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A spatially coherent achromatic fourier transformer
Volume 42, number 4
OPTICS COMMUNICATIONS
15 July 1982
A SPATIALLY COHERENT ACHROMATIC FOURIER TRANSFORMER R. F E R R I E R E and J.P. GOEDGEBUER Laboratoire de Physique Gdn~rale et Opaque, Assoei~ au CNRS no. 214, Holographie et Traitement Opaque des Signaux, Universit~ de Franche-Comt~, 25030 Besan¢on Cedex, France
Received 14 April 1982
A specific configuration is proposed wherein two zone plates combined with an achromatic objective permits to cancel the chromatic scale factor as broad-band illumination is used in the Fraunhofer diffraction process. The set-up acts as a wavelength-independent Fourier transformer yielding an achromatic diffraction pattern of any complex pupil. Experimental results are also presented and demonstrate the ability of the system to work over the visible spectrum.
I. Introduction One of the most promising characteristics of optics lies in its ability to perform 2-D analogical Fourier transforms in real-time by means of conventional lenses. However the process requires strict conditions upon the coherence of the illumination, sometimes turning out to drawbacks as far as the spectral broadness of the source is concerned. For several years, much work has been devoted to processors working with polychromatic or broad-band light [ 1 - 1 1 ]. Most o f them use gratings or zone plates as beamsplitters, or as particular achromatic illumination devices [2,11 ]. More recently, Morris and George have demonstrated matched spatial filtering using an achromatic optical transformation [ 5 - 7 ] . They also proposed a theoretical description o f a three-lens achromatic Fourier transform system where every lens is in fact a combination of a specially designed objective with a holographic zone lens [9,10]. In this paper we present an achromatic Fourier transformer where a couple of Fresnel zone plates in cascade with an achromat yields a wavelength-independent Fraunhofer diffraction pattern of any pupil [8]. The first zone plate works simultaneously as a Fourier transformer for the spatial frequencies and as a chromatic dispersor for the spectral components of light. The second one, combined with an achromat, insures a chromatic recombination of the spectral 0 030-4018/82/0000-0000/$ 02.75 © 1982 North-Holland
components and acts as an imaging system for the spatial frequencies. Such a device gives the achromatic Fourier transform o f any complex pupil, independently of the spectral content of the source. The broadband operation o f the system is shown with experimental results.
2. Principle of operation The diffraction pattern Go(x2o/f, y2ohO of. a puPil g(xl, Yl ) illuminated in monochromatic light of wavenumber o and observed at the back focal plane (x2, Y2) o f a lens is:
Go(X2°/f'Y2O/f) =
g(Xl ' Y l ) _oo
!
X exp
{-j2zt(XlX2Off+ylY2O]f)}dXl dYl[
2 ,
(1)
m
where f i s the focal length of the lens. With broadband illumination o f power-spectrum B(o) ranging from o 1 to 02, the distribution o f energy at the rear focal plane of the lens becomes: 02
Gl(X2,Y2) = f
B(o) Go(x2o]f,Y2O/f)do.
(2)
etI
223
Volume
42, number 4
OPTICSCOMMUNICATIONS
It results in a chromatic blurring of the diffraction patterns Go(x2o/f , Y2 off) the latters varying proportionally in size with wavelength according to a chromatic scale factor S(e) =fie. One notesthat the chromatic difference of scale is cancelled as the scale factor S(o) does not depend on e any more, e.g. if the focal length f of the lens arrangement is proportional to o:f= ko. This requirement is fulfilled with a holographic zone plate whose focal length is defined by f(o) = Ro/oo, where o0 and R are the wavenumber and the radius of curvature of the spherical wave producing the zone plate at the recording process. Under this condition, eq. (2) reads:
G2(x2,Y2) =
f
B(o)
01
l;;
g(xl,Yl)
_oa
2 X expl-j27r ~°° (XlX 2 +ylY2) 1 dx 1 dy I
=
g ( x l , Y l ) exp
27r
do
IX2 +YlY2
--oo
× dx 1 dy 1
B(a)do al o2
= GO(x2oo/R,Y2oo[R) f
B(o)do.
