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Computer generated holograms constructed with slanted fringes
Volume 64, number
OPTICS
4
COMPUTER
GENERATED
Antti SAUKKONEN
15 November
COMMUNICATIONS
HOLOGRAMS
‘, Jarmo TOLVANEN
CONSTRUCTED and Raima
WITH SLANTED
1987
FRINGES
SILVENNOINEN
Department of Physics, University ofJoensuu, SF-80100 Joensuu IO. Finland Received
24 November
1986
Constructing methods for computer generated holograms using only the real part of the Fourier transform are discussed. A diagonal, double diagonal and a “unidirectional” construction method are proposed and investigated. These are compared with the Nagashima and Asakura type computer generated hologram.
investigation photographic
1. Introduction
The most significant tool of making computer generated holograms (CGH) is the discrete Fourier transform (DFT), more accurately: fast Fourier transform (FFT). The use of FFT is a common feature in most applications of CGH, but there exists a great number of different recording methods. The classical recording method, introduced by Brown and Lohmann [ 11, tends to the so called detour phase hologram. This type of hologram uses both the real and imaginary parts of the DFT. The hologram itself consists of an array of apertures whose height and position are related with the amplitude and phase of the object wavefront. A remarkable simplification on recording and calculation techniques was reported by Nagashima and Asakura [2,3]. Their method utilizes only the real part of DFT. Lee [4] has proposed that the CGH structure should be visualized as horizontal fringes. Our work is based on the idea of using the real part of DFT only and plotting the CGH structure as a network of slanted fringes. The quality of the images given by such CGHs are also compared with a Nagashima and Asakura type hologram. In connection with photographs of the reconstructed images, portions of the holograms photographed through a microscope are presented. This process is for an ’ Lappeenranta peenranta,
of the structure deformation due to the process used in CGH reduction.
2. On the calculation of the hologram data The calculation method includes the usual procedure of coding the image into a two dimensional matrix and its Fourier transform. According to the familiar DFT definition, F(m,n)=
N-l
N-l
1 k=O
c h=O
f(kh)
Xexp[2rri(km+hn)lN], m, n=O, 1,2, .... N-l,
where F( m, n) represents an element of the Fourier transformed matrix, f(k, h) is an element of the object matrix and i is the imaginary unit. In the present paper a matrix size NX N is assumed. A diffuse illumination of the object is simulated by the use of random phase coding so that, f(k,
h)=A
cos(y,)+iA
of Technology,
PL20,
53851
Lap-
0 030-4018/87/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
sin(fj7),
(2)
where A is the amplitude of the object matrix element and phase angle a, is a random number between the limits O
University Finland
(1)
In general, our treatment B.V.
(3) uses values 0 or 1 for the 317
Volume 64. number
4
OPTICS
COMMUNICATIONS
amplitude A so that background elements have the value 0 and the elements of the object have the value 1. The Fourier transform of a plane object f(x, y) is briefly presented as
Y)l>
F=Wx,
(4)
where 9 is the Fourier transform operation defined in the discrete case as in eq. (1). Nagashima and Asakura expressed the hologram data as H’(m, n)=Re[F(m,
n)] .
The values of H’( m, n) are then digitized according to the following statement, H(m, n)=O =l
ifH’(m,
n)IO,
1987
CGH, which we call diagonal because of the plotting method. As an example we shall consider the case (a) in fig. 2. Now the fringes are assumed to be mostly parallel with the diagonal from (N, 1) to (1, N). The neighbouring elements of the FFT operated and digitized matrix are connected with a line in this direction if they both have value 1. The result of reconstruction and a portion of the CGH itself is seen in fig. 3. The hologram structure is practically unchanged.
(5) to 0 or 1
(6)
ifH’(m,n)>O.
It should be noted that this kind of classifying the H(m, n) values is not exactly the same as Nagashima and Asakura. Anyway, data values determined by eq. (6) are used in all the following examples. Actually in all our examples we have a common data matrix, which has the size of 128 x 128 elements.
3. Nagashima and Asakura type hologram According to the ref. [ 31, the drawing is carried out by an x-y plotter. For H(m, n) =O, we draw a piece of a horizontal line and for H( m, n) = 1 a triangular line is drawn. Figs. 1a and b show the reconstructed image and the hologram structure. No significant changes in the hologram structure are obtained.
