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Image contrast and resolution in holographic multiplexing by spatial division
Volume 10, number 3
OPTICS COMMUNICATIONS
IMAGE CONTRAST
AND RESOLUTION
IN HOLOGRAPHIC
March 1974
MULTIPLEXING
BY SPATIAL DIVISION P. H A R I H A R A N '~ and Z.S. HEGEDUS
CSIR O Division oj Physics, National Standards Laboratory, Sydney, Australia 2008 Received 28 December 1973 An analysis of the effects of two different techniques of spatial-division multiplexing on the quality of the image reconstructed by the hologram of a diffusing object is presented, along with some experimental results obtained with a coherent optical system under diffraction-limited conditions. These show that the use of random masks for spatial division in the cases studied gives much poorer resolution than simple sub-division of the hologram area, because of the serious loss of image contrast with a random mask.
1. Introduction As is well known, N wavefronts can be multiplexed on a single hologram plate by using the entire area of the plate and changing the angle at which the reference beam is incident for each exposure [ 1]. Another method (spatial-division multiplexing) is to divide tile plate into N separate regions and record individual holograms in each o f these regions. The former has the disadvantage that the signal-to-noise ratio of each image varies as N -2, while, in the latter, the available aperture size for each hologranl is proportional to N 1/2, resulting in a loss of resolution and an increase in the average speckle size. To avoid these problems, an alternative technique of spatial-division naultiplexing has been described, which does not result in a reduction in the size o f each hologram. In this, a series of spatially complementary masks are used in contact with the recording medium, having an equal number of identical, small apertures at random locations, so that each hologram is recorded on a spatially distinct area, but this area is distributed over the entire plate [2]. The present paper is an attempt at a comparison, based on linear filter theory, of the effects of these two techniques of spatial-division multiplexing on the
quality o f the image of an extended, diffusing object. Some experimental results are also given which support the theoretical conclusions.
2. Theory If the reconstruction of a Fraunhofer hologram is considered, the introduction o f an aperture in the hologram plane (spatial sampling of the hologram) can be seen to be equivalent to sampling the spatial frequency spectrum of the coherently illuminated object. The analysis of this problem can therefore be simplified by studying the coherent imaging system shown in fig. 1, whose impulse response is determined by a sampling aperture H(u) in the Fourier plane, u being a vector (~, r/) corresponding to the reduced spatial frequency (u = v/X]), where v is the actual position vector in the Fourier plane, f i s the focal length of the lenses, and ~. is the wavelength. If the object is a transparency with an ampfitude transmittance t(r) (r being a vector ( x , y ) in the object plane), which is illuminated through a diffuser d(r) by a laser, so that the object amplitude f(r) is f ( r ) = t(r) d(r).
(1)
The image amplitude g(r) is given by the relation On leave from the Indian Institute of Science, Bangalore 560012, India. 238
g(r) = f ( r ) * h(r),
(2)
Volume 10, number 3
OPTICS COMMUNICATIONS
Object ptane
For sinrplicity, a spatial frequency u along one of the principal axes of the square aperture, say the axis, will be considered. The incoherent transfer function of the system can then be written as
Fourier
YI~
/2"1 x
L,
prone
Image
~..
pto~
March 1974
H(u) =H(u) * H*(-u) =f
f
X X(a,~ ) Fig. 1. Schematic diagram of a coherent imaging system. The transparency T in the object plane is illuminated through the diffuser D by a beam L from a laser. A mask H can be introduced in the Fourier plane to sample the spatial frequency spectrum of the object. where h(r) is the two-dimensional Fourier transform of H(u). The average power spectrum Si(u) of the image illuminance can then be expressed as the sum of two terms [3]
rect(a) rect(c~+u)rect(/3)rect(/3)
X(a+u,~) d(~ d~
(s)
and since the effect of the first four terms in this
integral is merely to impose finite limits on it, this becomes
1/2-u 1/2 H(u)= fl/X(c~'13)X(a+u'13)dc~dl3"2 -; :2
(9)
where ~21(u,u ) is the power spectrum of the signal, and fZ 2 (u, u) is the power spectrum of the speckle. The power spectrum of the signal is [3]
Apart from the finite limits on the integral, eq. (9) is the same as the autocorrelation function ofX(~, rT), which drops rapidly from its peak value of 1IN when u = 0, to a constant value of 1IN2 for values of u greater than the dimensions of the individual apertures in the mask. If the relatively narrow peak at low spatial frequencies is neglected, the incoherent transfer function of the system is
£21(u,u ) = II(u) H(u)l 2,
H(u) = (1
g1(u) = s2~(u,u)
+ fz2 (u,u) ,
(3)
(4)
where ~(u) is the Fourier transform of the illuminance of the object, and H(u) is the incoherent transfer function of the optical system, which is determined by the mask in the Fourier plane. It is assumed that the mask H(u) consists of a large number of very small apertures distributed at random within a square of unit dimensions (reduced units). The transmittance of this mask can then be written as
H(u) = rect(~)
rect(r/) X(~, ~),
(5)
where rect(~)= l, for I ~ l < w1 , = O, for
1 t~l >~-,
(6)
u)/N 2 .
