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Spatially coherent white light correlator
Volume 27, number 1
OPTICS ~OMMUNICATIONS
October 1978
SPATIALLY COHERENT WHITE LIGHT CORRELATOR Jean-Pierre GOEDGEBUER and Richard GAZEU Laboratoire de Physique G~n~rale et Optique *, Universitd de Franche-Comtd, 2.5030 Besanfon Cedex, France Received 30 June 1978
A spatial-temporal correlator in which the spectral components of light behave as independent information carriers working simultaneously is described. The use of spatially coherent light permits to process the amplitude and the phase of an imput data through a chromatic Fourier Hologram acting as a matched filter. A rough estimation of the S/N ratio shows a great increase compared to that obtained in conventional monochromatic correlators. It could be applied in pattern recognition to reduce the so-called false alarms.
1. Introduction and principle of operation The demonstration brought out in recent papers [ i ' 2 ] about the use of spatial and temporal frequencies of light in holography, interferometry and speckle phenomena also apply to convolution and correlation aiaalog techniques. We propose a white light processor p~rforming 1-D convolution and correlation products ~ t w e e n an input message l a n d a signal g stored in a chromatic Fourier hologram. Compared to other correlators working in incoherent white light [3,4, 5], the present correlator takes advantage of the well-known * Associe'au CNRS no. 214.
input plane x /
,U__
flexibility and data-handling capacity of coherent systems, while retaining a S/N ratio comparable to that obtained with incoherent processors. The basic experimental filtering device is similar to that of conventional coherent correlators. The set-up is sketched in fig. 1. It mainly consists in two spectroscopic devices (P1, L3) and (L 4, P2) adjusted in cascade so that their dispersion law cancel each other. The principle of operation can be summarized as follows. Any beam of white light incident onto the system is dispersed in the spectral plane (u', ?~). As one notices that the variables x and u' are reciprocal through a Fourier transform which is performed by the lenses (L1, L2, L3) , one realizes that the spectral
filtering IIl|ne
FT ---~/Y'
'
Fig. 1. Multi-wavelength filtering device. The input and output planes are conjugated through the lenses L1, L2, L3 and L4. Dispersors P1 and P2 are adjusted so that any incident beam of white light is dispersed in the filtering plane, and then recombined to Yield white light at the output. 53
Volume 27, number 1
OPTICS COMMUNICATIONS
plane (u', X) can be considered as a filtering plane for each wavelength X. The next step consists in the recombination of the spectral components to yield white light. It is achieved by the second part (L4, P2) of the correlator, that is symmetrical to the first one, as in coherent correlators. The lens (L4) Fourier transforms the amplitude distribution in the filtering plane and the resulting polychromatic responses are gathered at the output plane (x',y'), by means of the dispersor P2" The filtering operation has to be carried out with suitable masks. Both amplitude and phase filters can be used. The example we give in the following deals with a chromatic Fourier Hologram, that permits convolution and correlation integrals to be computed.
2. Recording of a chromatic Fourier hologram as a matched filter The holographic filter is synthesized with the help of the first part of the correlator, just as in monochromatic processors. The set-up, which has been put into operations for several years in our laboratory, was described for another application [6]. In fig. 2a the technique is illustrated by the recording of the chromatic Fourier hologram of a rectangular aperture g(x). The reference is a thin slit located at x = x 0 and emitting a wave of white light of amplitude A. The pattern at the plane (u', X) is a sequence of 1-Dpolychromatic Fourier holograms lying along the u'-direction and displayed along the X-axis. The energy distribution H(u', X) at the output plane of the spectroscope is proportional to [6]:
,
t~
/
HI
*.,,,
V
Xexp - j 2 ~ - ~ x c¢A2+ G +G
,
()/ /,/
dx
exp j21r .
U
/
~-~ exp -j2rr~-~x 0
x0 (1)
where G(u'/Xf) = FT{g(x)} is the spatial Fourier transform ofg(x), 6(x) is the Dirac impulse, and * denotes a complex conjugate. Fig. 3a and 3b are photographs of spectro-holograms recorded in different conditions. Such chromatic Fourier holograms may be inserted in the processing system illustrated previously in fig. 1. For simplicity's sake, we will assume that the amplitude transmittance t(u', X) of the spectro-hologram is proportional to the intensity distribution H(u', X) incident during exposure.
3. Processing the input data
Let f(x) be the input data to be filtered. The chromatic Fourier hologram is replaced in its recording position in the filtering plane (u', X). For a radiation
L3
I*
Fig. 2. Principle of recording a chromatic Fourier hologram (after ref. [6]). The various spectral components of light generate monochromatic holograms repeated along the h-axis.
54
{g(x) + Aa(x - Xo)}
.