4. The diagonal method Let us first consider the positioning of the object in our object matrix as presented in fig. 2. If we think of the formation of a hologram as an interference between an object and reference wave, it would seem intuitively obvious that in cases (a) or (b) of fig. 2 the interference fringes in the hologram would appear rather in slanted directions than in horizontal (or vertical) direction. Based on this way of deduction we have made a 318
15 November
5. The double diagonal method A little bit more advanced method than in the previous chapter is to check if any trend appears in the direction parallel to the diagonal from (1, 1) to (N, N). This means that if both neighbouring elements in this direction have value 1, they are also connected with a line. When the object is of the type of case (a) in fig. 2, this kind of trend is not dominant, but some orientation yet exists. If we use, say, the whole upper half of the matrix of fig. 2 for the object, there will exist a certain trend parallel to the both diagonals. The reconstructed image and a portion of the corresponding hologram are presented in fig. 4. The hologram structure is strongly deformated and resembles a grating of an “ordinary” hologram. The drawing method itself can not be recognized in this case.
6. The “unidirectional”
method
As a final step in our evaluation process we shall consider the case, where the neighbouring elements are connected in both diagonal directions but also in vertical and horizontal directions if they both have simultaneously the value 1. The reconstructed image and a portion of the corresponding hologram are presented in fig. 5. As in the previous case, the hologram structure is dramatically deformated. Here too the plotting method is not recognized.
7. Discussion All holograms were reduced by photographing a size about 5 mm x 5 mm. As recording material
to we
Volume 64, number 4
OPTICS COMMUNICATIONS
15 November 1987
~
Fig. 1. Nagashima and Asakura type CGH. Reconstructed image (a) and a photograph from the CGH structure (b). 1,li
1 ___
..
\1,11
Fig. 2. The position of the object in the object matrix. In case (a) the dc-term corresponds to (I,1 ) and in case (b) to (I ,N). In all examples the a-case is used.
have used Agfa-gevaert HDU3 film, which has extremely high contrast and good resolution. Reduced holograms were then reconstructed with a helium-neon laser light. A plane wave was focussed through the holograms and the images were photographed from the image plane. The following conclusions can be made: (A) In fig. 1 the Nagashima and Asakura type hologram gives a noisy cloud around the dc-term and
several diffracted orders are obtained. In fact, the second order images appear stronger than the first order ones. The presence of some low-pass effect causes unsharpening of the image. (B) As expected, the diagonal method in fig. 3 causes a clear orientation of the weak higher order images. A slight low-pass effect gives still some unsharpness. The noise level is remarkably smaller than in case A. (C) The double diagonal method in fig. 4 generates more higher order images than appears in case B, and here we observed some unsharpness too. The noise level referred to the B-case is slightly higher. (D) The unidirectional method in fig. 5 has the best properties of the B- and C-cases, namely low noise level and quite modest higher order images. The image quality can be hold relatively good. The diagonal method may be respected as a preliminary study because with this method we can use only a quarter of the object matrix for object coding. The double diagonal and the unidirectional methods might have some potential advantages because of their simplicity in some applications, like in complex spatial filtering.
319
Volume 64, number
4
Fig. 3. The diagonal
OPTICS
method.
Fig. 4. The double diagonal
320
Reconstructed
method.
Reconstructed
COMMUNICATIONS
image (a) and a photograph
image (a) and a photograph
15 November
from the CGH structure
from the CGH structure
(b).
(b)
I987
Volume 64. number 4
OPTICS COMMUNICATIONS
15 November 1987
Fig. 5. The unidirectional method. Reconstructed image (a) and a photograph from the CGH structure (b).
References
[ 31 K. Nagashima and T. Asakura, Opt. Laser Tech. 10 (1978)
[ 1 ] B.R. Brown and A.W. Lohmann, Appl. Optics 5 (1966) 967. [ 21 K. Nagashima and T. Asakura, Optics Comm. 17 (1976)273.
[ 41 Wai-Hon Lee, Appl. Optics 18 (1979) 366 1.
