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Optical Wigner distribution and ambiguity function for complex signals and images
Volume 67, number 3
OPTICS COMMUNICATIONS
1 July 1988
OPTICAL W I G N E R D I S T R I B U T I O N AND A M B I G U I T Y F U N C T I O N
FOR C O M P L E X SIGNALS AND I M A G E S Yao LI, George E I C H M A N N and Michael CONNER
DepartmentofElectricalEngineering, The CityCollegeof The City UnviersityofNew York,New York,NY 10031, USA Received 15 February 1988
Using a degenerateoptical phase conjugation device, a new real-time coherent optical Wigner distribution and ambiguityfunction implementation method for both one- and two-dimensionalcomplexsignals is suggested. For a one dimensional (1D) s i g n a l f ( x ) , the Wigner distribution ( W D ) W(x, ~) and ambiguity function (AF) A(x', ~) are defined as
W(x, ~)= f f ( x + x ' / 2 ) f * ( x - x ' / 2 ) --
oo
X e x p ( - i 2 n x ' ~ ) dx' ,
(la)
and
7
A(x', ~)= j f ( x + x , / 2 ) f . ( x - x , / 2 ) ×exp(-i2zcx~) dx,
(lb)
where * denotes complex conjugation. Similarly, the WD and AF for a two dimensional (2D) i m a g e f ( x , y) are defined as
×f*(x-x'/2, y-y'/2) × e x p [ - i 2 n ( x ' ~ + y ' t / ) ] dx' d y ' ,
(2a)
and
xf*(x-x'/2, y-y'/2) × e x p [ -i2n(x~+yr/) ] d x d y .
(2b)
The WD and AF can also equivalently be defined using frequency-domain signals [ 1-3]. The use of Wigner Distribution ( W D ) and Ambiguity Function (AF) in signal analysis, communication, image processing and pattern recognition has been studied [ 1-3 ]. To save computation time and memory space, optical implementations of both WD and AF for both 1D and 2D real signals and images have been described [4-14]. To display the optical WD ( O W D ) and AF (OAF) for a 1D real signal, at a system input plane, two identical signal transparencies, each being rotated+ ( - ) 4 5 °, were superposed [4,5,7]. Using a spherical and cylindrical lens pair, along the two orthogonal directions at a system output plane, the Fourier transform and image of the composite input are obtained. It has been noted [ 7 ] that by rotating a cylindrical lens by 90 °, the OWD (OAF) system can be used to generate an OAF (OWD). A somewhat modified OWD (OAF) processor that requires only a single transparency has also been constructed [ 6]. In an alternative approach, the OWD for a 1D real signal was implemented using a Fourier lens and a pair of synchronized stepping motors [ 8 ]. These OWD (OAF) techniques can be generalized as the 2D optical filtering of 1D signals [ 9 ]. Along the same line, the OWD (OAF) for a 2-D real image has also been studied. Since for a 2D input, the generated OWD (OAF) will be of four dimensions (4D), techniques that sectionally display the 4D OWD (OAF) outputs have been used. With a pair of Fourier lenses, a mirror and a beamsplitter, both single and double transparency OWD (OAF)
0 030-4018/88/$03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
177
Volume 67, number 3
OPTICSCOMMUNICATIONS
methods were reported [10-13]. To increase the display efficiency, using a grating-based multiple imaging system, a multiple-section OWD (OAF) display technique has been suggested [12] and implemented [ 13 ]. Recently, an extension from a monochromatic to a pseudocolor OWD display has been demonstrated [ 14]. All of the above-mentioned methods use real, either 1D or 2D, inputs. However, in many practical situations, 1D and 2D complex signals are often encountered. A proposed ID complex signal OWD method uses a modified Michelson interferometer followed by an imaging/Fourier-transforming system for display [ 15 ]. The interference of two input beams always contains, at its + / - first order, a product of a signal and its complex conjugate term. This product is used to generate the OWD. A second method uses a hologram to represent the complex signal for a OWD processing [ 15 ]. However, for each OWD, time consuming