- Email: [email protected]
An optical implementation for the estimation of the fractional-Fourier order
I May 1997
OPTICS COMMUNICATIONS ELSEVIER
Optics Communications
137 (1997) 214-218
An optical implementation for the estimation of the fractional-Fourier order Sumiyoshi Abe a, John T. Sheridan b a College of Science and Techitology, Nihon Unioersiy, Funabushi, Chiba 274, Japan b Physics Department. Dublin Institute of Technology. KeGn Street, Dublin, Ireland Received
19 September
1996; accepted 4 December
1996
Abstract Using the theory of optimal estimation of a displacement estimating the order of a fractional-Fourier-transform system.
1. Introduction In recent years, much attention has been paid to the extension of the classical Fourier transform to the fractionalized one and its optical implementations [l-8]. Now it is known that there are at least two ways to optically realize the fractional Fourier transform (FRT) of a given input wavefunction. One uses quadratic graded-index (GRIN) media [ 1,2]. The other is the combination of the operations of lenses and free-space propagations [3,5,7] ‘. Suppose that the refractive index of a GRIN medium of an optical fractional Fourier system is not completely specified. Then the following problem may be posed: How the fractional order of this system can be determined. The problem of this kind has been studied in the context of quantum theory by Helstrom [9], who has developed the theory of parameter estimation which is also applicable to wave optics. The purpose of this article is to introduce this theory to the
’ In Ref. [5], Eqs. (9) and (10) should follows:
uFRT(x) =
d&
exp[(i/2).x’cot
read respectively
as
parameter,
we illustrate
an optical
implementation
for
FRT optics and to present an implementation for the estimation of the order of an unspecified FRT system. The FRT of an input field $ is defined by [lo-121
[F,*l(-r)
Xexp(-if[++I(t)]+fX’cot@)
x/dx’exp
kx”cot0-ixx’cosecH (
$(x’).
(1)
i
Here 0 is the order of the FRT ‘. The symbol I(A) denotes the largest integer smaller than A. Some basic properties of the FRT operator are in order: (a) the Abelian group property; FH,F02 = FH, + t)2, (b) the cyclicity; Fs+?,, = FH, (c) Fzn = F,, = 1 (the identity), (d) F_ H = Ftim’ (the inverse), (e) the standard Fourier transform corresponds to F rr,2. From these properties, one sees that the FRT is a sort of rotational operation with the rotation angle 0. In fact, the FRT (1) is known to be equivalent to the &rotation of the Wigner distribution function associated with I,!J
0
+ix(n-mcott’)]/dx’exp[(i/2)x”cotO _ i(x-m).~‘cosecO]
u(x’),
~LENS(~~)=exp[(i/2)~x2-i(m5-n)x]u(x-m)
0030.4018/97/$17.00 Copyright PII SOO30-4018(96)00787-O
’ The order P of the FRT is usually defined by P = 0/(7r/2). In this paper, 0 is also referred to as the order of the FRT.
0 1997 Elsevier Science B.V. All rights reserved
S. Abe. J.T. Sheridan/Optics
Commurzications
215
137 (1997) 214-218
in phase space [3,12]. Usually the range of 0 is restricted to [O, 2%-l. This paper is organized as follows. In Section 2, the general theory is mathematically reviewed and applied to the estimation of the order of an unknown FRT system. An optical implementation based on the method developed in Section 2 is presented and discussed in Section 3. In Section 4, a numerical examination of the proposed theory and a determination of its primary characteristics are given. Section 5 is devoted to concluding remarks regarding the generality of these results.
