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A simple realization of fractional Fourier transform and relation to harmonic oscillator Green's function
1 August 1994
OPTICS COMMUNICATIONS Optics Communications 110 ( 1994) 23-26
ELSEVIER
A simple realization of fractional Fourier transform and relation to harmonic oscillator Green’s function G.S. Agarwal School of Physics, University of Hyderabad, Hyderabad - 500 134, India
R. Simon Institute ofMathematical Sciences, CT Campus, Madras
600 113, India
Received 25 January 1994
Abstract A simple relation between the fractional Fourier transform (FRACFT) and the Green’s function for the harmonic oscillator is demonstrated. This relation enables us to understand easily the characteristics of FRACFT and leads to an elementary realization of FRACR in terms of Fresnel diffraction.
1. Introduction W(x,p) Recently the concept of fractional Fourier transform (FRACFT for brevity) is gaining importance in various areas of optics and optical information processing [ 1,2 1. There are different methods of introducing the FRACFT. An attractive approach is based on the Wigner function [ 3 1. Consider a field b(x). The variable x may refer either to the spatial coordinate or to time. Let us introduce the variable 0 conjugate to 2 such that [$fi]=i,
h-+-i&,
.&x
.
(1)
In terms of physical variables if x represents spatial coordinate (time), then p represents momentum (frequency). For simplicity we will assume that all the variables have been scaled appropriately so that all are dimensionless quantities. The Wigner function can be defined by
=
s 8(x+X/2) -cc
&*(x-X/2)
e-‘@dX
(2) +cO
=-
1 b(p+P/2) 272 s --03
where 8(p)
&xl=
b(p-P/2)
is the Fourier transform
&j-b(P)
eixp dp .
eiPxdP,
(3)
of b(x), (4)
Note that s
WX,P)
dp=zxl
b(x)
W(x, P) (kc= I b(P)
I2
I2 .
>
(5) (6)
The FRACFT is now defined by the following equations - the field B(x) is transformed into a new
0030-4018/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SsDr0030-4018(94)00235-M
24
G.S. Agarwal, R. Simon /Optics Communications 110 (1994) 23-26
field V(x) (which is to be determined, below)
see Eq. ( 10)
8(x) + V(x)
(7)
=W(xcos(o-psinyl,pcos~+xsin~;O). (13)
such that the Wigner function
transforms
as
3
w(x,P)+@(x,P)
(8)
Thus the relation (9) corresponds to the time evolution given by ( 11) and hence we conclude that 1V) =e-1Q~18)
where @(x,p)=W(xcos~-psinyl,pcosp+xsinyl). (9) The relation (9) should enable one to reconstruct V in terms of 8. Lohmann has discussed this procedure.
V(x)=(xl~)=(xle-‘“~‘I)
v(x) = j K,(x>Y) g”(y) dy .
s
= =
In this paper we present a very elementary way to relate V to 8. We expect the following linear relation between the two
(10)
--co
The kernel K,(x. y) for which (9) holds will define the FRACFT. Note that if K(x, y ) c exp ( - ixy), then (10) represents the usual Fourier transform. To derive the kernel Kq we use our knowledge from quantum optical literature, where one knows the general structure of the Hamiltonian leading to (9). At the operator level let us write
(14)
On projecting both sides of ( 14) and using the completeness of Ix) we get
= 2. FRACFT and the harmonic oscillator Green’s function
.
s
dy (xle-iACly)
(yl X)
dy (xle-“qly)R(y)
s
dyK,(x,y)fi(y)
>
(15a)
where K,(x,y)=
(_ule-‘“vly)
(15b)
From ( 15b) we immediately see that the FRACFT is the same as the transformation by a kernel which is just the Green’s function for the harmonic oscillator problem. The latter is very well known [ 5 ] and hence
Kp(x,y)= x exp
J
1
~ 27ri sin p [(x2+v2)
COS~-2XYl
(16) (17)
8(x) = (xl 8) .