precalculation and recording process is needed. Recently, an alternative complex signal OWD method has been proposed [ 16 ]. First, using a reflective phase filter in a Fourier plane, a product of the original input with its complex conjugate was obtained. Using one of the conventional OWD systems, in a second step, the corresponding OWD for a complex input was obtained. The drawback of the technique is that the input phase must be small, i.e. 0(x) << n. In this letter, a new complex signal OWD (OAF) is proposed. To generate the required product of the input and its complex conjugate, optical phase conjugation (OPC) via nonlinear four-wave mixing [ 17 ] process is used. The proposed method can process either 1D or 2D large phase variation signals and images. In fig. 1, a schematic of an OPC-based OWD (OAF) is depicted. The input is a collimated monochromatic plane wave. Using the beamsplitters B~ and B2, the input is split into three parts: with El, E2 traveling in counterpropagating while E3 in an offangle direction. All three beams are directed to a third order optical nonlinear material. Upon nonlinear interaction, the fourth beam E4, generated in an opposite direction to E3, can be expressed as
E4 = CE1EzE'~ , where C denotes a proportionality constant. Using this OPC property, the OWD (OAF) for complex 178
1July 1988
output- I-
M L ~;~
L~I ~
/ O
\E,
P
C
input
mGerid
~z L5
Dk
Fig. 1. A schematicof an OPC-basedOWD (OAF) displaysetup: M, mirror; B, beamsplitter; D, Doveprism; LI-LT,identical focal length (J) spherical lenses, and Ls, a 45 ° rotated cylindrical lens of focal lengths Together with a plane wave, the two, either rotated (for a 1D case) or shifted (for a 2D case), copies of the input are imaged to the OPC cell to generate an output signal carries the required triple product term for a OWD (OAF) implementation.
signals and images can be generated. As an example, using the setup shown in fig. 1, the implementation of a OWD for a 1D complex signal is first described. Let a complex 1D signalf(x) be inserted between B1 and B2. This 1D signal is then split into two copies; with one copy being imaged onto the OPC material via the beam path L3-B3-L4 while the other copy is rotated 90 ° and then imaged onto the same OPC cell via the beam path L s - D r M 2 - D 2 - L 6 . In a 45 ° rotated cartesian coordinates (x~, x2), the two thus generated signals are E2(x,, x2) = f [ (x,
+x2)/~/21,
E3(x,, x2) = f [ (x, - x 2 ) / x f l 2 l .
(4)
Together with an uniform plane wave El = E, the OPC signal E4 is E4 (Xl, x2 )
=Df[(x, +x2)/xf2lf*[(x,-x2)/x/2l
,
(5)
where D is a constant. This signal, after passing through B3 and a composite lens, i.e. L7 and L8, is imaged along x~ and Fourier transformed in the x2 direction. Thus, at the output plane, the light amplitude distribution is
Volume 67, number 3
Eo(x,, ~ ) = D
J
f[(x,
OPTICS COMMUNICATIONS
+x2)/x/~]
--oo
× f * [ (x~ - x 2 ) / x / ~ ] e x p ( - i 2 n ~ x 2 ) dx2 = (O/x/2)
W(x,/x/2, ¢/x/~).
(6)
Using slight modifications, the O P C system can also be used to display on O A F for a 1D complex signal. In this case, the cylindrical lens L8 is rotated ninety degrees. This rotation exchanges the imaging and Fourier transforming directions. Thus, the corresponding output light distribution is
Eo(x2,¢)=(Olx/~)A(x:lx/~,¢l,,/~).
l July 1988
such as Bil2SiO2o (BSO), can be employed. Using BSO, optical correlation of both real and complex (with phase much larger than n) images have been experimentally demonstrated [ 19 ]. For autocorrelation, the observed processing speed is typically about few hundred microsconds. On the other hand, with an intensity ratio modification, the cross-correlation speed is orders o f magnitude faster [20]. Using this speed advantage, a real-time cross-WD and cross-AE of complex signals and images can be realized. This work was supported in part by a grant from the Air Force Office of Scientific Research.