generates the FRT [12]. This factor could be subtracted from the definition of the FRT. In any case, it gives no effects to the following discussion. From Eqs. (5) and (6). one sees that Eq. (2) can also be written as follows:
2. Theory for the FRT order estimation
‘“+“de^~*(x,~)Qi(x’,~)=6(x-x’) /a
Assume the situation that the fractional order 0 of the FRT system is not known. Then, with a given input I/J(X), how can 0 be estimated through measurements of the output sample q( x, 0) propagated through the unspecified FRT system? A quantum-mechanical analogue of this problem has been studied in Ref. [9]. According to the quantum theory of parameter estimation, which turns out to be also applicable to classical wave optics with the paraxial approximation, the optimal estimation in the present case can be supplied by the conditional distribution
P(&)
= j=
--L
dx@‘(x,ij)p(x.@)
‘,
p(&)=
1% dx+(x)*(x.H-6)‘. -CC
This distribution depends only on the difference tJ - 6. In quantum mechanics, the function @(x, e^) in Eq. (3) is called the (complex conjugate of) phase state [14]. In addition to cyclicity @(x, e^+ 2~7) = @(x, I?), it has also the following properties:
(completeness),
(9)
= dx@‘(x,@D(x,8) / --x
(Y in Eq. (9) is a window parameter, which is an arbitrary real constant. It is conveniently chosen for each problem. From Eq. (9), it follows that = jr dx]‘P(x,O)]* - -A
2”+“de^f(H^10) /a
which is expected to have a peak at a certain value of the estimate e^. In the above, the function @(x, 6) is given by
(3) where u, is the normalized order n [13]
Hermite-Gaussian
U,(X) = n-‘/“(2”n!)-“‘exp(
-x2/2)
function of
H,(x).
(4)
Note that the Hermite-Gaussian function, and is the eigenfunction
function (4) is a real of the FRT operator Fi:
[F;~.](x)=exp[-iO^(n++)]
un(x).
(5)
This fact makes it possible to construct the function @(x, e^>by using the FRT: @(x,f?)=exp(ifj/2)[Fi+](x), @*(x,i)=exp(-i8/2)[F_,-4](x), where 4 functions
is the infinite
(6) sum of the Hermite-Gaussian
= /
= &&Lx).
(7)
The factor l/2 appearing in the phase factors in Eqs. (5) and (6) is related to the ground-state energy of the quantum-mechanical harmonic oscillator. whose Hamiltonian
= dxl$(x)12. -cc
(11)
Here the second equality is the generalized Parseval equality which the FRT satisfies. The conditional distribution P( 610) turns out to be peaked at the value of 8, showing its estimation. The variance of e^ with respect to this distribution (Ae^)‘= (A)=
(8)
-(i)‘,
(12)
j'"'"dt?A(e^)P(e^lS). a
(13)
gives the measure of confidence of the estimated value. However unfortunately, such an estimation cannot be physical. This is because, as clearly seen from Eq. (lo), the total power of the field @ is infinitely large. To avoid this difficulty, here we propose to use, instead of Eq. (71, the following truncated sum of the HermiteGaussian modes: N 4N(X)
cb(x) = @(x*0)
(8)
=
&
Accordingly
nIzo%w
the conditional
(14)
distribution
becomes
(15)
S. Abe, J.T. Sheridan/Optics
216
where a, are the expansion coefficients with respect to the basis functions (4) a, = ; dxu,(x) --cc
3. An optical implementation
of the input field
ij(x>.
Now we discuss an optical implementation of this scheme for estimating the unknown order of the FRT system. To implement the scheme, we use an FRT device whose order could conveniently be varied over a range which forms a sufficiently large subinterval of [O, z?T]. Such a device is supplied by combining the operations of fake zoom lenses and free-space propagations [ 151. In Fig. la, its schematic diagram is presented showing the required elements. From left to right, the diagram shows the input field, $(r), before and after passage through some system of an unknown FRT order 0. The output field P(x, 0) then passes through a variable FRT system of order - 6. The resulting field is then multiplied by a screen of transmittance c#+(.Y) in Eq. (14). A collection lens which sums over ?I might be introduced. in front of the detector. to perform the integration if a photodetector of sufficiently large area is not available. The output
(16)
Due to the truncation, this distribution does not satisfy Eq. (11) any longer. Instead the following equality holds 2rr+ad&‘,v( /a
610) = t la,/‘, n=O
137 f 19971214-218
Communications
Cl’)
which approaches /“,dxl+( x)12 in the optimal limit N + 00. Note that this integral is still independent of the unknown FRT order 0. Therefore the normalization manipulation
does not affect the estimated value of 0.
. . T \1
P(x,&8) i s.(x)Y(xJy
.. . ..-..