(10)
If, then, I ( u ) = 1 within the range of interest, the power spectrum of the signal becomes gZl(U,U)=(1
u)Z/N 4.
(11)
The power spectrum of the speckle, under the same conditions, can be written as [3]
~22(u,u)=2[Rtt(O)[ 2 [H(u)l 2 *
IH*(-u)[ 2,
(12)
where Rtt (r) is the spatial autocorrelation function of t(r) the transmittance of the object. Since H(u) only takes the values 0 or 1, IH(u)[ 2 = H(u), and if 2 [Rtt(0)[ 2 is taken equal to unity, the power spectrum of the speckle becomes
and X(~, r/) is a random variable which takes the values
~z2 (u,u) = H ( u ) • H * ( - u ) X(~,r/) = 1, over the openings, = O, elsewhere.
= (1-
u)/N2.
(13)
(7)
The fraction of the area of the random mask covered by the apertures in it is 1IN.
Eqs. (l 1) and (13) can be normalized by multiplying them by a factor o f N 2 so that the peak value of the power spectrum of the speckle (at u = 0) is equal 239
Volume 10, number 3
OPTICS COMMUNICATIONS
to unity. Tlle normalized power spectrum of tile signal is then ~21(u,u) = (1
u)2/:V 2
(14)
while the normalized power spectrum of the speckle becomes gZ2(u,u) = ( l - - u ) .
(15)
It is interesting to compare eqs. (14) and (15) with the results which would be obtained if the random mask is replaced with a square aperture having the same effective area (side = N 1/2). hi this case, tire normalized power spectrum of the signal is ~-2'I ( u , u ) = ( 1 .
NI/2u) 2
(16)
while tile normalized power spectrum of the speckle is
P.2(u,u)=(1
Nl/2u).
(17)
The theoretical power spectra of tire signal and the speckle for these two types of masks, when N = 4 (i.e. for the situation corresponding to %ur hologranrs multiplexed on tile same plate), are plotted in figs. 2a and 2b. From these curves it is apparent that while the power spectra of the signal and the speckle extend to higher spatial frequencies with the random mask, the power spectrum of the signal is attenuated in this case by a factor of N 2, resulting in a considerable loss of image contrast.
March 1974
3.Experimentalresults The effect o f this, in practice, can be seen from figs. 4a and 4b which are photographs of a resolution test chart made on fine-grain film using two well-corrected lenses ( j = 1 m) in an optical system similar to that shown in fig. 1. Fig. 4a was made with a mask in tile Fourier plane provided with 100 apertures, 1 mm X 1 mm in size, distributed at random within a 2 cnr × 2 cm square (N = 4), while fig. 4b was made with a single square aperture having the same effective area ( 1 cm × 1 cm). These parameters of the optical system were chosen so that its perfornmnce was strictly diffraction-limited. For comparison, fig. 3a shows part o f the original target (the largest test pattern actually had a spatial frequency of 2.6 lines/nun)and fig. 3b shows the image obtained with a 2 cm square aperture in tire Fourier plane. Because of the high noise level due to speckle, the resolution actually attained with the 1 cm square aperture is only of tire order o 1 0 . 4 0 f U m a x , the theoretical cut-off spatial frequency, which in this case is around 16 lines/rain (Urea x = b/~.f, where b is the width of the aperture). It is also apparent that the random mask does not give better resolution, even though the power spectrum of the signal extends to much higher spatial frequencies with it. This is presumably because of the adverse effect which the loss in contrast of the image has on the signal-to-noise ratio.