P
''
I+/
Sp
AI
6qx-~afA t l
H(u', x) ~
October 1978
Fig. 3. Chromatic Fourier holograms of a slit recorded at the output of the set-up of fig. 2. As in the recording of conventional matched filters, characteristic frequency bands can be enhanced by controlling the time-exposure. In a, the recording is approximately linear in energy; in b, the hologram is matched to high frequencies by saturating the low frequencies.
OPTICS COMMUNICATIONS
Volume 27, number 1
of wavelength X, the amplitude distribution U(x') at the output plane,is the convolution product o f f ( x ) with the spatial impulse response of the correlator:
U(x') = f ( x ' ) ® FT(t(u', X)} = A2f(x ') + f(x') ® {g(x') * g(x')} + 8(x' + x o) ® {g(x') ® f(x')} + ~ ( x ' - x o) ~ {f(x') • g(x')},
(2)
where ® stands for a convolution integral, and * for a correlation product. Eq. (2) describes the well-known responses of a coherent correlator, as a Fourier hologram is used as a filter [7]. One gets two lateral terms located at x' = - x 0 and x' = x 0. They correspond to convolution and cross-correlation products respectively. One should note that eq. (2) is independent of the wavelength. It means that the various spectral components of white light generate the same responses, the latters being superimposed at the output of the system since all the radiations are recombined by the dispersor P2"
a.
b.
Fig. 4. Responses obtained at the output plane of the correlator with the filters of fig. 3. The zero order term of diffraction represents the input message somwhat altered. The +_1 orders are images of the processed information (convolution and correlation by a rectangular signal). In b, the use of the Fourier hologram of fig. 3b yields a derivative effect along x'-axis.
October 1978
Fig. 4 gives examples of the responses obtained at the output of the correlator as the Fourier holograms of fig. 3 are used as chromatic filters. The input message is a set of 2-D letters displayed along the y-axis. The filtering operation concerns the u'-direction only, orthogonal to the k-axis. In fig. 4a, the lateral images correspond to convolution and correlation of the set of letters with a rectangular message. It results in a thickening of the characters along the x'-direction. In fig. 4b, the aspect of dedoubling is caused by the derivative effect introduced by the saturation of the low frequencies in the filter shown in fig. 3b.
4. Conclusion The system can be termed a multiplexing correlator. Multiplexing refers to the fact that various chromatic components carry the same information simultaneously. Correlations and convolutions are performed with the help of chromatic Fourier holograms as spatial and spectral filters. However the technique is limited to 1-D filtering as far as the recording of the filter is concerned. As we announced it in the introduction, the present correlator essentially differs from other types working in white light [3-5] in the sense it takes advantage of the spatially coherent illumination, that permits (i) to enhance characteristic frequency bands with matched filters, (ii) to process both the amplitude and the phase of an input data. One has to note that the processed signal obtained at the output of the correlator results from the superposition of the responses generated by the spectral components of light - without X - distorsions. An increase of S/N ratio can be expec{ed from such a situation, compared to that obtained in conventional monochromatic correlators. A rough idea of it can be given. Following the approach exposed in ref. [8] about incoherent optics and generalizing the results to white light optics, the potential increase of S/N ratio might be equal to x/~, where N is the number of channels, i.e. the number of spectral elements resolved by the system. As the resolvance power of the spectroscopic devices is R -- X/SX = 106, the number N of the resolved spectral components in the visible spectrum (AX ~ 0.4/am) is N ~ AX/SX = (0.4 X 106)/0.6 = ~- X 106, that would correspond to S/N ~, 103 . Such an order of magnitude has to be verified. A 55
Volume 27, number 1
OPTICS COMMUNICATIONS
next paper will deal with an experimentaland theoretical study of the S/N ratio in white light devices.
Acknowledgement The authors wish to thank Prof. J.-Ch. Vi6not for his guidance and fruitful suggestions, including in the editing of the manuscript, and the C.N.R.S. for its financial support in the frame o f the "Action Th6matique Programm6e: Traitement des Images".
56
October 1978
References [ 1] J.-Ch. Vienot, J.P. Goedgebuer and A. Lacourt, Applied Optics 16 (1977) 454. [2] J.P. Goedgebuer and J.-Ch. Vienot, Optics Commun. 19 (1976) 229. [3] J.D. Armitage, A. Lohmann and D.P. Paris, Japanese J. of Applied Physics, 4 suppl. 1 (1965). [4] A.W. Lohmann and H. Bartelt, CNRS-DFG Meeting on Image processing, Bad Honnef (W. Germany) March 1978. [5] A. Lacourt, Optics Commun. 27 (1978) 47. [6] J. Calatroni, Optics Commun. 19 (1976) 49. [7] A.B. Vanderlugt, IEEE Transf. Inf. Theory 10 41964) 139. [8] P. Chavel and S. Lowenthal, J. Opt. Soc. of America, to be published.