310.
321
OPTICS
4
COMPUTER
GENERATED
Antti SAUKKONEN
15 November
COMMUNICATIONS
HOLOGRAMS
‘, Jarmo TOLVANEN
CONSTRUCTED and Raima
WITH SLANTED
1987
FRINGES
SILVENNOINEN
Department of Physics, University ofJoensuu, SF-80100 Joensuu IO. Finland Received
24 November
1986
Constructing methods for computer generated holograms using only the real part of the Fourier transform are discussed. A diagonal, double diagonal and a “unidirectional” construction method are proposed and investigated. These are compared with the Nagashima and Asakura type computer generated hologram.
investigation photographic
1. Introduction
The most significant tool of making computer generated holograms (CGH) is the discrete Fourier transform (DFT), more accurately: fast Fourier transform (FFT). The use of FFT is a common feature in most applications of CGH, but there exists a great number of different recording methods. The classical recording method, introduced by Brown and Lohmann [ 11, tends to the so called detour phase hologram. This type of hologram uses both the real and imaginary parts of the DFT. The hologram itself consists of an array of apertures whose height and position are related with the amplitude and phase of the object wavefront. A remarkable simplification on recording and calculation techniques was reported by Nagashima and Asakura [2,3]. Their method utilizes only the real part of DFT. Lee [4] has proposed that the CGH structure should be visualized as horizontal fringes. Our work is based on the idea of using the real part of DFT only and plotting the CGH structure as a network of slanted fringes. The quality of the images given by such CGHs are also compared with a Nagashima and Asakura type hologram. In connection with photographs of the reconstructed images, portions of the holograms photographed through a microscope are presented. This process is for an ’ Lappeenranta peenranta,
of the structure deformation due to the process used in CGH reduction.
2. On the calculation of the hologram data The calculation method includes the usual procedure of coding the image into a two dimensional matrix and its Fourier transform. According to the familiar DFT definition, F(m,n)=
N-l
N-l
1 k=O
c h=O
f(kh)
Xexp[2rri(km+hn)lN], m, n=O, 1,2, .... N-l,
where F( m, n) represents an element of the Fourier transformed matrix, f(k, h) is an element of the object matrix and i is the imaginary unit. In the present paper a matrix size NX N is assumed. A diffuse illumination of the object is simulated by the use of random phase coding so that, f(k,
h)=A
cos(y,)+iA
of Technology,
PL20,
53851
Lap-
0 030-4018/87/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
sin(fj7),
(2)
where A is the amplitude of the object matrix element and phase angle a, is a random number between the limits O
University Finland
(1)
In general, our treatment B.V.
(3) uses values 0 or 1 for the 317
Volume 64. number
4
OPTICS
COMMUNICATIONS
amplitude A so that background elements have the value 0 and the elements of the object have the value 1. The Fourier transform of a plane object f(x, y) is briefly presented as
Y)l>
F=Wx,
(4)
where 9 is the Fourier transform operation defined in the discrete case as in eq. (1). Nagashima and Asakura expressed the hologram data as H’(m, n)=Re[F(m,
n)] .
The values of H’( m, n) are then digitized according to the following statement, H(m, n)=O =l
ifH’(m,
n)IO,
1987
CGH, which we call diagonal because of the plotting method. As an example we shall consider the case (a) in fig. 2. Now the fringes are assumed to be mostly parallel with the diagonal from (N, 1) to (1, N). The neighbouring elements of the FFT operated and digitized matrix are connected with a line in this direction if they both have value 1. The result of reconstruction and a portion of the CGH itself is seen in fig. 3. The hologram structure is practically unchanged.
(5) to 0 or 1
(6)
ifH’(m,n)>O.
It should be noted that this kind of classifying the H(m, n) values is not exactly the same as Nagashima and Asakura. Anyway, data values determined by eq. (6) are used in all the following examples. Actually in all our examples we have a common data matrix, which has the size of 128 x 128 elements.
3. Nagashima and Asakura type hologram According to the ref. [ 31, the drawing is carried out by an x-y plotter. For H(m, n) =O, we draw a piece of a horizontal line and for H( m, n) = 1 a triangular line is drawn. Figs. 1a and b show the reconstructed image and the hologram structure. No significant changes in the hologram structure are obtained.