(7)
In addition, using the OPC-based system, O W D ( O A F ) for a 2D complex image can be implemented. To optically generate eqs. (2a), using a sectional display method [ 10 ], the product o f the 2D image and its 180 ° rotated complex conjugate, each of which is spatially translated by an amount + (Xo, Yo), must be performed. This product is then 2D Fourier transformed to generate a sample, i.e. W(xo, Y0; ~, q), of the 4D OWD. With various spatial shifts, different O W D samples can be obtained. During the ,implementation, the 180 ° image rotation spatial shifts can be obtained via the Dove prisms D~ and D2 as well as the mirror and beamsplitter MI, B3, respectively. Also, the lenses L7 and L8 should be removed. Similarly, samples o f an O A F for a 2D complex image can also be obtained. In this case, the only difference in the implementation procedure is that the image complex conjugate should not be rotated and the Dove prisms DI and D2 are not required. In addition, the OPC-device allows for the generation o f O W D ( O A F ) using frequency domain signals and images. By replacing a space- with a frequency-domain signal in the above described systems, an identical O W D ( O A F ) can be obtained. The above described O W D ( O A F ) systems are similar to those devices proposed for real-time image convolution and correlation [ 18-20 ], and thus, can be physically constructed with the same type o f materials. In particular, for an OPC-based image correlation implementation, a photorefractive material
References [ 1] E. Wigner, Phys. Rev. 40 (1932) 749. [ 2 ] P.W. Woodward, Probability and information theory, with application to radar (Pergamon, Oxford, 1963). [3] T.A.C.M. Claasen and W.F.G. Mecklenbrauker, Philips J. Res. 35 (1980) 217, 372. [4] R.J. Marks II, J.F. Walkup and T.F. Krile, Appl. Optics 16 (1977) 746. [5] M.J. Bastiaans, Optics Comm. 25 (1978) 26. [ 6 ] R.J. Marks II and M.W. Hall, Appl. Optics 18 ( 1979) 2539. [7] M.J. Bastiaans, Appl. Optics 19 (1980) 192. [8] H.O. BarteR, K.-H. Brenner and A.W. Lohmann, Optics Comm. 32 (1980) 32. [9] G. Eichmann and B.Z. Dong, Appl. Optics 21 (1982) 3152. [ 10] R. Bamler and H. Glunder, Proc. 10th Intern. Opt. Compt. Conf., IEEE Compt. Soc. (1983) p. 117. [ 11 ] H.H. Szu and J.A. Blodgett, in: Optics in four dimensions, 1980, AIP Conf. Proc. 65 ( 1981 ) p. 292. [12 ] M. Conner and Y. Li, Appl. Optics 24 (1985) 3825. [13IT. lwai, A.K. Gupta and T. Asakura, Optics Comm. 58 (1986) 15. [ 14 ] S.I. Grosz, W.D. Furlan, E.E. Sicre and M. Garavaglia, Appl. Optics 26 (1987) 971. [15] K.-H. Brenner and A.W. Lohmann, Optics Comm. 42 (1982) 310. [ 16] T. Mateeva and P. Sharlandjiev, Optics Comm. 57 (1986) 153. [17] R.A. Fisher, ed., Optical phase conjugation (Academic Press, New York, 1984). [ 18 ] D.M. Peper, J.A. Yeung, D. Fekete and A. Yariv, Optics Lett. 3 (1978) 7. [ 19] J.O. White and A. Yariv, Appl. Phys. Lett. 37 (1980) 5. [20] P.D. Foote, T.J. Hall and L.M. Connors, Opt. Laser Technology, Feb. (1986) 39.
