x u2 Ž x 2 , y2 . s y
i
HHu Ž x , y . exp lB 1
1
1
½
ip
A Ž x 12 q y 12 . q D Ž x 22 q y 22 . y 2 Ž x 1 x 2 q y 1 y 2 .
lB
5
d x 1 d y1 ,
Ž1.
where, A, B, C and D are the elements of the ray transfer matrix of the optical system, l the wavelength. Applying a Fourier transform on both sides of Eq. Ž1. yields
HHu Ž x 2
2 , y2
=
HHexp
. exp yi2p Ž nx 2 x 2 q n y 2 y 2 . d x 2 d y 2 s y ip
½
D Ž x 22 q y 22 . y 2 Ž x 1 x 2 q y1 y 2 .
lB
5
i
HHu Ž x , y . exp lB 1
1
1
ip
lB
A Ž x 12 q y 12 .
exp yi2p Ž nx 2 x 2 q n y 2 y 2 . d x 2 d y 2 d x 1 d y 1 ,
the term on the left-hand side of the above equation is just the output angle spectrum, then we have A 2 Ž nx 2 , n y 2 . s
1 D
exp y
=exp
ipl B D
ip
½ ž lB
Ž nx22 q ny22 . HHu1Ž x 1 , y1 .
Ay
1 D
/Ž
x 12 q y 12 . y
2lB D
Ž nx 2 x 1 q n y 2 y1 .
5
d x 1 d y1 .
Ž2.
Writing u1Ž x 1, y 1 . in terms of the input angle spectrum as u1Ž x 1 , y 1 . s
HH A Ž n 1
x1 , n y1
. exp i2p Ž nx1 x 1 q n y1 y1 . d nx1 d n y1 ,
Ž3.
and substituting Eq. Ž3. into Ž2., we get A 2 Ž nx 2 , n y 2 . s
il
HH A Ž n C 1
x1 , n y1
½
. exp y
ipl
2 2 D Ž nx1 q n y1 . q A Ž nx22 q ny22 . y 2 Ž nx1nx 2 q ny1ny 2 .
C
5
d nx1 d n y1 .
Ž4. The above equation is just the wanted Collins formula in frequency-domain, which gives the relationship between the input and output angle spectra through a general ABCD optical system. One can know from Eq. Ž4. that for a general ABCD system, the angle spectrum propagation follows a law which is very similar to that of the complex amplitude propagation, but the changes in the position of the matrix elements should be especially noted. Eq. Ž4. can be more convenient for studies on light propagation and transform in frequency-domain. Now special attention should be given to those situations where the matrix elements A, B, C and D are equal to zero, respectively. For C s 0, using the property of the d-function, Eq. Ž4. becomes A 2 Ž nx 2 , n y 2 . s
1 D
exp y
ipl B D
Ž nx22 q ny22 .
A1
ž
nx 2 n y2 , . D D
/
Ž5.
It can be seen that there exists a simple input–output relationship of the angle spectrum. In view of the similarity between Eq. Ž5. and that in space-domain when B s 0 Žcorresponding to the image-forming system in space-domain., we can call the system where C s 0 the image-forming in frequency-domain. For D s 0, from Eq. Ž4. one can find A 2 Ž nx 2 , n y 2 . s y
i
lB
exp ipl AB Ž nx22 q n y22 .
HH A Ž n 1
x1 , n y1
. exp yi2pl B Ž nx 2 nx1 q n y 2 n y1 . d nx1 d n y1 .
Ž6.
Obviously, the above equation can be rewritten as A 2 Ž nx 2 , n y 2 . s y
i
lB
exp ipl AB Ž nx22 q n y22 . u1 Ž yl Bnx 2 ,y l Bn y 2 . ,
Ž7.
where u1Ž x 1, y 1 . is the complex amplitude distribution of the input field. In this case, the input plane is just the equivalent object focal plane of the optical system.
Z. Liu et al.r Optics Communications 155 (1998) 7–11
9
For B s 0, one can get il
A 2 Ž nx 2 , n y 2 . s
HH A Ž n C 1
x1 , n y1
½
. exp y
ipl C
2 2 D Ž nx1 q n y1 .q
1 D
Ž nx22 q ny22 . y 2 Ž nx1nx 2 q ny1ny 2 .
5
d nx1 d n y1 .
Ž8. It is known that the case when B s 0 is related to the image-forming in space-domain.
3. Fractional Fourier transforms in the frequency-domain It is well-known that the fractional Fourier transform ŽFRT. performs the following operation w5x, u2 Ž x 2 , y2 . s
HHu Ž x , y . exp 1
1
1
ip
l f1
Ž x 2 q y12 q x 22 q y 22 . tan f 1
exp y
i2p
l f 1 sin f
Ž x 1 x 2 q y1 y 2 . d x 1 d y1 ,
Ž9.
where f 1 is called the standard focal length which decides the scale factor of the coordinates, the order is given by P s 2 frp , and when P s 1 the transform goes back to the conventional Fourier transform. Now let us consider how to implement the FRT by using the ABCD optical system. We note that if we assume that the following relations hold A s D s cos f ,
yf 1C s Brf 1 s sin f ,
Ž 10 .
substituting into Eq. Ž1., we can rewritte Eq. Ž1. as u2 Ž x 2 , y2 . s y
ip
i
l f1
HHu Ž x , y . exp sin f 1
=exp y
i2p
l f 1 sin f
1
1
l f 1 tan f
Ž x 12 q y12 q x 22 q y 22 .