(11)
For quadratic Hamiltonian it is well known #’ that the time evolution of the Wigner function is same as the evolution of the classical trajectories i.e. W(x( p), P(P), V) = W(x, P; 0) X
x(v)
cos $0 -sina,
CD>6 >( (q)
=
sina, x cos(p )6>
)
(12)
and hence #’ The transformation linear transformations
of the Wigner function is discussed in Ref. [ 41.
under
arbitrary
Eqs. ( 10) and ( 15) define the Fractional Fourier transform. In the limit p-n/2, the FRACFT goes over the Fourier transform as indicated by Eq. ( 17 ) . It should be borne in mind that Kv is defined upto an overall phase. This is because the Wigner function is quadratic in field amplitudes and thus a definition hke (9) will lead to a relation between field amplitudes which would be arbitrary upto a phase. The above analysis is easily generalized to two spatial dimensions. Let us connect the above to the wave propagation in paraxial approximation in a medium characterized by a parabolic refractive index profile i.e., with a dielectric function E
G.S. Aganval, R. Simon /Optics Communications 110 (1994) 23-26
c=n;[1 -rl(x:+x:)] The wave equation as
(18) in paraxial
approximation
E(r,l)=b(r)exp(i~n,;-iwl),
ad
a2 I-a(kz) + 2 a(kX,y .
reads
(19)
I
+
25
known [ 7 ] in cases where the exponential involves a linear function of _? and 8. Let us now examine the meaning of each term in Eq. (2 1): The term on the extreme right corresponds to a free propagation over a “distance” tan p. The term on the extreme left corresponds to a propagation through a thin lens of focal length kf= cotan q. The middle term in (2 1) corresponds to a magnifier as exp[-ilogcosryq)]W(x)
[ (kx,)2+(kx2)2]&=0. (22) Eq. (20) is equivalent to ( 11) if we choose q/k2 = 1. Thus fractional Fourier transforms will be generated for all lengths / of the medium such that kl< n/2.
3. Fractional Fourier transformations from Fresnel diffraction Having realized the connection of FRACFT with the harmonic oscillator problem in quantum mechanics we use the relation ( 15 ) to present alternate realizations of FRACFT. In particular we will show the connection of FRACFT to Fresnel diffraction. To see this we see that p2/2 part of H gives the free propagation (which would correspond to Fresnel transform) and that x2/2 part corresponds to a lens. Thus we would like to decompose the exponential appearingin (15) as exp[-ipg
+ ii2)]
Xexp
-i logcosq
_rxp(
-i tan
9:)
(.@+$‘) 2
( Xexp(-iqtan(o).
>
(21)
Here the coefficients on the right hand side are determined from the Iwasawa decomposition for the SU ( 1, 1) group. A simplified derivation of Iwasawa decomposition #2 for our problem is given in the Appendix. The Iwasawa decomposition might be viewed upon as analog of Baker-Hausdorffidentity which is so well ‘* For a discussion
of Iwasawa more detailed group theoretical
decomposition see Ref. [ 6 1; for description see Ref. [ 7 1.
The Iwasawa decomposition thus enables us to derive the important result - the FRACFT can be synthesized by a combination of three optical elements in the following order (a) free propagation over a distance tan q,; (b) a magnifier which changes the field b(x) into b(xlcos V); (c) propagation through a thin lens of focal length f= ( X/2n)cotan q. It should be further noted that on combining ( 14), (21) and (22) we get the interesting relation V(x)=(xlV)=exp(-itany;)& X(x/cosylIexp(-itfi2tanv)I&).
(23)
The relation (23) enables one to interpret the FRACFT as the Fresnel transform [ 8 ] because (i ) the prefactor is just the phase factor and would not (ii) appear in any intensity measurements (x 1exp ( - ifb’ tan p) IS) just represents the Fresnel transform in the operator version. Thus to conclude we have shown (i) The relation between the FRACFT and the Green’s function of the harmonic oscillator (in units for which mass and frequency are unity). (ii) How the FRACFT can be realized in terms of the standard optical elements. This is achieved by using Iwasawa’s decomposition. As a byproduct of this work we have been able to obtain a rather simple derivation of the Iwasawa decomposition. Finally the present work suggests the possibility of introducing generalized version of FRACFT by using a kernel which corresponds to v/k2 # 1 (cf the discussion following Eq. (20) ). One of us (GSA) thanks the NSF grant number
G.S. Aganval, R. Simon /Optics Communications I1 0 (1994) 23-26
26
INT 9100685 for a travel grant to USA which resulted in this work.