(3)
tr 1
The distribution of energy described by eq. (3) is now that of a wavelength-independent diffraction pattern GO(x 2 o 0 JR, y 2 °o/R) whose size is determined by the scale factor R/o 0 (depending on the recording param: eters of the zone plate), and weighted by the energy of the source. In fact, due to the chromatic dispersion of a zone plate, the energy distribution G2(x2, Y2) is displayed along the optical axis z of the zone plate, the variable o in the volume integral (3) being linked to the longitudinal variable z by the dispersion law of the zone plate: z = Ro/o 0 for the first order of diffraction. In other words, the energy distribution G2(x2,Y2) takes the form of wavelength-independent diffraction patterns GO(x 2 o0 ]R, y 2 °O]R ) pile d along the optical axis z of the zone plate. The overall pattern is a volume of diffraction. Such a situation is encountered in the set224
15 July 1982
Fig. i Experimental set-up;g(xl,Yl): input pupil; ZP1 and ZP 2 : holographic zone plates; O1 and 02 : achromatic lenses; G2(x2, Y2): achromatic diffraction pattern of g(x 1, Y 1). The light grey and dark grey regions represent the optical path for two spectral components, e.g. red and blue respectively. up shown in fig. 1. The input pupil g(x 1 ,Yl) is illuminated with a collimated beam of polychromatic light. The on-axis zone plate ZP 1 yields the wavelengthindependent Fraunhofer diffraction patterns Go(x2°o[R,Y2°o [R) ° f g ( x l ,Yl) for every spectral component emitted by the source. As mentioned above, these diffraction patterns are displayed in depth along the optical axis; their abscissa z from ZP1 is z = Ro/o O. The overall pattern is the volume of diffraction V discussed so far, with parallel generating lines, i.e. a cylinder of diffraction. The key yielding the achromatic diffraction pattern G2(x2, Y2) is then to recombine in a single plane the patterns Go(X2oo/R,Y2oo]R ) displayed along zaxis. It is achieved by the lens combination (O1, ZP2, 02) whose disposition is governed by the lens combination formulas of geometrical optics. The achromatic doublet O1, of focal length f l , images the cylinder V at V'. The doublet O 1 is located at l = 2R + f l from ZP 1 so that the axial and longitudinal geometrical magnifications between V and V' are quasi linear in wavenumber o. The second zone plate ZP 2 acts on V' as an imaging system. Its focal length is f2(o) = f21a/Ro O. Its image focal planes are superimposed with V', wavenumber by wavenumber. Under this condition and after the lens combination formulas, it can be shown that the image of V' through ZP2 is compressed in a single plane set at infinity, yielding an achromatic diffraction pattern well-corrected chromatically. Such a pattern can be observed at the back focal plane of O2 (focal length f l ) ; it is described by eq. (3).
Volume 42, number 4
OPTICS COMMUNICATIONS
15 July 1982
• arc lamp (400--600 rim). For comparison, the diffraction patterns o f the same pupils are also shown (left) as the achromatic arrangement (ZPI, O1, ZP2)is removed from the set-up, resulting in a chromatic blurring of the patterns. From a general point o f view, any 2-D pupil can be Fourier transformed, whether it be an amplitude or a phase pupil. Several parameters remain to be studied, such as the chromatic aberrations or the spectral resolution o f the set-up. These parameters determine the accuracy of coincidence of the various monochromatic patterns that are superimposed at the output plane, and therefore the accuracy of the achromatic Fourier transform. However preliminary results indicate a fairly good chromatic compensation yielding a Fourier transformation that is well chromatically corrected over a wide range of wavelengths; the error on the achromatic Fourier transformation is estimated to be of about 2% for the visible spectrum ( 4 0 0 - 6 0 0 nm).