1.0
1-0
(b)
0.s ~~,u)
0'5
"v 0.5 tl
~-~-I=--
1-0
I
O.fi U ~
i
I 1.0
Fig. 2. Normalized, theoretical power spectra of file signal and the speckle in tile image of a coherently illuminated, diffusing object with (a) a random sampling mask (iV = 4) contained within a square of unit dimensions, and (b) a single square aperture having the stone effective area (side = 3/-1/2), in the Fourier plane. 240
V o l u m e 10, n u m b e r 3
OPTICS COMMUNICATIONS
March 1974
[l[tOlllliLl31JoIoiLli 111111111 jlljlljl]l]lLlj]Ljjllj ("
,
R
'3
150
600
Fig. 3. (a) A section of the resolution target used as the test object, along with lb) the image obtained with diffuse, coherent illumination with a 2 cm square aperture in the Fourier plane. (The largest test pattern s h o w n has a spatial frequency of 2.6 lines/mm.)
(a)
(b)
Fig. 4. Images obtained with diffuse, coherent illumination with (a) a r a n d o m sampling mask (N = 4) contained within a 2 cm square aperture, and (b) a single 1 cm square aperture, in the Fourier plane. 241
Volume 10, nunlber 3
OPTICS COMMUNICATIONS
~a)
March 1974
!b)
Fig. 5. Images obtained with diffuse, coherent illumination with (a) a random sampling mask (N = 16) contained within a 2 cm square aperture, and (b) a single 5 mm square aperture, in the 1 ourier plane. This is s u p p o r t e d by figs. 5a and 5b, which show the results o b t a i n e d with a r a n d o m mask having 25 apertures, 1 m m × 1 m m , within a 2 cm square ( N = 16), and a square aperture (5 m m × 5 ram) having the same transmittance. In this case, the loss of image contrast with the r a n d o m mask is even more severe, and the resolution is m u c h lower than that obtained with a single, square aperture of equal transmittance,
242
The author wish to t h a n k Dr. W.H. Steel for helpful discussions.
References [11 P.J. van tteerden, Appl. Opt. 2 (1963) 387. [21 H.J.Caulfield, Appl. Opt. 9 (1970) 1218. [3] S. Lowenthal and H. Arsenault, J. Opt. Soc. Am. 60 (1970) 1478.
OPTICS COMMUNICATIONS
IMAGE CONTRAST
AND RESOLUTION
IN HOLOGRAPHIC
March 1974
MULTIPLEXING
BY SPATIAL DIVISION P. H A R I H A R A N '~ and Z.S. HEGEDUS
CSIR O Division oj Physics, National Standards Laboratory, Sydney, Australia 2008 Received 28 December 1973 An analysis of the effects of two different techniques of spatial-division multiplexing on the quality of the image reconstructed by the hologram of a diffusing object is presented, along with some experimental results obtained with a coherent optical system under diffraction-limited conditions. These show that the use of random masks for spatial division in the cases studied gives much poorer resolution than simple sub-division of the hologram area, because of the serious loss of image contrast with a random mask.
1. Introduction As is well known, N wavefronts can be multiplexed on a single hologram plate by using the entire area of the plate and changing the angle at which the reference beam is incident for each exposure [ 1]. Another method (spatial-division multiplexing) is to divide tile plate into N separate regions and record individual holograms in each o f these regions. The former has the disadvantage that the signal-to-noise ratio of each image varies as N -2, while, in the latter, the available aperture size for each hologranl is proportional to N 1/2, resulting in a loss of resolution and an increase in the average speckle size. To avoid these problems, an alternative technique of spatial-division naultiplexing has been described, which does not result in a reduction in the size o f each hologram. In this, a series of spatially complementary masks are used in contact with the recording medium, having an equal number of identical, small apertures at random locations, so that each hologram is recorded on a spatially distinct area, but this area is distributed over the entire plate [2]. The present paper is an attempt at a comparison, based on linear filter theory, of the effects of these two techniques of spatial-division multiplexing on the
quality o f the image of an extended, diffusing object. Some experimental results are also given which support the theoretical conclusions.