Appendix
On using (22 ) we can reexpress
Xexp[-ilogcosp~~)]K{~$(x;~).
Simplified derivation of Iwasawa decomposition Eq.
On combining
(26 ) as
(27)
(24) and (27 ) we get the result (2 1).
(21) The Green’s function ( 16) can be used to derive the results (2 1) by rewriting the factors in the exponent. For this purpose we note that the kernel K(O) for free propagation is given by K(O)(x r ’ y)=
=&
(xlexp( exp($
-iq@*/2) (x-y)‘).
1~)
(24) (25)
We rewrite ( 16 )
=J&exp[+tanyrx’
+ icotanq(y-
-&--I
= ~9exp(~tanq.x2)K~.$(xsec~,y).
(26)
References [ 1 ] H.M. Ozaktas and D. Mendlovic, Optics Comm. 101 ( 1993) 163; D. Mendlovic and H.M. Ozaktas, J. Opt. Sot. Am. AI0 (1993) 1875. [2] A.W. Lohmann, J. Opt. Sot. Am. A10 (1993) 2818; and references therein. [ 31 E. Wigner, Phys. Rev. 40 (1932) 749. [4] G.S. Agarwal, J. Modern Optics 34 (1987) 909; R. Simon, E.C.G. Sudershan and N. Mukunda, Phys. Rev. A 29 (1984) 3273; A 31 (1985) 2419; A36 (1987) 3868; A37 (1988) 3028; see also M.J. Bastiaans, Optics Comm. 25 (1978) 26; J. Opt. Sot. Am. 69 (1979) 1710. [ 5 ] See for example E. Merzbacher, Quantum Mechanics (Wiley, NewYork, 1961) p. 159. [ 61 R. Simon and N. Mukunda, Optics Comm. 95 (1993) 39. [7] S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces (Academic, New York, 1978). [ 81 Cf. A. Messiah, Quantum Mechanics, Vol. 1 (Wiley, New York, 1961) p. 442. [9] J.W. Goodman, Introduction to Fourier Optics (McGraw Hill, New York, 1968).
OPTICS COMMUNICATIONS Optics Communications 110 ( 1994) 23-26
ELSEVIER
A simple realization of fractional Fourier transform and relation to harmonic oscillator Green’s function G.S. Agarwal School of Physics, University of Hyderabad, Hyderabad - 500 134, India
R. Simon Institute ofMathematical Sciences, CT Campus, Madras
600 113, India
Received 25 January 1994
Abstract A simple relation between the fractional Fourier transform (FRACFT) and the Green’s function for the harmonic oscillator is demonstrated. This relation enables us to understand easily the characteristics of FRACFT and leads to an elementary realization of FRACR in terms of Fresnel diffraction.
1. Introduction W(x,p) Recently the concept of fractional Fourier transform (FRACFT for brevity) is gaining importance in various areas of optics and optical information processing [ 1,2 1. There are different methods of introducing the FRACFT. An attractive approach is based on the Wigner function [ 3 1. Consider a field b(x). The variable x may refer either to the spatial coordinate or to time. Let us introduce the variable 0 conjugate to 2 such that [$fi]=i,
h-+-i&,
.&x
.
(1)
In terms of physical variables if x represents spatial coordinate (time), then p represents momentum (frequency). For simplicity we will assume that all the variables have been scaled appropriately so that all are dimensionless quantities. The Wigner function can be defined by
=
s 8(x+X/2) -cc
&*(x-X/2)
e-‘@dX
(2) +cO
=-
1 b(p+P/2) 272 s --03
where 8(p)
&xl=
b(p-P/2)
is the Fourier transform
&j-b(P)
eixp dp .
eiPxdP,
(3)
of b(x), (4)
Note that s
WX,P)
dp=zxl
b(x)
W(x, P) (kc= I b(P)
I2
I2 .