179
OPTICS COMMUNICATIONS
1 July 1988
OPTICAL W I G N E R D I S T R I B U T I O N AND A M B I G U I T Y F U N C T I O N
FOR C O M P L E X SIGNALS AND I M A G E S Yao LI, George E I C H M A N N and Michael CONNER
DepartmentofElectricalEngineering, The CityCollegeof The City UnviersityofNew York,New York,NY 10031, USA Received 15 February 1988
Using a degenerateoptical phase conjugation device, a new real-time coherent optical Wigner distribution and ambiguityfunction implementation method for both one- and two-dimensionalcomplexsignals is suggested. For a one dimensional (1D) s i g n a l f ( x ) , the Wigner distribution ( W D ) W(x, ~) and ambiguity function (AF) A(x', ~) are defined as
W(x, ~)= f f ( x + x ' / 2 ) f * ( x - x ' / 2 ) --
oo
X e x p ( - i 2 n x ' ~ ) dx' ,
(la)
and
7
A(x', ~)= j f ( x + x , / 2 ) f . ( x - x , / 2 ) ×exp(-i2zcx~) dx,
(lb)
where * denotes complex conjugation. Similarly, the WD and AF for a two dimensional (2D) i m a g e f ( x , y) are defined as
×f*(x-x'/2, y-y'/2) × e x p [ - i 2 n ( x ' ~ + y ' t / ) ] dx' d y ' ,
(2a)
and
xf*(x-x'/2, y-y'/2) × e x p [ -i2n(x~+yr/) ] d x d y .
(2b)
The WD and AF can also equivalently be defined using frequency-domain signals [ 1-3]. The use of Wigner Distribution ( W D ) and Ambiguity Function (AF) in signal analysis, communication, image processing and pattern recognition has been studied [ 1-3 ]. To save computation time and memory space, optical implementations of both WD and AF for both 1D and 2D real signals and images have been described [4-14]. To display the optical WD ( O W D ) and AF (OAF) for a 1D real signal, at a system input plane, two identical signal transparencies, each being rotated+ ( - ) 4 5 °, were superposed [4,5,7]. Using a spherical and cylindrical lens pair, along the two orthogonal directions at a system output plane, the Fourier transform and image of the composite input are obtained. It has been noted [ 7 ] that by rotating a cylindrical lens by 90 °, the OWD (OAF) system can be used to generate an OAF (OWD). A somewhat modified OWD (OAF) processor that requires only a single transparency has also been constructed [ 6]. In an alternative approach, the OWD for a 1D real signal was implemented using a Fourier lens and a pair of synchronized stepping motors [ 8 ]. These OWD (OAF) techniques can be generalized as the 2D optical filtering of 1D signals [ 9 ]. Along the same line, the OWD (OAF) for a 2-D real image has also been studied. Since for a 2D input, the generated OWD (OAF) will be of four dimensions (4D), techniques that sectionally display the 4D OWD (OAF) outputs have been used. With a pair of Fourier lenses, a mirror and a beamsplitter, both single and double transparency OWD (OAF)
0 030-4018/88/$03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
177
Volume 67, number 3
OPTICSCOMMUNICATIONS
methods were reported [10-13]. To increase the display efficiency, using a grating-based multiple imaging system, a multiple-section OWD (OAF) display technique has been suggested [12] and implemented [ 13 ]. Recently, an extension from a monochromatic to a pseudocolor OWD display has been demonstrated [ 14]. All of the above-mentioned methods use real, either 1D or 2D, inputs. However, in many practical situations, 1D and 2D complex signals are often encountered. A proposed ID complex signal OWD method uses a modified Michelson interferometer followed by an imaging/Fourier-transforming system for display [ 15 ]. The interference of two input beams always contains, at its + / - first order, a product of a signal and its complex conjugate term. This product is used to generate the OWD. A second method uses a hologram to represent the complex signal for a OWD processing [ 15 ]. However, for each OWD, time consuming precalculation and recording process is needed. Recently, an alternative complex