4. The diagonal method Let us first consider the positioning of the object in our object matrix as presented in fig. 2. If we think of the formation of a hologram as an interference between an object and reference wave, it would seem intuitively obvious that in cases (a) or (b) of fig. 2 the interference fringes in the hologram would appear rather in slanted directions than in horizontal (or vertical) direction. Based on this way of deduction we have made a 318
15 November
5. The double diagonal method A little bit more advanced method than in the previous chapter is to check if any trend appears in the direction parallel to the diagonal from (1, 1) to (N, N). This means that if both neighbouring elements in this direction have value 1, they are also connected with a line. When the object is of the type of case (a) in fig. 2, this kind of trend is not dominant, but some orientation yet exists. If we use, say, the whole upper half of the matrix of fig. 2 for the object, there will exist a certain trend parallel to the both diagonals. The reconstructed image and a portion of the corresponding hologram are presented in fig. 4. The hologram structure is strongly deformated and resembles a grating of an “ordinary” hologram. The drawing method itself can not be recognized in this case.
6. The “unidirectional”
method
As a final step in our evaluation process we shall consider the case, where the neighbouring elements are connected in both diagonal directions but also in vertical and horizontal directions if they both have simultaneously the value 1. The reconstructed image and a portion of the corresponding hologram are presented in fig. 5. As in the previous case, the hologram structure is dramatically deformated. Here too the plotting method is not recognized.
7. Discussion All holograms were reduced by photographing a size about 5 mm x 5 mm. As recording material
to we
Volume 64, number 4
OPTICS COMMUNICATIONS
15 November 1987
~
Fig. 1. Nagashima and Asakura type CGH. Reconstructed image (a) and a photograph from the CGH structure (b). 1,li
1 ___
..
\1,11
Fig. 2. The position of the object in the object matrix. In case (a) the dc-term corresponds to (I,1 ) and in case (b) to (I ,N). In all examples the a-case is used.
have used Agfa-gevaert HDU3 film, which has extremely high contrast and good resolution. Reduced holograms were then reconstructed with a helium-neon laser light. A plane wave was focussed through the holograms and the images were photographed from the image plane. The following conclusions can be made: (A) In fig. 1 the Nagashima and Asakura type hologram gives a noisy cloud around the dc-term and
several diffracted orders are obtained. In fact, the second order images appear stronger than the first order ones. The presence of some low-pass effect causes unsharpening of the image. (B) As expected, the diagonal method in fig. 3 causes a clear orientation of the weak higher order images. A slight low-pass effect gives still some unsharpness. The noise level is remarkably smaller than in case A. (C) The double diagonal method in fig. 4 generates more higher order images than appears in case B, and here we observed some unsharpness too. The noise level referred to the B-case is slightly higher. (D) The unidirectional method in fig. 5 has the best properties of the B- and C-cases, namely low noise level and quite modest higher order images. The image quality can be hold relatively good. The diagonal method may be respected as a preliminary study because with this method we can use only a quarter of the object matrix for object coding. The double diagonal and the unidirectional methods might have some potential advantages because of their simplicity in some applications, like in complex spatial filtering.
319
Volume 64, number
4
Fig. 3. The diagonal
OPTICS
method.
Fig. 4. The double diagonal
320
Reconstructed
method.
Reconstructed
COMMUNICATIONS
image (a) and a photograph
image (a) and a photograph
15 November
from the CGH structure
from the CGH structure
(b).
(b)
I987
Volume 64. number 4
OPTICS COMMUNICATIONS
15 November 1987
Fig. 5. The unidirectional method. Reconstructed image (a) and a photograph from the CGH structure (b).
References
[ 31 K. Nagashima and T. Asakura, Opt. Laser Tech. 10 (1978)
[ 1 ] B.R. Brown and A.W. Lohmann, Appl. Optics 5 (1966) 967. [ 21 K. Nagashima and T. Asakura, Optics Comm. 17 (1976)273.
[ 41 Wai-Hon Lee, Appl. Optics 18 (1979) 366 1.
310.
321