>
(5) (6)
The FRACFT is now defined by the following equations - the field B(x) is transformed into a new
0030-4018/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SsDr0030-4018(94)00235-M
24
G.S. Agarwal, R. Simon /Optics Communications 110 (1994) 23-26
field V(x) (which is to be determined, below)
see Eq. ( 10)
8(x) + V(x)
(7)
=W(xcos(o-psinyl,pcos~+xsin~;O). (13)
such that the Wigner function
transforms
as
3
w(x,P)+@(x,P)
(8)
Thus the relation (9) corresponds to the time evolution given by ( 11) and hence we conclude that 1V) =e-1Q~18)
where @(x,p)=W(xcos~-psinyl,pcosp+xsinyl). (9) The relation (9) should enable one to reconstruct V in terms of 8. Lohmann has discussed this procedure.
V(x)=(xl~)=(xle-‘“~‘I)
v(x) = j K,(x>Y) g”(y) dy .
s
= =
In this paper we present a very elementary way to relate V to 8. We expect the following linear relation between the two
(10)
--co
The kernel K,(x. y) for which (9) holds will define the FRACFT. Note that if K(x, y ) c exp ( - ixy), then (10) represents the usual Fourier transform. To derive the kernel Kq we use our knowledge from quantum optical literature, where one knows the general structure of the Hamiltonian leading to (9). At the operator level let us write
(14)
On projecting both sides of ( 14) and using the completeness of Ix) we get
= 2. FRACFT and the harmonic oscillator Green’s function
.
s
dy (xle-iACly)
(yl X)
dy (xle-“qly)R(y)
s
dyK,(x,y)fi(y)
>
(15a)
where K,(x,y)=
(_ule-‘“vly)
(15b)
From ( 15b) we immediately see that the FRACFT is the same as the transformation by a kernel which is just the Green’s function for the harmonic oscillator problem. The latter is very well known [ 5 ] and hence
Kp(x,y)= x exp
J
1
~ 27ri sin p [(x2+v2)
COS~-2XYl
(16) (17)
8(x) = (xl 8) .
(11)
For quadratic Hamiltonian it is well known #’ that the time evolution of the Wigner function is same as the evolution of the classical trajectories i.e. W(x( p), P(P), V) = W(x, P; 0) X
x(v)
cos $0 -sina,
CD>6 >( (q)
=
sina, x cos(p )6>
)
(12)
and hence #’ The transformation linear transformations
of the Wigner function is discussed in Ref. [ 41.
under
arbitrary
Eqs. ( 10) and ( 15) define the Fractional Fourier transform. In the limit p-n/2, the FRACFT goes over the Fourier transform as indicated by Eq. ( 17 ) . It should be borne in mind that Kv is defined upto an overall phase. This is because the Wigner function is quadratic in field amplitudes and thus a definition hke (9) will lead to a relation between field amplitudes which would be arbitrary upto a phase. The above analysis is easily generalized to two spatial dimensions. Let us connect the above to the wave propagation in paraxial approximation in a medium characterized by a parabolic refractive index profile i.e., with a dielectric function E
G.S. Aganval, R. Simon /Optics Communications 110 (1994) 23-26
c=n;[1 -rl(x:+x:)] The wave equation as
(18) in paraxial
approximation
E(r,l)=b(r)exp(i~n,;-iwl),
ad
a2 I-a(kz) + 2 a(kX,y .
reads
(19)
I
+
25
known [ 7 ] in cases where the exponential involves a linear function of _? and 8. Let us now examine the meaning of each term in Eq. (2 1): The term on the extreme right corresponds to a free propagation over a “distance” tan p. The term on the extreme left corresponds to a propagation through a thin lens of focal length kf= cotan q. The middle term in (2 1) corresponds to a magnifier as exp[-ilogcosryq)]W(x)
[ (kx,)2+(kx2)2]&=0. (22) Eq. (20) is equivalent to ( 11) if we choose q/k2 = 1. Thus fractional Fourier transforms will be generated for all lengths / of the medium such that kl< n/2.
3. Fractional Fourier transformations from Fresnel diffraction Having realized the connection of FRACFT with the harmonic oscillator problem in quantum mechanics we use the relation ( 15 ) to present alternate realizations of FRACFT. In particular we will show the connection of FRACFT to Fresnel diffraction. To see this we see that p2/2 part of H gives the free propagation (which would correspond to Fresnel transform) and that x2/2 part corresponds to a lens. Thus we would like to decompose the exponential appearingin (15) as exp[-ipg
+ ii2)]
Xexp
-i logcosq
_rxp(
-i tan
9:)
(.@+$‘) 2
( Xexp(-iqtan(o).