signal OWD method has been proposed [ 16 ]. First, using a reflective phase filter in a Fourier plane, a product of the original input with its complex conjugate was obtained. Using one of the conventional OWD systems, in a second step, the corresponding OWD for a complex input was obtained. The drawback of the technique is that the input phase must be small, i.e. 0(x) << n. In this letter, a new complex signal OWD (OAF) is proposed. To generate the required product of the input and its complex conjugate, optical phase conjugation (OPC) via nonlinear four-wave mixing [ 17 ] process is used. The proposed method can process either 1D or 2D large phase variation signals and images. In fig. 1, a schematic of an OPC-based OWD (OAF) is depicted. The input is a collimated monochromatic plane wave. Using the beamsplitters B~ and B2, the input is split into three parts: with El, E2 traveling in counterpropagating while E3 in an offangle direction. All three beams are directed to a third order optical nonlinear material. Upon nonlinear interaction, the fourth beam E4, generated in an opposite direction to E3, can be expressed as
E4 = CE1EzE'~ , where C denotes a proportionality constant. Using this OPC property, the OWD (OAF) for complex 178
1July 1988
output- I-
M L ~;~
L~I ~
/ O
\E,
P
C
input
mGerid
~z L5
Dk
Fig. 1. A schematicof an OPC-basedOWD (OAF) displaysetup: M, mirror; B, beamsplitter; D, Doveprism; LI-LT,identical focal length (J) spherical lenses, and Ls, a 45 ° rotated cylindrical lens of focal lengths Together with a plane wave, the two, either rotated (for a 1D case) or shifted (for a 2D case), copies of the input are imaged to the OPC cell to generate an output signal carries the required triple product term for a OWD (OAF) implementation.
signals and images can be generated. As an example, using the setup shown in fig. 1, the implementation of a OWD for a 1D complex signal is first described. Let a complex 1D signalf(x) be inserted between B1 and B2. This 1D signal is then split into two copies; with one copy being imaged onto the OPC material via the beam path L3-B3-L4 while the other copy is rotated 90 ° and then imaged onto the same OPC cell via the beam path L s - D r M 2 - D 2 - L 6 . In a 45 ° rotated cartesian coordinates (x~, x2), the two thus generated signals are E2(x,, x2) = f [ (x,
+x2)/~/21,
E3(x,, x2) = f [ (x, - x 2 ) / x f l 2 l .
(4)
Together with an uniform plane wave El = E, the OPC signal E4 is E4 (Xl, x2 )
=Df[(x, +x2)/xf2lf*[(x,-x2)/x/2l
,
(5)
where D is a constant. This signal, after passing through B3 and a composite lens, i.e. L7 and L8, is imaged along x~ and Fourier transformed in the x2 direction. Thus, at the output plane, the light amplitude distribution is
Volume 67, number 3
Eo(x,, ~ ) = D
J
f[(x,
OPTICS COMMUNICATIONS
+x2)/x/~]
--oo
× f * [ (x~ - x 2 ) / x / ~ ] e x p ( - i 2 n ~ x 2 ) dx2 = (O/x/2)
W(x,/x/2, ¢/x/~).
(6)
Using slight modifications, the O P C system can also be used to display on O A F for a 1D complex signal. In this case, the cylindrical lens L8 is rotated ninety degrees. This rotation exchanges the imaging and Fourier transforming directions. Thus, the corresponding output light distribution is
Eo(x2,¢)=(Olx/~)A(x:lx/~,¢l,,/~).
l July 1988
such as Bil2SiO2o (BSO), can be employed. Using BSO, optical correlation of both real and complex (with phase much larger than n) images have been experimentally demonstrated [ 19 ]. For autocorrelation, the observed processing speed is typically about few hundred microsconds. On the other hand, with an intensity ratio modification, the cross-correlation speed is orders o f magnitude faster [20]. Using this speed advantage, a real-time cross-WD and cross-AE of complex signals and images can be realized. This work was supported in part by a grant from the Air Force Office of Scientific Research.