” .. . . ... . . .. .. .. .. . .. .. . .. .. . . .
> Z
ml I
1Unknown
FB System
II
Variable
F-2 = F-p,+;,,]
~
Transmittance Function
Detector
FiIter
Systems
-%,t ‘,
ftan (2
fsin(7r - g,,,,)
f tan + i
!
Fig. I. (a) Schematic of the proposed system for estimating the order of an unknown FRT system. (b) A simple optical implementation of the Fourier transform. (c) Possible implementation of continuously variable FRT systems I and II containing two Fourier transform subsystems. At least two such systems must be used in order to span the entire range of 6.
217
S. Abe. LT. Sheridan/ Optics Communications 137 (1997) 214-218
voltage of the detector then has a value proportional to the incident intensity, which is proportional to F,,,(e^l0). In Fig. lb, a standard optical set-up to implement a Fourier transform is shown. Such a system is employed in Fig. lc and referred to as F,,?. A fractional Fourier transformer of variable order can be implemented following the scheme given in Ref. [1.5]. An inverse FRT which spans the entire range of orders can be produced by reinterpreting and modifying the “type I” geometry described in this reference. In general at least two of these modified systems must be used as shown in Fig. la. Based on the Abelian property and 2rr-cyclicity of the FRT. it can be shown that Fp,Fp,, = FPkP,, = Fzn_c~,ig,,b = F - (6, + 6,,) where PLII = 37- %I, and 0 < 0,,,, I 7~. This leads to the schematic representation in Fig. lc. Closing this section, we note that this optical implementation is meant to illustrate the proposed estimation approach and that much work remains to be done characterizing and implementing an optimized system. For example, non-paraxial effects and effects due to non-ideal transmittance function implementation have been completely neglected in our discussion.
Fig;?. Plots of the normalized distribution p,(e/ff) as a function of 0 (in radians) for N = 1, 5, 10. The FRT order to he estimated is 0 = VT/~. The input Gaussian field is centered at x = 3. i.e., [=3.
ante, a function independent of 0 is found. These results are presented in the following equations: (e^>!v= 0,
4. Numerical
(21)
results
In order to gain a feel for the behavior of the FRT order estimation method proposed in the preceding section, let us examine numerical results calculated for the example of a input Gaussian field (19) where 5 is a given real constant. In this case, the normalized conditional distribution is calculated to be
This distribution has a single peak over any 277 radian range and is symmetric about it. The peak corresponds to the unknown FRT order 0. Then the window parameter LY, appearing in Eq. (13) with FN(610) instead of P(e^lQ>, can be advantageously chosen so as to make the expectation value (e^>v = lUzr+ a d%~v(6l01 equal 0. This is simply demonstrated by setting LY= 0 - 7r and carrying out the resulting analysis. An indication of the accuracy with which the FRT order 0 can be estimated using FJ010) is given by the variance of $. Fixing (Y= 0 - r and calculating the corresponding expression for the vari-
In Fig. 2, plots of the conditional distribution given in Eq. (20) for various numbers of Hermite-Gaussian modes retained in the calculation are presented. A value of [= 3 and an FRT order of 0 = 7r/4 were used in the numerical calculations. Increasing the number of modes retained decreases the variance of 6 and therefore improves the estimation of 8. In Fig. 3, plots of the variance cf 6 as a
-40
20
40
5 Fig. 3. The variance of 6 is plotted as a function of 5, for N = 1, 5. 10. 15.
218
S. Abe, J.T. Sheridan /Optics
function of 5 for various values of N are presented. For finite values of N, asymptotic behavior occurs as 5 goes to km. This behavior can be identified as a characteristic of the finite nature of the sums in Eq. (22). Clearly an estimation procedure can now be discerned. First Fig. 3 is examined and a suitable (N, 5) pair is identified to achieve a suitable small variance for identification of the FRT order. Second the position of the input Gaussian field is adjusted, and the transmittance function refined so as to produce the required estimation peak.
