Abstract

A new type of phase-only filter is described for wave-front construction, in which both the amplitude and phase information necessary to construct an arbitrary image over a limited field are encoded. It is shown that this phase-only filter can duplicate the performance of an ideal complex-valued spatial filter (a filter that controls both amplitude and phase transmittance). This phase-only filter controls the amplitude transmittance by the use of a modulated high-frequency phase carrier that diffracts a controlled amount of light into the image. This type of filter is particularly useful in the implementation of computational wave-front construction, because the maximum spatial frequency that must be plotted is associated with the image and not the carrier. The performance of the filter is examined both numerically and experimentally.

© 1971 Optical Society of America

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References

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  1. E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 52, 1123 (1962).
    [Crossref]
  2. A. Vander Lugt, IEEE Trans. IT-10, 139 (1964).
  3. W. T. Cathey, Appl. Opt. 9, 1478 (1970).
    [Crossref] [PubMed]
  4. In general, complicated wave fronts will be filtered. When used to form an image, the incident wave front will usually be plane or spherical.
  5. R. M. Bracewell, The Fourier Transform and Its Applications (McGraw –Hill, New York, 1965).
  6. If Xj=∑n=0N-1An exp(2πijn/N) then An=N-1∑j=0N-1Xj exp(-2πijn/N).
  7. H. M. Smith, J. Opt. Soc. Am. 58, 533 (1968).
    [Crossref]
  8. The system we used accommodated 256-point transforms.
  9. APL/360 is available from IBM as a program product.
  10. G. C. Danielson and C. Lanczos, J. Franklin Inst. 233, 365 (1942).
    [Crossref]
  11. J. W. Cooley and J. W. Tukey, Math. Comp. 19, 297 (1965).
    [Crossref]
  12. L. B. Lesem, P. M. Hirsch, and J. A. Jordan, IBM J. Res. Develop. 13, 150 (1969).
    [Crossref]
  13. D. Kermisch, J. Opt. Soc. Am. 60, 15 (1970).
    [Crossref]

1970 (2)

1969 (1)

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, IBM J. Res. Develop. 13, 150 (1969).
[Crossref]

1968 (1)

1965 (1)

J. W. Cooley and J. W. Tukey, Math. Comp. 19, 297 (1965).
[Crossref]

1964 (1)

A. Vander Lugt, IEEE Trans. IT-10, 139 (1964).

1962 (1)

1942 (1)

G. C. Danielson and C. Lanczos, J. Franklin Inst. 233, 365 (1942).
[Crossref]

Bracewell, R. M.

R. M. Bracewell, The Fourier Transform and Its Applications (McGraw –Hill, New York, 1965).

Cathey, W. T.

Cooley, J. W.

J. W. Cooley and J. W. Tukey, Math. Comp. 19, 297 (1965).
[Crossref]

Danielson, G. C.

G. C. Danielson and C. Lanczos, J. Franklin Inst. 233, 365 (1942).
[Crossref]

Hirsch, P. M.

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, IBM J. Res. Develop. 13, 150 (1969).
[Crossref]

Jordan, J. A.

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, IBM J. Res. Develop. 13, 150 (1969).
[Crossref]

Kermisch, D.

Lanczos, C.

G. C. Danielson and C. Lanczos, J. Franklin Inst. 233, 365 (1942).
[Crossref]

Leith, E. N.

Lesem, L. B.

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, IBM J. Res. Develop. 13, 150 (1969).
[Crossref]

Smith, H. M.

Tukey, J. W.

J. W. Cooley and J. W. Tukey, Math. Comp. 19, 297 (1965).
[Crossref]

Upatnieks, J.

Vander Lugt, A.

A. Vander Lugt, IEEE Trans. IT-10, 139 (1964).

Appl. Opt. (1)

IBM J. Res. Develop. (1)

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, IBM J. Res. Develop. 13, 150 (1969).
[Crossref]

IEEE Trans. (1)

A. Vander Lugt, IEEE Trans. IT-10, 139 (1964).

J. Franklin Inst. (1)

G. C. Danielson and C. Lanczos, J. Franklin Inst. 233, 365 (1942).
[Crossref]

J. Opt. Soc. Am. (3)

Math. Comp. (1)

J. W. Cooley and J. W. Tukey, Math. Comp. 19, 297 (1965).
[Crossref]

Other (5)

The system we used accommodated 256-point transforms.

APL/360 is available from IBM as a program product.

In general, complicated wave fronts will be filtered. When used to form an image, the incident wave front will usually be plane or spherical.

R. M. Bracewell, The Fourier Transform and Its Applications (McGraw –Hill, New York, 1965).

If Xj=∑n=0N-1An exp(2πijn/N) then An=N-1∑j=0N-1Xj exp(-2πijn/N).

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Figures (13)

Fig. 1
Fig. 1

Amplitude of the scattering function

Fig. 2
Fig. 2

Phase of the scattering function

Fig. 3
Fig. 3

Scattering function for saw tooth given by

Fig. 4
Fig. 4

One period of the amplitude transmittance and phase modulation of the composite filter given by Φ(v) = T(v) exp(v).