OPTICS ~OMMUNICATIONS
October 1978
SPATIALLY COHERENT WHITE LIGHT CORRELATOR Jean-Pierre GOEDGEBUER and Richard GAZEU Laboratoire de Physique G~n~rale et Optique *, Universitd de Franche-Comtd, 2.5030 Besanfon Cedex, France Received 30 June 1978
A spatial-temporal correlator in which the spectral components of light behave as independent information carriers working simultaneously is described. The use of spatially coherent light permits to process the amplitude and the phase of an imput data through a chromatic Fourier Hologram acting as a matched filter. A rough estimation of the S/N ratio shows a great increase compared to that obtained in conventional monochromatic correlators. It could be applied in pattern recognition to reduce the so-called false alarms.
1. Introduction and principle of operation The demonstration brought out in recent papers [ i ' 2 ] about the use of spatial and temporal frequencies of light in holography, interferometry and speckle phenomena also apply to convolution and correlation aiaalog techniques. We propose a white light processor p~rforming 1-D convolution and correlation products ~ t w e e n an input message l a n d a signal g stored in a chromatic Fourier hologram. Compared to other correlators working in incoherent white light [3,4, 5], the present correlator takes advantage of the well-known * Associe'au CNRS no. 214.
input plane x /
,U__
flexibility and data-handling capacity of coherent systems, while retaining a S/N ratio comparable to that obtained with incoherent processors. The basic experimental filtering device is similar to that of conventional coherent correlators. The set-up is sketched in fig. 1. It mainly consists in two spectroscopic devices (P1, L3) and (L 4, P2) adjusted in cascade so that their dispersion law cancel each other. The principle of operation can be summarized as follows. Any beam of white light incident onto the system is dispersed in the spectral plane (u', ?~). As one notices that the variables x and u' are reciprocal through a Fourier transform which is performed by the lenses (L1, L2, L3) , one realizes that the spectral
filtering IIl|ne
FT ---~/Y'
'
Fig. 1. Multi-wavelength filtering device. The input and output planes are conjugated through the lenses L1, L2, L3 and L4. Dispersors P1 and P2 are adjusted so that any incident beam of white light is dispersed in the filtering plane, and then recombined to Yield white light at the output. 53
Volume 27, number 1
OPTICS COMMUNICATIONS
plane (u', X) can be considered as a filtering plane for each wavelength X. The next step consists in the recombination of the spectral components to yield white light. It is achieved by the second part (L4, P2) of the correlator, that is symmetrical to the first one, as in coherent correlators. The lens (L4) Fourier transforms the amplitude distribution in the filtering plane and the resulting polychromatic responses are gathered at the output plane (x',y'), by means of the dispersor P2" The filtering operation has to be carried out with suitable masks. Both amplitude and phase filters can be used. The example we give in the following deals with a chromatic Fourier Hologram, that permits convolution and correlation integrals to be computed.
2. Recording of a chromatic Fourier hologram as a matched filter The holographic filter is synthesized with the help of the first part of the correlator, just as in monochromatic processors. The set-up, which has been put into operations for several years in our laboratory, was described for another application [6]. In fig. 2a the technique is illustrated by the recording of the chromatic Fourier hologram of a rectangular aperture g(x). The reference is a thin slit located at x = x 0 and emitting a wave of white light of amplitude A. The pattern at the plane (u', X) is a sequence of 1-Dpolychromatic Fourier holograms lying along the u'-direction and displayed along the X-axis. The energy distribution H(u', X) at the output plane of the spectroscope is proportional to [6]:
,
t~
/
HI
*.,,,
V
Xexp - j 2 ~ - ~ x c¢A2+ G +G
,
()/ /,/
dx
exp j21r .
U
/
~-~ exp -j2rr~-~x 0
x0 (1)
where G(u'/Xf) = FT{g(x)} is the spatial Fourier transform ofg(x), 6(x) is the Dirac impulse, and * denotes a complex conjugate. Fig. 3a and 3b are photographs of spectro-holograms recorded in different conditions. Such chromatic Fourier holograms may be inserted in the processing system illustrated previously in fig. 1. For simplicity's sake, we will assume that the amplitude transmittance t(u', X) of the spectro-hologram is proportional to the intensity distribution H(u', X) incident during exposure.
3. Processing the input data
Let f(x) be the input data to be filtered. The chromatic Fourier hologram is replaced in its recording position in the filtering plane (u', X). For a radiation
L3
I*
Fig. 2. Principle of recording a chromatic Fourier hologram (after ref. [6]). The various spectral components of light generate monochromatic holograms repeated along the h-axis.
54
{g(x) + Aa(x - Xo)}
.