5. Concluding remarks We have used the theory of parameter estimation to propose a novel optical implementation suitable for the identification of the order of any unknown FRT system. The implementation requires the use of an input Gaussian field, a variable inverse FRT system and a screen containing a fixed transmittance function. Numerical results based on the mathematical procedure, which illustrates the method of FRT order estimation, have been presented. In particular, a discussion of the use of the variance of the estimate with respect to the conditional distribution to identify the most suitable number of Hermite-Gaussian modes and the corresponding input field geometry, has been developed. The theory of parameter estimation can be used to produce generalized identification systems. In this way, analogous treatments of systems whose eigenfunctions (modes) are clearly definable can be carried out. For example, transformation parameters within systems with cylindrical or spherical symmetries can be dealt with using the Bessel or Legendre functions, respectively. Nore added. Quite recently the following two references have been brought to our attention: Ref. [I61 proposes techniques which generate the FRT of variable order. Ref. [ 171discusses reconstruction of the GRIN-media parameters, which is a specific issue of optical-parameter determination different from the present one.
Communications
137 (1997) 214-218
Acknowledgements S.A. was supported in part by the Atomic Energy Research Institute of Nihon University. J.T.S. would like to thank
C. Coutsomitros,
A. Lucia,
and G. Volta
for their
supports.
References [I] D. Mendlovic and H.M. Ozaktas. 5. Opt. Sot. Am. A 10 (1993) 1875. [2] H.M. Ozaktas and D. Mendlovic. J. Opt. Sot. Am. A 10 (1993) 2522. [3] A.W. Lohmann, J. Opt. Sot. Am. A 10(1993)2181. [4] H.M. Ozaktas, B. Barshan. D. Mendlovic and L. Onural. J. Opt. Sot. Am. A II (1994) 547. [5] S. Abe and J.T. Sheridan, Opks Lett. 19 (1994) 1801. [6] H.M. Ozaktas and D. Mendlovic. J. Opt. Sot. Am. A 12 (1995) 743. [7] S. Abe and J.T. Sheridan. J. Mod. Optics 42 (1995) 2373. (81 Status Report on: The Fractional Fourier Transform, Vol. 1. (Faculty of Engineering, Tel-Aviv University, 1995). eds. A.W. Lohmann. D. Mendlovic and Z. Zalevsky. This report contains many relevant references. [9] C.W. Helstrom, Int. J. Theor. Phys. 11 (1974) 357; Quantum Detection and Estimation Theory (Academic Press, New York. 1976). [IO] V. Namias, J. Inst. Math. Appl. 25 (1980) 241. [I I] A.C. McBride and F.H. Kerr, IMA J. Appi. Math. 39 (1987) 159. [12] S. Abe and J.T. Sheridan, J. Phys. A 27 (1994) 4179; Corrigenda A 27 (1994) 7937. [I31 L. Mandel and E. Wolf. Optical Coherence and Quantum Optics (Cambridge University Press. Cambridge, 1995) Ch. 5, p. 261. [ 141 P. Carruthers and M.M. Nieto. Rev. Mod. Phys. 40 (1968) 411. [15] A.W. Lohmann. Optics Comm. 115 (1995) 437. [16] R.G. Dorsch, Appl. Optics 34 (1995) 6016. [ 171 T. Alieva and F. Agulh-Lopez, Optics Comm. I14 (1995) 161.
OPTICS COMMUNICATIONS ELSEVIER
Optics Communications
137 (1997) 214-218
An optical implementation for the estimation of the fractional-Fourier order Sumiyoshi Abe a, John T. Sheridan b a College of Science and Techitology, Nihon Unioersiy, Funabushi, Chiba 274, Japan b Physics Department. Dublin Institute of Technology. KeGn Street, Dublin, Ireland Received
19 September
1996; accepted 4 December
1996
Abstract Using the theory of optimal estimation of a displacement estimating the order of a fractional-Fourier-transform system.