2. Theory If the reconstruction of a Fraunhofer hologram is considered, the introduction o f an aperture in the hologram plane (spatial sampling of the hologram) can be seen to be equivalent to sampling the spatial frequency spectrum of the coherently illuminated object. The analysis of this problem can therefore be simplified by studying the coherent imaging system shown in fig. 1, whose impulse response is determined by a sampling aperture H(u) in the Fourier plane, u being a vector (~, r/) corresponding to the reduced spatial frequency (u = v/X]), where v is the actual position vector in the Fourier plane, f i s the focal length of the lenses, and ~. is the wavelength. If the object is a transparency with an ampfitude transmittance t(r) (r being a vector ( x , y ) in the object plane), which is illuminated through a diffuser d(r) by a laser, so that the object amplitude f(r) is f ( r ) = t(r) d(r).
(1)
The image amplitude g(r) is given by the relation On leave from the Indian Institute of Science, Bangalore 560012, India. 238
g(r) = f ( r ) * h(r),
(2)
Volume 10, number 3
OPTICS COMMUNICATIONS
Object ptane
For sinrplicity, a spatial frequency u along one of the principal axes of the square aperture, say the axis, will be considered. The incoherent transfer function of the system can then be written as
Fourier
YI~
/2"1 x
L,
prone
Image
~..
pto~
March 1974
H(u) =H(u) * H*(-u) =f
f
X X(a,~ ) Fig. 1. Schematic diagram of a coherent imaging system. The transparency T in the object plane is illuminated through the diffuser D by a beam L from a laser. A mask H can be introduced in the Fourier plane to sample the spatial frequency spectrum of the object. where h(r) is the two-dimensional Fourier transform of H(u). The average power spectrum Si(u) of the image illuminance can then be expressed as the sum of two terms [3]
rect(a) rect(c~+u)rect(/3)rect(/3)
X(a+u,~) d(~ d~
(s)
and since the effect of the first four terms in this
integral is merely to impose finite limits on it, this becomes
1/2-u 1/2 H(u)= fl/X(c~'13)X(a+u'13)dc~dl3"2 -; :2
(9)
where ~21(u,u ) is the power spectrum of the signal, and fZ 2 (u, u) is the power spectrum of the speckle. The power spectrum of the signal is [3]
Apart from the finite limits on the integral, eq. (9) is the same as the autocorrelation function ofX(~, rT), which drops rapidly from its peak value of 1IN when u = 0, to a constant value of 1IN2 for values of u greater than the dimensions of the individual apertures in the mask. If the relatively narrow peak at low spatial frequencies is neglected, the incoherent transfer function of the system is
£21(u,u ) = II(u) H(u)l 2,
H(u) = (1
g1(u) = s2~(u,u)
+ fz2 (u,u) ,
(3)
(4)
where ~(u) is the Fourier transform of the illuminance of the object, and H(u) is the incoherent transfer function of the optical system, which is determined by the mask in the Fourier plane. It is assumed that the mask H(u) consists of a large number of very small apertures distributed at random within a square of unit dimensions (reduced units). The transmittance of this mask can then be written as
H(u) = rect(~)
rect(r/) X(~, ~),
(5)
where rect(~)= l, for I ~ l < w1 , = O, for
1 t~l >~-,
(6)
u)/N 2 .
(10)
If, then, I ( u ) = 1 within the range of interest, the power spectrum of the signal becomes gZl(U,U)=(1
u)Z/N 4.
(11)
The power spectrum of the speckle, under the same conditions, can be written as [3]
~22(u,u)=2[Rtt(O)[ 2 [H(u)l 2 *
IH*(-u)[ 2,
(12)
where Rtt (r) is the spatial autocorrelation function of t(r) the transmittance of the object. Since H(u) only takes the values 0 or 1, IH(u)[ 2 = H(u), and if 2 [Rtt(0)[ 2 is taken equal to unity, the power spectrum of the speckle becomes
and X(~, r/) is a random variable which takes the values
~z2 (u,u) = H ( u ) • H * ( - u ) X(~,r/) = 1, over the openings, = O, elsewhere.
= (1-
u)/N2.
(13)
(7)
The fraction of the area of the random mask covered by the apertures in it is 1IN.
Eqs. (l 1) and (13) can be normalized by multiplying them by a factor o f N 2 so that the peak value of the power spectrum of the speckle (at u = 0) is equal 239
Volume 10, number 3
OPTICS COMMUNICATIONS
to unity. Tlle normalized power spectrum of tile signal is then ~21(u,u) = (1
u)2/:V 2
(14)
while the normalized power spectrum of the speckle becomes gZ2(u,u) = ( l - - u ) .