Ž x 1 x 2 q y1 y 2 . d x 1 d y1 .
Ž 11.
Obviously, Eq. Ž11. is just a FRT except for an unimportant scaling constant. Then it can be concluded that any ABCD optical system satisfying the relation Ž10. can implement a FRT. On the other hand, let us recall the concept of reciprocally symmetric optical system ŽRSOS. in geometric optics, the RSOS is a system that performs the same operation when used either in positive direction or in its converse along the optical axis. If using the terms of matrix optics, we can find the following conditions for an ABCD system to be a RSOS A s D,
AD y BC s 1,
Ž 12.
where the latter relation requires that the input and output medium is the same, which is usually satisfied. Thus the former relation A s D becomes relatively more important. A simple comparison between Eqs. Ž10. and Ž12. allows us to find that conditions Ž10. must lead to conditions Ž12.. Consequently, we can say that an arbitrary RSOS can implement a FRT. It is worthwhile to point out that from relations Ž10. there exist FRTs with not only real but also complex orders according to the ray matrix elements A, D and the standard focal length f 1. Especially for FRTs with complex orders, one can find them in several references, such as Refs. w8x and w9x, etc. Now we focus our attention on the transform of the angle spectrum performed by a RSOS. Here in order to give a comparison between the operations in space-domain and frequency-domain, we assume that the RSOS satisfies the following relations, A s D s cos f ,
yCrl2 f 2 s l2 f 2 B s sin f .
Ž 13.
Using the above equations in Eq. Ž4. yields that A 2 Ž nx 2 , n y 2 . s y
i
l f 2 sin f
= exp y
HH A Ž n 1
i2p
l f 2 sin f
x1 , n y 2
. exp
ip
l f 2 tan f
Ž nx12 q ny12 q nx22 q ny22 .
Ž nx1 nx 2 q n y1 n y 2 . d nx1 d n y1 .
Ž 14.
Z. Liu et al.r Optics Communications 155 (1998) 7–11
10
Fig. 1. Setup for observing the FRT of the angle spectrum. The order is P s Ž2rp .arccosŽ1 y frZ ., where f is the focal length of the two lenses. By changing the distance Z, various orders can be obtained.
Obviously, this is a FRT describing the operation in frequency-domain, and it has the same order as that in space-domain by Eq. Ž11., although the scale factor may be different due to the difference between f 2 and f 1. We can conclude that an arbitrary RSOS can implement a FRT in both space-domain and frequency-domain with the same order. On the other hand, we introduced the concept of symmetric transform w12x, which is a kind of transform that implements the same operation at the same time in both space-domain and frequency-domain. Thus the FRT is, in other words, a symmetric transform. This feature can also be derived from the pure mathematical properties of the FRT as w6x F Ž a . F Žb . w u Ž x , y . x 4 s F Ž b . F Ža . w u Ž x , y . x 4 s F Ž a q b . w u Ž x , y . x ,
Ž 15 .
Ža .
where F represents the FRT with the order of a . Consider a FRT of function u1Ž x 1, y 1 . in space-domain with the order of P as follows, u 2 Ž x 2 , y 2 . s F Ž P . w u1Ž x 1 , y 1 . x . Then performing a conventional Fourier transform on both sides of the above equation, and taking into account relations Ž15., we can get F Ž1. w u 2 Ž x 2 , y 2 . x s F Ž1. F Ž P . w u1Ž x 1 , y 1 . x 4 s F Ž P . F Ž 1 . w u1Ž x 1 , y 1 . x 4 s F Ž P . A1Ž nx1 , n y1 . 4 , so we have A 2 Ž nx 2 , n y 2 . s F Ž P . A1 Ž nx1 , n y1 . , which is just the FRT in frequency-domain with the same order P. In order to observe the FRT of the angle spectrum, one can use the simple optical setup shown in Fig. 1. But it is worthwhile to note that the distributions on the Fourier plane and output plane are not directly the actual angle spectrum distributions, but after scaling the coordinates, one can get the actual angle spectra.
4. Conclusion The propagation and transfer of the plane-wave angle spectrum of the optical signal through an arbitrary ABCD optical system are found to be very similar to those of the complex amplitude, which can be described by the Collins diffraction integral formula in frequency-domain. In this paper, we have derived this frequency-domain Collins formula. In addition, by applying this formula we discussed the fractional Fourier transform in frequency-domain, it has been found that any arbitrary reciprocally symmetric ABCD optical system can implement simultaneously the FRT in both space-domain and frequencydomain. In other words, FRT belongs to such a kind of symmetric transform which can perform the same operation in both space-domain and frequency-domain.
Acknowledgements This work is supported by the key project of the National Natural Science Foundation and National Hi-Tech Foundation of China. The authors are grateful to the reviewers for useful comments.
Z. Liu et al.r Optics Communications 155 (1998) 7–11
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x
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