1. Introduction In recent years, much attention has been paid to the extension of the classical Fourier transform to the fractionalized one and its optical implementations [l-8]. Now it is known that there are at least two ways to optically realize the fractional Fourier transform (FRT) of a given input wavefunction. One uses quadratic graded-index (GRIN) media [ 1,2]. The other is the combination of the operations of lenses and free-space propagations [3,5,7] ‘. Suppose that the refractive index of a GRIN medium of an optical fractional Fourier system is not completely specified. Then the following problem may be posed: How the fractional order of this system can be determined. The problem of this kind has been studied in the context of quantum theory by Helstrom [9], who has developed the theory of parameter estimation which is also applicable to wave optics. The purpose of this article is to introduce this theory to the
’ In Ref. [5], Eqs. (9) and (10) should follows:
uFRT(x) =
d&
exp[(i/2).x’cot
read respectively
as
parameter,
we illustrate
an optical
implementation
for
FRT optics and to present an implementation for the estimation of the order of an unspecified FRT system. The FRT of an input field $ is defined by [lo-121
[F,*l(-r)
Xexp(-if[++I(t)]+fX’cot@)
x/dx’exp
kx”cot0-ixx’cosecH (
$(x’).
(1)
i
Here 0 is the order of the FRT ‘. The symbol I(A) denotes the largest integer smaller than A. Some basic properties of the FRT operator are in order: (a) the Abelian group property; FH,F02 = FH, + t)2, (b) the cyclicity; Fs+?,, = FH, (c) Fzn = F,, = 1 (the identity), (d) F_ H = Ftim’ (the inverse), (e) the standard Fourier transform corresponds to F rr,2. From these properties, one sees that the FRT is a sort of rotational operation with the rotation angle 0. In fact, the FRT (1) is known to be equivalent to the &rotation of the Wigner distribution function associated with I,!J
0
+ix(n-mcott’)]/dx’exp[(i/2)x”cotO _ i(x-m).~‘cosecO]
u(x’),
~LENS(~~)=exp[(i/2)~x2-i(m5-n)x]u(x-m)
0030.4018/97/$17.00 Copyright PII SOO30-4018(96)00787-O
’ The order P of the FRT is usually defined by P = 0/(7r/2). In this paper, 0 is also referred to as the order of the FRT.
0 1997 Elsevier Science B.V. All rights reserved
S. Abe. J.T. Sheridan/Optics
Commurzications
215
137 (1997) 214-218
in phase space [3,12]. Usually the range of 0 is restricted to [O, 2%-l. This paper is organized as follows. In Section 2, the general theory is mathematically reviewed and applied to the estimation of the order of an unknown FRT system. An optical implementation based on the method developed in Section 2 is presented and discussed in Section 3. In Section 4, a numerical examination of the proposed theory and a determination of its primary characteristics are given. Section 5 is devoted to concluding remarks regarding the generality of these results.
generates the FRT [12]. This factor could be subtracted from the definition of the FRT. In any case, it gives no effects to the following discussion. From Eqs. (5) and (6). one sees that Eq. (2) can also be written as follows:
2. Theory for the FRT order estimation
‘“+“de^~*(x,~)Qi(x’,~)=6(x-x’) /a
Assume the situation that the fractional order 0 of the FRT system is not known. Then, with a given input I/J(X), how can 0 be estimated through measurements of the output sample q( x, 0) propagated through the unspecified FRT system? A quantum-mechanical analogue of this problem has been studied in Ref. [9]. According to the quantum theory of parameter estimation, which turns out to be also applicable to classical wave optics with the paraxial approximation, the optimal estimation in the present case can be supplied by the conditional distribution
P(&)
= j=
--L
dx@‘(x,ij)p(x.@)
‘,
p(&)=
1% dx+(x)*(x.H-6)‘. -CC
This distribution depends only on the difference tJ - 6. In quantum mechanics, the function @(x, e^) in Eq. (3) is called the (complex conjugate of) phase state [14]. In addition to cyclicity @(x, e^+ 2~7) = @(x, I?), it has also the following properties:
(completeness),
(9)
= dx@‘(x,@D(x,8) / --x
(Y in Eq. (9) is a window parameter, which is an arbitrary real constant. It is conveniently chosen for each problem. From Eq. (9), it follows that = jr dx]‘P(x,O)]* - -A
2”+“de^f(H^10) /a
which is expected to have a peak at a certain value of the estimate e^. In the above, the function @(x, 6) is given by
(3) where u, is the normalized order n [13]
Hermite-Gaussian
U,(X) = n-‘/“(2”n!)-“‘exp(
-x2/2)
function of
H,(x).