(15)
It is interesting to compare eqs. (14) and (15) with the results which would be obtained if the random mask is replaced with a square aperture having the same effective area (side = N 1/2). hi this case, tire normalized power spectrum of the signal is ~-2'I ( u , u ) = ( 1 .
NI/2u) 2
(16)
while tile normalized power spectrum of the speckle is
P.2(u,u)=(1
Nl/2u).
(17)
The theoretical power spectra of tire signal and the speckle for these two types of masks, when N = 4 (i.e. for the situation corresponding to %ur hologranrs multiplexed on tile same plate), are plotted in figs. 2a and 2b. From these curves it is apparent that while the power spectra of the signal and the speckle extend to higher spatial frequencies with the random mask, the power spectrum of the signal is attenuated in this case by a factor of N 2, resulting in a considerable loss of image contrast.
March 1974
3.Experimentalresults The effect o f this, in practice, can be seen from figs. 4a and 4b which are photographs of a resolution test chart made on fine-grain film using two well-corrected lenses ( j = 1 m) in an optical system similar to that shown in fig. 1. Fig. 4a was made with a mask in tile Fourier plane provided with 100 apertures, 1 mm X 1 mm in size, distributed at random within a 2 cnr × 2 cm square (N = 4), while fig. 4b was made with a single square aperture having the same effective area ( 1 cm × 1 cm). These parameters of the optical system were chosen so that its perfornmnce was strictly diffraction-limited. For comparison, fig. 3a shows part o f the original target (the largest test pattern actually had a spatial frequency of 2.6 lines/nun)and fig. 3b shows the image obtained with a 2 cm square aperture in tire Fourier plane. Because of the high noise level due to speckle, the resolution actually attained with the 1 cm square aperture is only of tire order o 1 0 . 4 0 f U m a x , the theoretical cut-off spatial frequency, which in this case is around 16 lines/rain (Urea x = b/~.f, where b is the width of the aperture). It is also apparent that the random mask does not give better resolution, even though the power spectrum of the signal extends to much higher spatial frequencies with it. This is presumably because of the adverse effect which the loss in contrast of the image has on the signal-to-noise ratio.
1.0
1-0
(b)
0.s ~~,u)
0'5
"v 0.5 tl
~-~-I=--
1-0
I
O.fi U ~
i
I 1.0
Fig. 2. Normalized, theoretical power spectra of file signal and the speckle in tile image of a coherently illuminated, diffusing object with (a) a random sampling mask (iV = 4) contained within a square of unit dimensions, and (b) a single square aperture having the stone effective area (side = 3/-1/2), in the Fourier plane. 240
V o l u m e 10, n u m b e r 3
OPTICS COMMUNICATIONS
March 1974
[l[tOlllliLl31JoIoiLli 111111111 jlljlljl]l]lLlj]Ljjllj ("
,
R
'3
150
600
Fig. 3. (a) A section of the resolution target used as the test object, along with lb) the image obtained with diffuse, coherent illumination with a 2 cm square aperture in the Fourier plane. (The largest test pattern s h o w n has a spatial frequency of 2.6 lines/mm.)
(a)
(b)
Fig. 4. Images obtained with diffuse, coherent illumination with (a) a r a n d o m sampling mask (N = 4) contained within a 2 cm square aperture, and (b) a single 1 cm square aperture, in the Fourier plane. 241
Volume 10, nunlber 3
OPTICS COMMUNICATIONS
~a)
March 1974
!b)
Fig. 5. Images obtained with diffuse, coherent illumination with (a) a random sampling mask (N = 16) contained within a 2 cm square aperture, and (b) a single 5 mm square aperture, in the 1 ourier plane. This is s u p p o r t e d by figs. 5a and 5b, which show the results o b t a i n e d with a r a n d o m mask having 25 apertures, 1 m m × 1 m m , within a 2 cm square ( N = 16), and a square aperture (5 m m × 5 ram) having the same transmittance. In this case, the loss of image contrast with the r a n d o m mask is even more severe, and the resolution is m u c h lower than that obtained with a single, square aperture of equal transmittance,
242
The author wish to t h a n k Dr. W.H. Steel for helpful discussions.
References [11 P.J. van tteerden, Appl. Opt. 2 (1963) 387. [21 H.J.Caulfield, Appl. Opt. 9 (1970) 1218. [3] S. Lowenthal and H. Arsenault, J. Opt. Soc. Am. 60 (1970) 1478.