P
''
I+/
Sp
AI
6qx-~afA t l
H(u', x) ~
October 1978
Fig. 3. Chromatic Fourier holograms of a slit recorded at the output of the set-up of fig. 2. As in the recording of conventional matched filters, characteristic frequency bands can be enhanced by controlling the time-exposure. In a, the recording is approximately linear in energy; in b, the hologram is matched to high frequencies by saturating the low frequencies.
OPTICS COMMUNICATIONS
Volume 27, number 1
of wavelength X, the amplitude distribution U(x') at the output plane,is the convolution product o f f ( x ) with the spatial impulse response of the correlator:
U(x') = f ( x ' ) ® FT(t(u', X)} = A2f(x ') + f(x') ® {g(x') * g(x')} + 8(x' + x o) ® {g(x') ® f(x')} + ~ ( x ' - x o) ~ {f(x') • g(x')},
(2)
where ® stands for a convolution integral, and * for a correlation product. Eq. (2) describes the well-known responses of a coherent correlator, as a Fourier hologram is used as a filter [7]. One gets two lateral terms located at x' = - x 0 and x' = x 0. They correspond to convolution and cross-correlation products respectively. One should note that eq. (2) is independent of the wavelength. It means that the various spectral components of white light generate the same responses, the latters being superimposed at the output of the system since all the radiations are recombined by the dispersor P2"
a.
b.
Fig. 4. Responses obtained at the output plane of the correlator with the filters of fig. 3. The zero order term of diffraction represents the input message somwhat altered. The +_1 orders are images of the processed information (convolution and correlation by a rectangular signal). In b, the use of the Fourier hologram of fig. 3b yields a derivative effect along x'-axis.
October 1978
Fig. 4 gives examples of the responses obtained at the output of the correlator as the Fourier holograms of fig. 3 are used as chromatic filters. The input message is a set of 2-D letters displayed along the y-axis. The filtering operation concerns the u'-direction only, orthogonal to the k-axis. In fig. 4a, the lateral images correspond to convolution and correlation of the set of letters with a rectangular message. It results in a thickening of the characters along the x'-direction. In fig. 4b, the aspect of dedoubling is caused by the derivative effect introduced by the saturation of the low frequencies in the filter shown in fig. 3b.
4. Conclusion The system can be termed a multiplexing correlator. Multiplexing refers to the fact that various chromatic components carry the same information simultaneously. Correlations and convolutions are performed with the help of chromatic Fourier holograms as spatial and spectral filters. However the technique is limited to 1-D filtering as far as the recording of the filter is concerned. As we announced it in the introduction, the present correlator essentially differs from other types working in white light [3-5] in the sense it takes advantage of the spatially coherent illumination, that permits (i) to enhance characteristic frequency bands with matched filters, (ii) to process both the amplitude and the phase of an input data. One has to note that the processed signal obtained at the output of the correlator results from the superposition of the responses generated by the spectral components of light - without X - distorsions. An increase of S/N ratio can be expec{ed from such a situation, compared to that obtained in conventional monochromatic correlators. A rough idea of it can be given. Following the approach exposed in ref. [8] about incoherent optics and generalizing the results to white light optics, the potential increase of S/N ratio might be equal to x/~, where N is the number of channels, i.e. the number of spectral elements resolved by the system. As the resolvance power of the spectroscopic devices is R -- X/SX = 106, the number N of the resolved spectral components in the visible spectrum (AX ~ 0.4/am) is N ~ AX/SX = (0.4 X 106)/0.6 = ~- X 106, that would correspond to S/N ~, 103 . Such an order of magnitude has to be verified. A 55
Volume 27, number 1
OPTICS COMMUNICATIONS
next paper will deal with an experimentaland theoretical study of the S/N ratio in white light devices.
Acknowledgement The authors wish to thank Prof. J.-Ch. Vi6not for his guidance and fruitful suggestions, including in the editing of the manuscript, and the C.N.R.S. for its financial support in the frame o f the "Action Th6matique Programm6e: Traitement des Images".
56
October 1978
References [ 1] J.-Ch. Vienot, J.P. Goedgebuer and A. Lacourt, Applied Optics 16 (1977) 454. [2] J.P. Goedgebuer and J.-Ch. Vienot, Optics Commun. 19 (1976) 229. [3] J.D. Armitage, A. Lohmann and D.P. Paris, Japanese J. of Applied Physics, 4 suppl. 1 (1965). [4] A.W. Lohmann and H. Bartelt, CNRS-DFG Meeting on Image processing, Bad Honnef (W. Germany) March 1978. [5] A. Lacourt, Optics Commun. 27 (1978) 47. [6] J. Calatroni, Optics Commun. 19 (1976) 49. [7] A.B. Vanderlugt, IEEE Transf. Inf. Theory 10 41964) 139. [8] P. Chavel and S. Lowenthal, J. Opt. Soc. of America, to be published.