(4)
Note that the Hermite-Gaussian function, and is the eigenfunction
function (4) is a real of the FRT operator Fi:
[F;~.](x)=exp[-iO^(n++)]
un(x).
(5)
This fact makes it possible to construct the function @(x, e^>by using the FRT: @(x,f?)=exp(ifj/2)[Fi+](x), @*(x,i)=exp(-i8/2)[F_,-4](x), where 4 functions
is the infinite
(6) sum of the Hermite-Gaussian
= /
= &&Lx).
(7)
The factor l/2 appearing in the phase factors in Eqs. (5) and (6) is related to the ground-state energy of the quantum-mechanical harmonic oscillator. whose Hamiltonian
= dxl$(x)12. -cc
(11)
Here the second equality is the generalized Parseval equality which the FRT satisfies. The conditional distribution P( 610) turns out to be peaked at the value of 8, showing its estimation. The variance of e^ with respect to this distribution (Ae^)‘= (A)=
(8)
-(i)‘,
(12)
j'"'"dt?A(e^)P(e^lS). a
(13)
gives the measure of confidence of the estimated value. However unfortunately, such an estimation cannot be physical. This is because, as clearly seen from Eq. (lo), the total power of the field @ is infinitely large. To avoid this difficulty, here we propose to use, instead of Eq. (71, the following truncated sum of the HermiteGaussian modes: N 4N(X)
cb(x) = @(x*0)
(8)
=
&
Accordingly
nIzo%w
the conditional
(14)
distribution
becomes
(15)
S. Abe, J.T. Sheridan/Optics
216
where a, are the expansion coefficients with respect to the basis functions (4) a, = ; dxu,(x) --cc
3. An optical implementation
of the input field
ij(x>.
Now we discuss an optical implementation of this scheme for estimating the unknown order of the FRT system. To implement the scheme, we use an FRT device whose order could conveniently be varied over a range which forms a sufficiently large subinterval of [O, z?T]. Such a device is supplied by combining the operations of fake zoom lenses and free-space propagations [ 151. In Fig. la, its schematic diagram is presented showing the required elements. From left to right, the diagram shows the input field, $(r), before and after passage through some system of an unknown FRT order 0. The output field P(x, 0) then passes through a variable FRT system of order - 6. The resulting field is then multiplied by a screen of transmittance c#+(.Y) in Eq. (14). A collection lens which sums over ?I might be introduced. in front of the detector. to perform the integration if a photodetector of sufficiently large area is not available. The output
(16)
Due to the truncation, this distribution does not satisfy Eq. (11) any longer. Instead the following equality holds 2rr+ad&‘,v( /a
610) = t la,/‘, n=O
137 f 19971214-218
Communications
Cl’)
which approaches /“,dxl+( x)12 in the optimal limit N + 00. Note that this integral is still independent of the unknown FRT order 0. Therefore the normalization manipulation
does not affect the estimated value of 0.
. . T \1
P(x,&8) i s.(x)Y(xJy
.. . ..-..
” .. . . ... . . .. .. .. .. . .. .. . .. .. . . .
> Z
ml I
1Unknown
FB System
II
Variable
F-2 = F-p,+;,,]
~
Transmittance Function
Detector
FiIter
Systems
-%,t ‘,
ftan (2
fsin(7r - g,,,,)
f tan + i
!
Fig. I. (a) Schematic of the proposed system for estimating the order of an unknown FRT system. (b) A simple optical implementation of the Fourier transform. (c) Possible implementation of continuously variable FRT systems I and II containing two Fourier transform subsystems. At least two such systems must be used in order to span the entire range of 6.