>
(21)
Here the coefficients on the right hand side are determined from the Iwasawa decomposition for the SU ( 1, 1) group. A simplified derivation of Iwasawa decomposition #2 for our problem is given in the Appendix. The Iwasawa decomposition might be viewed upon as analog of Baker-Hausdorffidentity which is so well ‘* For a discussion
of Iwasawa more detailed group theoretical
decomposition see Ref. [ 6 1; for description see Ref. [ 7 1.
The Iwasawa decomposition thus enables us to derive the important result - the FRACFT can be synthesized by a combination of three optical elements in the following order (a) free propagation over a distance tan q,; (b) a magnifier which changes the field b(x) into b(xlcos V); (c) propagation through a thin lens of focal length f= ( X/2n)cotan q. It should be further noted that on combining ( 14), (21) and (22) we get the interesting relation V(x)=(xlV)=exp(-itany;)& X(x/cosylIexp(-itfi2tanv)I&).
(23)
The relation (23) enables one to interpret the FRACFT as the Fresnel transform [ 8 ] because (i ) the prefactor is just the phase factor and would not (ii) appear in any intensity measurements (x 1exp ( - ifb’ tan p) IS) just represents the Fresnel transform in the operator version. Thus to conclude we have shown (i) The relation between the FRACFT and the Green’s function of the harmonic oscillator (in units for which mass and frequency are unity). (ii) How the FRACFT can be realized in terms of the standard optical elements. This is achieved by using Iwasawa’s decomposition. As a byproduct of this work we have been able to obtain a rather simple derivation of the Iwasawa decomposition. Finally the present work suggests the possibility of introducing generalized version of FRACFT by using a kernel which corresponds to v/k2 # 1 (cf the discussion following Eq. (20) ). One of us (GSA) thanks the NSF grant number
G.S. Aganval, R. Simon /Optics Communications I1 0 (1994) 23-26
26
INT 9100685 for a travel grant to USA which resulted in this work.
Appendix
On using (22 ) we can reexpress
Xexp[-ilogcosp~~)]K{~$(x;~).
Simplified derivation of Iwasawa decomposition Eq.
On combining
(26 ) as
(27)
(24) and (27 ) we get the result (2 1).
(21) The Green’s function ( 16) can be used to derive the results (2 1) by rewriting the factors in the exponent. For this purpose we note that the kernel K(O) for free propagation is given by K(O)(x r ’ y)=
=&
(xlexp( exp($
-iq@*/2) (x-y)‘).
1~)
(24) (25)
We rewrite ( 16 )
=J&exp[+tanyrx’
+ icotanq(y-
-&--I
= ~9exp(~tanq.x2)K~.$(xsec~,y).
(26)
References [ 1 ] H.M. Ozaktas and D. Mendlovic, Optics Comm. 101 ( 1993) 163; D. Mendlovic and H.M. Ozaktas, J. Opt. Sot. Am. AI0 (1993) 1875. [2] A.W. Lohmann, J. Opt. Sot. Am. A10 (1993) 2818; and references therein. [ 31 E. Wigner, Phys. Rev. 40 (1932) 749. [4] G.S. Agarwal, J. Modern Optics 34 (1987) 909; R. Simon, E.C.G. Sudershan and N. Mukunda, Phys. Rev. A 29 (1984) 3273; A 31 (1985) 2419; A36 (1987) 3868; A37 (1988) 3028; see also M.J. Bastiaans, Optics Comm. 25 (1978) 26; J. Opt. Sot. Am. 69 (1979) 1710. [ 5 ] See for example E. Merzbacher, Quantum Mechanics (Wiley, NewYork, 1961) p. 159. [ 61 R. Simon and N. Mukunda, Optics Comm. 95 (1993) 39. [7] S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces (Academic, New York, 1978). [ 81 Cf. A. Messiah, Quantum Mechanics, Vol. 1 (Wiley, New York, 1961) p. 442. [9] J.W. Goodman, Introduction to Fourier Optics (McGraw Hill, New York, 1968).