Acknowledgement
Fig. 2. (a) Diffraction pattern of an amplitude grid illuminated in white light. On the left, the diffraction pattern is recorded at the back focal of a conventional lens (chromatic blurring); on the right, the pattern is recorded at the output plane x2 of the set-up of fig. 1. (b) Young's experiment in white light. On the left, the contrast of the fringes decreases rapidly on both sides of the zero-order fringe; on the right, the interference pattern is well-corrected chromatically: the contrast of the fringes is high and constant throughout the pattern.
3. Experimental results and conclusion Eq. (3) indicates that the size of the achromatic diffraction pattern observed at the output plane of the set-up is determined by the scale factor oo/R, depending on the recording parameters of ZP 1 . The experimental results shown in fig. 2 have been obtained with a 0 = 1.94/am - 1 (5145 A ) R = 30 cm and with f l = 30 cm. On the right, they are photographs of achromatic diffraction patterns of pupils (a 2-D grid and two slits) illuminated in white light by a xenon
The authors wish to thank Prof. J.Ch. Vidnot, Dr. N. Aebischer and Dr. G. Tribillon for their fruitful discussions in the writing of the manuscript and the DRET for its financial support.
References [1] R. Katyl, Appl. Optics 11 (1972) 622. [2] E. Leith and J. Roth, Appl. Optics 16 (1977) 2565. [3] J.P. Goedgebuer and R. Gazeu, Optics Comm. 27 (1978) 53. [4] R. Ferri6re, J.P. Goedgebuer and J.Ch. Vidnot, Optics Comm. 31 (1979) 285. [5] G. Morris and N. George, Optics Lett. 5 (1980) 202. [6] G. Morris and N. George, Optics Lett. 5 (1980) 446. [7] G. Morris and N. George, Appl. Optics 19 (1980) 3843. [8] R. Ferri6re, J.P. Goedgebuer and J.Ch. Vi6not, Confdfence Europ~enne d'Optique 1980, Horizons de Foptique, April 1980. [9] G. Morris, Appl. Optics 20 (1981) 2017. [10] N. George and G. Morris, ICO XII, eds. F.T. Arteechi and F.R. Aussenegg, Graz (Austria, September 1981); Current trends in optics. [11] G. Collins, Appl. Optics 20 (1981) 3109.
225
OPTICS COMMUNICATIONS
15 July 1982
A SPATIALLY COHERENT ACHROMATIC FOURIER TRANSFORMER R. F E R R I E R E and J.P. GOEDGEBUER Laboratoire de Physique Gdn~rale et Opaque, Assoei~ au CNRS no. 214, Holographie et Traitement Opaque des Signaux, Universit~ de Franche-Comt~, 25030 Besan¢on Cedex, France
Received 14 April 1982
A specific configuration is proposed wherein two zone plates combined with an achromatic objective permits to cancel the chromatic scale factor as broad-band illumination is used in the Fraunhofer diffraction process. The set-up acts as a wavelength-independent Fourier transformer yielding an achromatic diffraction pattern of any complex pupil. Experimental results are also presented and demonstrate the ability of the system to work over the visible spectrum.
I. Introduction One of the most promising characteristics of optics lies in its ability to perform 2-D analogical Fourier transforms in real-time by means of conventional lenses. However the process requires strict conditions upon the coherence of the illumination, sometimes turning out to drawbacks as far as the spectral broadness of the source is concerned. For several years, much work has been devoted to processors working with polychromatic or broad-band light [ 1 - 1 1 ]. Most o f them use gratings or zone plates as beamsplitters, or as particular achromatic illumination devices [2,11 ]. More recently, Morris and George have demonstrated matched spatial filtering using an achromatic optical transformation [ 5 - 7 ] . They also proposed a theoretical description o f a three-lens achromatic Fourier transform system where every lens is in fact a combination of a specially designed objective with a holographic zone lens [9,10]. In this paper we present an achromatic Fourier transformer where a couple of Fresnel zone plates in cascade with an achromat yields a wavelength-independent Fraunhofer diffraction pattern of any pupil [8]. The first zone plate works simultaneously as a Fourier transformer for the spatial frequencies and as a chromatic dispersor for the spectral components of light. The second one, combined with an achromat, insures a chromatic recombination of the spectral 0 030-4018/82/0000-0000/$ 02.75 © 1982 North-Holland
components and acts as an imaging system for the spatial frequencies. Such a device gives the achromatic Fourier transform o f any complex pupil, independently of the spectral content of the source. The broadband operation o f the system is shown with experimental results.