Fig. 5
Fig. 5

Image produced by filter shown in Fig. 4.

Fig. 6
Fig. 6

One period of the kinoform filter given by Φ(v) = exp(v).

Fig. 7
Fig. 7

Image produced by the kinoform shown in Fig. 6.

Fig. 8
Fig. 8

One period of the zero-order phase-only filter where the phase is given by

Fig. 9
Fig. 9

Image produced by the filter shown in Fig. 8.

Fig. 10
Fig. 10

One cycle of the blazed first-order phase-only filter where the phase is given by

Fig. 11
Fig. 11

Image produced by the filter shown in Fig. 10.

Fig. 12
Fig. 12

Profilometer trace of one cycle of the surface profile of the experimental model of filter shown in Fig. 8.

Fig. 13
Fig. 13

Image produced by reflection from the filter shown in Fig. 12 and recorded on Kodak Plus X along with a microdensitometer trace of this record.

Equations (35)

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Φ ( v ) = T ( v ) exp i ϕ ( v ) .
Φ ( v ) T ( v ) cos [ ω v + ϕ ( v ) ]
Φ ( v ) exp i { T ( v ) cos [ ω v + ϕ ( v ) ] } ,
Φ ( v ) exp i [ ϕ ( v ) + h ( v ) β ( v ) ] ,
Ψ ( u ) = A j δ ( u - j Δ u ) ,
Φ ( v ) = A j exp ( 2 π i v j Δ u ) = T ( v ) exp [ i ϕ ( v ) ] ,
Ψ ( u ) = - L / 2 + L / 2 Φ ( v ) exp ( - 2 π i u v ) d v ,
Ψ ( u ) = A j sinc [ L ( u - j Δ u ) ] ,
A j = L Δ u 0 P Φ ( v ) exp ( - 2 π i j v Δ u ) d v , P = 1 / Δ u , sinc ( x ) = sin π x / π x .
A j = L Δ u n = 0 N - 1 ( n - 1 2 ) Δ v ( n + 1 2 ) Δ v T n exp [ i ( ϕ n - 2 π j v Δ u ) ] d v = L N - 1 sinc ( j / N ) n = 0 N - 1 T n exp [ i ( ϕ n - 2 π j n / N ) ] ,
T n = T ( n Δ v ) ,             ϕ n = ϕ ( n Δ v ) ,             N = 1 / Δ v Δ u .
A j = L sinc ( j / N ) A j ,
Φ ( v ) = exp { i [ ϕ ( v ) + h ( v ) β ( v ) ] } ,
A j = L Δ u n = 0 N - 1 ( n - 1 2 ) Δ v ( n + 1 2 ) Δ v exp { i ϕ n + i [ h n β ( 2 π v / Δ v ) - 2 π j v Δ u ] } d v = L N - 1 n = 0 N - 1 exp ( i ϕ n - 2 π i j n / N ) Δ v - 1 × - Δ v / 2 Δ v / 2 exp { i [ h n β ( 2 π v / Δ v ) - 2 π j v Δ u ] } d v .
A j = L N - 1 n = 0 N - 1 S n ( j / N ) exp [ i ( ϕ n - 2 π j n / N ) ] ,
S n ( u ) = ( 2 π ) - 1 - π π exp { i [ h n β ( y ) - u y ] } d y .
S n ( 0 ) = T n .
S n ( k ) = T n .
S n ( u ) = P = 0 u P S n ( P ) ( 0 ) / P ! ,
S n ( P ) ( 0 ) = ( 2 π ) - 1 ( - i ) p - π π exp [ i h n β ( y ) ] y p d y .
S n ( j / N ) S n ( 0 ) + j S n ( 1 ) ( 0 ) / N .
J 0 ( h n ) = S n ( 0 ) = T n .
S n ( u ) = 1 π 0 π exp ( i h n cos y ) cos u y d y .
S n ( u ) = sinc ( u - h n ) .
S n ( 1 ) = sinc ( 1 - h n ) = T n .
h n = 1 - sinc - 1 ( T n ) .
S n ( 1 ) = J 1 ( h n ) = T n .
S ( u ) = 1 π 0 π exp ( i h cos y ) cos u y d y .
S ( u ) = 1 π 0 π exp ( i h cos y ) cos u y d y .
S n ( u ) = sinc ( u - h ) .
ϕ ( v ) + h ( v ) cos 2 π v 32 Δ u ,             where             J 0 [ h ( v ) ] = T ( v ) .
ϕ ( v ) + h ( v ) β ( 2 π v 32 Δ v ) ,
h ( v ) = 1 - sin c - 1 [ T ( v ) ] ,             β ( y ) = y             ( - π < y < π ) .
Xj=n=0N-1Anexp(2πijn/N)
An=N-1j=0N-1Xjexp(-2πijn/N).