(7)
In addition, using the OPC-based system, O W D ( O A F ) for a 2D complex image can be implemented. To optically generate eqs. (2a), using a sectional display method [ 10 ], the product o f the 2D image and its 180 ° rotated complex conjugate, each of which is spatially translated by an amount + (Xo, Yo), must be performed. This product is then 2D Fourier transformed to generate a sample, i.e. W(xo, Y0; ~, q), of the 4D OWD. With various spatial shifts, different O W D samples can be obtained. During the ,implementation, the 180 ° image rotation spatial shifts can be obtained via the Dove prisms D~ and D2 as well as the mirror and beamsplitter MI, B3, respectively. Also, the lenses L7 and L8 should be removed. Similarly, samples o f an O A F for a 2D complex image can also be obtained. In this case, the only difference in the implementation procedure is that the image complex conjugate should not be rotated and the Dove prisms DI and D2 are not required. In addition, the OPC-device allows for the generation o f O W D ( O A F ) using frequency domain signals and images. By replacing a space- with a frequency-domain signal in the above described systems, an identical O W D ( O A F ) can be obtained. The above described O W D ( O A F ) systems are similar to those devices proposed for real-time image convolution and correlation [ 18-20 ], and thus, can be physically constructed with the same type o f materials. In particular, for an OPC-based image correlation implementation, a photorefractive material
References [ 1] E. Wigner, Phys. Rev. 40 (1932) 749. [ 2 ] P.W. Woodward, Probability and information theory, with application to radar (Pergamon, Oxford, 1963). [3] T.A.C.M. Claasen and W.F.G. Mecklenbrauker, Philips J. Res. 35 (1980) 217, 372. [4] R.J. Marks II, J.F. Walkup and T.F. Krile, Appl. Optics 16 (1977) 746. [5] M.J. Bastiaans, Optics Comm. 25 (1978) 26. [ 6 ] R.J. Marks II and M.W. Hall, Appl. Optics 18 ( 1979) 2539. [7] M.J. Bastiaans, Appl. Optics 19 (1980) 192. [8] H.O. BarteR, K.-H. Brenner and A.W. Lohmann, Optics Comm. 32 (1980) 32. [9] G. Eichmann and B.Z. Dong, Appl. Optics 21 (1982) 3152. [ 10] R. Bamler and H. Glunder, Proc. 10th Intern. Opt. Compt. Conf., IEEE Compt. Soc. (1983) p. 117. [ 11 ] H.H. Szu and J.A. Blodgett, in: Optics in four dimensions, 1980, AIP Conf. Proc. 65 ( 1981 ) p. 292. [12 ] M. Conner and Y. Li, Appl. Optics 24 (1985) 3825. [13IT. lwai, A.K. Gupta and T. Asakura, Optics Comm. 58 (1986) 15. [ 14 ] S.I. Grosz, W.D. Furlan, E.E. Sicre and M. Garavaglia, Appl. Optics 26 (1987) 971. [15] K.-H. Brenner and A.W. Lohmann, Optics Comm. 42 (1982) 310. [ 16] T. Mateeva and P. Sharlandjiev, Optics Comm. 57 (1986) 153. [17] R.A. Fisher, ed., Optical phase conjugation (Academic Press, New York, 1984). [ 18 ] D.M. Peper, J.A. Yeung, D. Fekete and A. Yariv, Optics Lett. 3 (1978) 7. [ 19] J.O. White and A. Yariv, Appl. Phys. Lett. 37 (1980) 5. [20] P.D. Foote, T.J. Hall and L.M. Connors, Opt. Laser Technology, Feb. (1986) 39.
179