2. Principle of operation The diffraction pattern Go(x2o/f, y2ohO of. a puPil g(xl, Yl ) illuminated in monochromatic light of wavenumber o and observed at the back focal plane (x2, Y2) o f a lens is:
Go(X2°/f'Y2O/f) =
g(Xl ' Y l ) _oo
!
X exp
{-j2zt(XlX2Off+ylY2O]f)}dXl dYl[
2 ,
(1)
m
where f i s the focal length of the lens. With broadband illumination o f power-spectrum B(o) ranging from o 1 to 02, the distribution o f energy at the rear focal plane of the lens becomes: 02
Gl(X2,Y2) = f
B(o) Go(x2o]f,Y2O/f)do.
(2)
etI
223
Volume
42, number 4
OPTICSCOMMUNICATIONS
It results in a chromatic blurring of the diffraction patterns Go(x2o/f , Y2 off) the latters varying proportionally in size with wavelength according to a chromatic scale factor S(e) =fie. One notesthat the chromatic difference of scale is cancelled as the scale factor S(o) does not depend on e any more, e.g. if the focal length f of the lens arrangement is proportional to o:f= ko. This requirement is fulfilled with a holographic zone plate whose focal length is defined by f(o) = Ro/oo, where o0 and R are the wavenumber and the radius of curvature of the spherical wave producing the zone plate at the recording process. Under this condition, eq. (2) reads:
G2(x2,Y2) =
f
B(o)
01
l;;
g(xl,Yl)
_oa
2 X expl-j27r ~°° (XlX 2 +ylY2) 1 dx 1 dy I
=
g ( x l , Y l ) exp
27r
do
IX2 +YlY2
--oo
× dx 1 dy 1
B(a)do al o2
= GO(x2oo/R,Y2oo[R) f
B(o)do.
(3)
tr 1
The distribution of energy described by eq. (3) is now that of a wavelength-independent diffraction pattern GO(x 2 o 0 JR, y 2 °o/R) whose size is determined by the scale factor R/o 0 (depending on the recording param: eters of the zone plate), and weighted by the energy of the source. In fact, due to the chromatic dispersion of a zone plate, the energy distribution G2(x2, Y2) is displayed along the optical axis z of the zone plate, the variable o in the volume integral (3) being linked to the longitudinal variable z by the dispersion law of the zone plate: z = Ro/o 0 for the first order of diffraction. In other words, the energy distribution G2(x2,Y2) takes the form of wavelength-independent diffraction patterns GO(x 2 o0 ]R, y 2 °O]R ) pile d along the optical axis z of the zone plate. The overall pattern is a volume of diffraction. Such a situation is encountered in the set224
15 July 1982
Fig. i Experimental set-up;g(xl,Yl): input pupil; ZP1 and ZP 2 : holographic zone plates; O1 and 02 : achromatic lenses; G2(x2, Y2): achromatic diffraction pattern of g(x 1, Y 1). The light grey and dark grey regions represent the optical path for two spectral components, e.g. red and blue respectively. up shown in fig. 1. The input pupil g(x 1 ,Yl) is illuminated with a collimated beam of polychromatic light. The on-axis zone plate ZP 1 yields the wavelengthindependent Fraunhofer diffraction patterns Go(x2°o[R,Y2°o [R) ° f g ( x l ,Yl) for every spectral component emitted by the source. As mentioned above, these diffraction patterns are displayed in depth along the optical axis; their abscissa z from ZP1 is z = Ro/o O. The overall pattern is the volume of diffraction V discussed so far, with parallel generating lines, i.e. a cylinder of diffraction. The key yielding the achromatic diffraction pattern G2(x2, Y2) is then to recombine in a single plane the patterns Go(X2oo/R,Y2oo]R ) displayed along zaxis. It is achieved by the lens combination (O1, ZP2, 02) whose disposition is governed by the lens combination formulas of geometrical optics. The achromatic doublet O1, of focal length f l , images the cylinder V at V'. The doublet O 1 is located at l = 2R + f l from ZP 1 so that the axial and longitudinal geometrical magnifications between V and V' are quasi linear in wavenumber o. The second zone plate ZP 2 acts on V' as an imaging system. Its focal length is f2(o) = f21a/Ro O. Its image focal planes are superimposed with V', wavenumber by wavenumber. Under this condition and after the lens combination formulas, it can be shown that the image of V' through ZP2 is compressed in a single plane set at infinity, yielding an achromatic diffraction pattern well-corrected chromatically. Such a pattern can be observed at the back focal plane of O2 (focal length f l ) ; it is described by eq. (3).
