Binary Fraunhofer Holograms, Generated by Computer

A. W. Lohmann; D. P. Paris

doi:10.1364/AO.6.001739

Binary Fraunhofer Holograms, Generated by Computer

A. W. Lohmann and D. P. Paris

Applied Optics
Vol. 6,
Issue 10,
pp. 1739-1748
(1967)
•https://doi.org/10.1364/AO.6.001739

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A. W. Lohmann and D. P. Paris, "Binary Fraunhofer Holograms, Generated by Computer," Appl. Opt. 6, 1739-1748 (1967)

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Abstract

When a hologram is desired from an object which does not exist physically but is known in mathematical terms, one can compute the hologram. An automatic plotter will make a drawing at a large scale which is then reduced photographically. Since the drawing can contain only black and white areas, we have developed a theory for binary holograms. They are equivalent in terms of image reconstruction with ordinary holograms. This has been proven theoretically and verified experimentally.

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References

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Author
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Publication

B. R. Brown, A. W. Lohmann, Appl. Opt. 5, 967 (1966).
[Crossref] [PubMed]
D. Hauk, A. Lohmann, Optik 15, 275 (1958).
C. A. Taylor, H. Lipson, Optical Transforms (Bell, London, 1964); G. Harburn, K. Walkley, C. A. Taylor, Nature 205, 1096 (1965).
[Crossref]
A. Kozma, D. L. Kelly, Appl. Opt. 4, 37 (1965).
[Crossref]
J. P. Waters, Appl. Phys. Letters 9, 405 (1957).
[Crossref]
M. Marquet, J. Tsujiuchi, Opt. Acta 8, 267 (1961).
[Crossref]
A. W. Lohmann, D. P. Paris, J. Opt. Soc. Am. 56, 537A (1966).
J. W. Cooley, J. W. Tukey, Math. Comp. 19, 297 (1965); IBM Share Libr. HARM 3425.
[Crossref]
E. N. Leith, J. Upatnieks, J. Opt. Soc. Am. 54, 1295 (1964).
[Crossref]
B. R. Brown, A. W. Lohmann, D. P. Paris, Opt. Acta 13, 377 (1966).
A. W. Lohmann, D. P. Paris, J. Opt. Soc. Am. 56, 1413 A (1966).
A. W. Lohmann, D. P. Paris, IEEE J. Quantum Electron. QE-2, LXV (1966).

1966 (5)

B. R. Brown, A. W. Lohmann, D. P. Paris, Opt. Acta 13, 377 (1966).

A. W. Lohmann, D. P. Paris, J. Opt. Soc. Am. 56, 1413 A (1966).

A. W. Lohmann, D. P. Paris, IEEE J. Quantum Electron. QE-2, LXV (1966).

B. R. Brown, A. W. Lohmann, Appl. Opt. 5, 967 (1966).
[Crossref] [PubMed]

A. W. Lohmann, D. P. Paris, J. Opt. Soc. Am. 56, 537A (1966).

1965 (2)

J. W. Cooley, J. W. Tukey, Math. Comp. 19, 297 (1965); IBM Share Libr. HARM 3425.
[Crossref]

A. Kozma, D. L. Kelly, Appl. Opt. 4, 37 (1965).
[Crossref]

1964 (1)

E. N. Leith, J. Upatnieks, J. Opt. Soc. Am. 54, 1295 (1964).
[Crossref]

1961 (1)

M. Marquet, J. Tsujiuchi, Opt. Acta 8, 267 (1961).
[Crossref]

1958 (1)

D. Hauk, A. Lohmann, Optik 15, 275 (1958).

1957 (1)

J. P. Waters, Appl. Phys. Letters 9, 405 (1957).
[Crossref]

Brown, B. R.

B. R. Brown, A. W. Lohmann, Appl. Opt. 5, 967 (1966).
[Crossref] [PubMed]

B. R. Brown, A. W. Lohmann, D. P. Paris, Opt. Acta 13, 377 (1966).

Cooley, J. W.