217
S. Abe. LT. Sheridan/ Optics Communications 137 (1997) 214-218
voltage of the detector then has a value proportional to the incident intensity, which is proportional to F,,,(e^l0). In Fig. lb, a standard optical set-up to implement a Fourier transform is shown. Such a system is employed in Fig. lc and referred to as F,,?. A fractional Fourier transformer of variable order can be implemented following the scheme given in Ref. [1.5]. An inverse FRT which spans the entire range of orders can be produced by reinterpreting and modifying the “type I” geometry described in this reference. In general at least two of these modified systems must be used as shown in Fig. la. Based on the Abelian property and 2rr-cyclicity of the FRT. it can be shown that Fp,Fp,, = FPkP,, = Fzn_c~,ig,,b = F - (6, + 6,,) where PLII = 37- %I, and 0 < 0,,,, I 7~. This leads to the schematic representation in Fig. lc. Closing this section, we note that this optical implementation is meant to illustrate the proposed estimation approach and that much work remains to be done characterizing and implementing an optimized system. For example, non-paraxial effects and effects due to non-ideal transmittance function implementation have been completely neglected in our discussion.
Fig;?. Plots of the normalized distribution p,(e/ff) as a function of 0 (in radians) for N = 1, 5, 10. The FRT order to he estimated is 0 = VT/~. The input Gaussian field is centered at x = 3. i.e., [=3.
ante, a function independent of 0 is found. These results are presented in the following equations: (e^>!v= 0,
4. Numerical
(21)
results
In order to gain a feel for the behavior of the FRT order estimation method proposed in the preceding section, let us examine numerical results calculated for the example of a input Gaussian field (19) where 5 is a given real constant. In this case, the normalized conditional distribution is calculated to be
This distribution has a single peak over any 277 radian range and is symmetric about it. The peak corresponds to the unknown FRT order 0. Then the window parameter LY, appearing in Eq. (13) with FN(610) instead of P(e^lQ>, can be advantageously chosen so as to make the expectation value (e^>v = lUzr+ a d%~v(6l01 equal 0. This is simply demonstrated by setting LY= 0 - 7r and carrying out the resulting analysis. An indication of the accuracy with which the FRT order 0 can be estimated using FJ010) is given by the variance of $. Fixing (Y= 0 - r and calculating the corresponding expression for the vari-
In Fig. 2, plots of the conditional distribution given in Eq. (20) for various numbers of Hermite-Gaussian modes retained in the calculation are presented. A value of [= 3 and an FRT order of 0 = 7r/4 were used in the numerical calculations. Increasing the number of modes retained decreases the variance of 6 and therefore improves the estimation of 8. In Fig. 3, plots of the variance cf 6 as a
-40
20
40
5 Fig. 3. The variance of 6 is plotted as a function of 5, for N = 1, 5. 10. 15.
218
S. Abe, J.T. Sheridan /Optics
function of 5 for various values of N are presented. For finite values of N, asymptotic behavior occurs as 5 goes to km. This behavior can be identified as a characteristic of the finite nature of the sums in Eq. (22). Clearly an estimation procedure can now be discerned. First Fig. 3 is examined and a suitable (N, 5) pair is identified to achieve a suitable small variance for identification of the FRT order. Second the position of the input Gaussian field is adjusted, and the transmittance function refined so as to produce the required estimation peak.
5. Concluding remarks We have used the theory of parameter estimation to propose a novel optical implementation suitable for the identification of the order of any unknown FRT system. The implementation requires the use of an input Gaussian field, a variable inverse FRT system and a screen containing a fixed transmittance function. Numerical results based on the mathematical procedure, which illustrates the method of FRT order estimation, have been presented. In particular, a discussion of the use of the variance of the estimate with respect to the conditional distribution to identify the most suitable number of Hermite-Gaussian modes and the corresponding input field geometry, has been developed. The theory of parameter estimation can be used to produce generalized identification systems. In this way, analogous treatments of systems whose eigenfunctions (modes) are clearly definable can be carried out. For example, transformation parameters within systems with cylindrical or spherical symmetries can be dealt with using the Bessel or Legendre functions, respectively. Nore added. Quite recently the following two references have been brought to our attention: Ref. [I61 proposes techniques which generate the FRT of variable order. Ref. [ 171discusses reconstruction of the GRIN-media parameters, which is a specific issue of optical-parameter determination different from the present one.
Communications
137 (1997) 214-218
Acknowledgements S.A. was supported in part by the Atomic Energy Research Institute of Nihon University. J.T.S. would like to thank
C. Coutsomitros,
A. Lucia,
and G. Volta
for their
supports.
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