Volume 42, number 4
OPTICS COMMUNICATIONS
15 July 1982
• arc lamp (400--600 rim). For comparison, the diffraction patterns o f the same pupils are also shown (left) as the achromatic arrangement (ZPI, O1, ZP2)is removed from the set-up, resulting in a chromatic blurring of the patterns. From a general point o f view, any 2-D pupil can be Fourier transformed, whether it be an amplitude or a phase pupil. Several parameters remain to be studied, such as the chromatic aberrations or the spectral resolution o f the set-up. These parameters determine the accuracy of coincidence of the various monochromatic patterns that are superimposed at the output plane, and therefore the accuracy of the achromatic Fourier transform. However preliminary results indicate a fairly good chromatic compensation yielding a Fourier transformation that is well chromatically corrected over a wide range of wavelengths; the error on the achromatic Fourier transformation is estimated to be of about 2% for the visible spectrum ( 4 0 0 - 6 0 0 nm).
Acknowledgement
Fig. 2. (a) Diffraction pattern of an amplitude grid illuminated in white light. On the left, the diffraction pattern is recorded at the back focal of a conventional lens (chromatic blurring); on the right, the pattern is recorded at the output plane x2 of the set-up of fig. 1. (b) Young's experiment in white light. On the left, the contrast of the fringes decreases rapidly on both sides of the zero-order fringe; on the right, the interference pattern is well-corrected chromatically: the contrast of the fringes is high and constant throughout the pattern.
3. Experimental results and conclusion Eq. (3) indicates that the size of the achromatic diffraction pattern observed at the output plane of the set-up is determined by the scale factor oo/R, depending on the recording parameters of ZP 1 . The experimental results shown in fig. 2 have been obtained with a 0 = 1.94/am - 1 (5145 A ) R = 30 cm and with f l = 30 cm. On the right, they are photographs of achromatic diffraction patterns of pupils (a 2-D grid and two slits) illuminated in white light by a xenon
The authors wish to thank Prof. J.Ch. Vidnot, Dr. N. Aebischer and Dr. G. Tribillon for their fruitful discussions in the writing of the manuscript and the DRET for its financial support.
References [1] R. Katyl, Appl. Optics 11 (1972) 622. [2] E. Leith and J. Roth, Appl. Optics 16 (1977) 2565. [3] J.P. Goedgebuer and R. Gazeu, Optics Comm. 27 (1978) 53. [4] R. Ferri6re, J.P. Goedgebuer and J.Ch. Vidnot, Optics Comm. 31 (1979) 285. [5] G. Morris and N. George, Optics Lett. 5 (1980) 202. [6] G. Morris and N. George, Optics Lett. 5 (1980) 446. [7] G. Morris and N. George, Appl. Optics 19 (1980) 3843. [8] R. Ferri6re, J.P. Goedgebuer and J.Ch. Vi6not, Confdfence Europ~enne d'Optique 1980, Horizons de Foptique, April 1980. [9] G. Morris, Appl. Optics 20 (1981) 2017. [10] N. George and G. Morris, ICO XII, eds. F.T. Arteechi and F.R. Aussenegg, Graz (Austria, September 1981); Current trends in optics. [11] G. Collins, Appl. Optics 20 (1981) 3109.
225