J. W. Cooley, J. W. Tukey, Math. Comp. 19, 297 (1965); IBM Share Libr. HARM 3425.
[Crossref]

Hauk, D.

D. Hauk, A. Lohmann, Optik 15, 275 (1958).

Kelly, D. L.

A. Kozma, D. L. Kelly, Appl. Opt. 4, 37 (1965).
[Crossref]

Kozma, A.

A. Kozma, D. L. Kelly, Appl. Opt. 4, 37 (1965).
[Crossref]

Leith, E. N.

E. N. Leith, J. Upatnieks, J. Opt. Soc. Am. 54, 1295 (1964).
[Crossref]

Lipson, H.

C. A. Taylor, H. Lipson, Optical Transforms (Bell, London, 1964); G. Harburn, K. Walkley, C. A. Taylor, Nature 205, 1096 (1965).
[Crossref]

Lohmann, A.

D. Hauk, A. Lohmann, Optik 15, 275 (1958).

Lohmann, A. W.

B. R. Brown, A. W. Lohmann, Appl. Opt. 5, 967 (1966).
[Crossref] [PubMed]

A. W. Lohmann, D. P. Paris, J. Opt. Soc. Am. 56, 537A (1966).

B. R. Brown, A. W. Lohmann, D. P. Paris, Opt. Acta 13, 377 (1966).

A. W. Lohmann, D. P. Paris, J. Opt. Soc. Am. 56, 1413 A (1966).

A. W. Lohmann, D. P. Paris, IEEE J. Quantum Electron. QE-2, LXV (1966).

Marquet, M.

M. Marquet, J. Tsujiuchi, Opt. Acta 8, 267 (1961).
[Crossref]

Paris, D. P.

A. W. Lohmann, D. P. Paris, J. Opt. Soc. Am. 56, 537A (1966).

A. W. Lohmann, D. P. Paris, IEEE J. Quantum Electron. QE-2, LXV (1966).

A. W. Lohmann, D. P. Paris, J. Opt. Soc. Am. 56, 1413 A (1966).

B. R. Brown, A. W. Lohmann, D. P. Paris, Opt. Acta 13, 377 (1966).

Taylor, C. A.

C. A. Taylor, H. Lipson, Optical Transforms (Bell, London, 1964); G. Harburn, K. Walkley, C. A. Taylor, Nature 205, 1096 (1965).
[Crossref]

Tsujiuchi, J.

M. Marquet, J. Tsujiuchi, Opt. Acta 8, 267 (1961).
[Crossref]

Tukey, J. W.

J. W. Cooley, J. W. Tukey, Math. Comp. 19, 297 (1965); IBM Share Libr. HARM 3425.
[Crossref]

Upatnieks, J.

E. N. Leith, J. Upatnieks, J. Opt. Soc. Am. 54, 1295 (1964).
[Crossref]

Waters, J. P.

J. P. Waters, Appl. Phys. Letters 9, 405 (1957).
[Crossref]

Appl. Opt. (2)

B. R. Brown, A. W. Lohmann, Appl. Opt. 5, 967 (1966).
[Crossref] [PubMed]

A. Kozma, D. L. Kelly, Appl. Opt. 4, 37 (1965).
[Crossref]

Appl. Phys. Letters (1)

J. P. Waters, Appl. Phys. Letters 9, 405 (1957).
[Crossref]

IEEE J. Quantum Electron. (1)

A. W. Lohmann, D. P. Paris, IEEE J. Quantum Electron. QE-2, LXV (1966).

J. Opt. Soc. Am. (3)

A. W. Lohmann, D. P. Paris, J. Opt. Soc. Am. 56, 1413 A (1966).

A. W. Lohmann, D. P. Paris, J. Opt. Soc. Am. 56, 537A (1966).

E. N. Leith, J. Upatnieks, J. Opt. Soc. Am. 54, 1295 (1964).
[Crossref]

Math. Comp. (1)

J. W. Cooley, J. W. Tukey, Math. Comp. 19, 297 (1965); IBM Share Libr. HARM 3425.
[Crossref]

Opt. Acta (2)

B. R. Brown, A. W. Lohmann, D. P. Paris, Opt. Acta 13, 377 (1966).

M. Marquet, J. Tsujiuchi, Opt. Acta 8, 267 (1961).
[Crossref]

Optik (1)

D. Hauk, A. Lohmann, Optik 15, 275 (1958).

Other (1)

C. A. Taylor, H. Lipson, Optical Transforms (Bell, London, 1964); G. Harburn, K. Walkley, C. A. Taylor, Nature 205, 1096 (1965).
[Crossref]

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Fig. 1

Setup for the construction of an image IMG from a binary hologram HOLO, illuminated by a point source x₀ in SOU.

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Fig. 2

Structure of the (n,m) cell in the binary hologram. The rectangular opening has a width cδν, a height W_nmδν, and it is displaced from the center of the cell at (nδν,mδν) by P_nmδν.

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Fig. 3

Binary Fraunhofer holography: (a) the hologram; (b) the image.

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Fig. 4

Simulation of a groundglass, i.e., object with random phase: (a) the hologram; (b) the image.

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Fig. 5

A holographic image with defects owing to zero-order approximation in the calculation.

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Fig. 6

Images from binary holograms of different orders: (a) M = 1; (b) M = 3.

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Equations (65)

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\tilde{u} (ν_{x}, ν_{y}) = \iint u (x, y) E (- x ν_{x} - y ν_{y}) d x d y .

const \tilde{u} (ν_{x}, ν_{y}) = H (ν_{x}, ν_{y}) E (x_{0} ν_{x}) .

\begin{array}{l} u (x, y) = 0 outside of ∣ x ∣ \leq Δ x / 2; ∣ y ∣ \leq Δ y / 2; \\ \tilde{u} (ν_{x}, ν_{y}) \approx 0 outside of ∣ ν_{x} ∣ \leq Δ ν / 2; ∣ v_{y} ∣ \leq Δ ν ν 2. \end{array}

h (x, y) = rect (x / Δ x) rect (y / Δ x) \iint H (ν_{x}, ν_{y}) E [(x + x_{0}) ν_{x} + y ν_{y}] d ν_{x} d ν; .

rect (z) = {\begin{array}{l} 1; in ∣ z ∣ \leq \frac{1}{2} \\ 0; otherwise \end{array}

h (x, y) = const u (x, y) .

N^{2} = (Δ x Δ y) / (δ x δ y) = {(Δ x / δ x)}^{2} = {(Δ x Δ ν)}^{2} = S W .

{(Δ ν / δ ν)}^{2} \geq {(Δ x / δ x)}^{2} = {(Δ x Δ ν)}^{2}; \to δ ν \leq 1 / Δ x .

\tilde{u} (ν_{x}, ν_{y}) = Σ Σ (n, m) \tilde{u} (n / Δ x, m / Δ x) \times sinc (ν_{x} Δ x - n) sinc (ν_{y} Δ x - m) .

∣ ν_{x} - (n + P_{n m}) δ ν ∣ \leq c δ ν / 2; ∣ ν_{y} - m δ ν ∣ \leq W_{n m} δ ν / 2.

\int_{(m - \frac{1}{2}) δ ν}^{(m + \frac{1}{2}) δ ν} H (ν_{x}, ν_{y}) d ν_{y} = H_{m} (ν_{x}); H_{m} [(n + P_{n m}) δ ν + μ] = H_{m} [(n + P_{n m}) δ ν - μ],

H (ν_{x}, ν_{y}) = Σ Σ (n, m) rect [\frac{ν_{x} - (n + P_{n m}) δ ν}{c δ ν}] rect [\frac{ν_{y} - m δ ν}{W_{n m} δ ν}] .

\iint H (ν_{x}, ν_{y}) E [(x + x_{0}) ν_{x} + y ν_{y}] d ν_{x} d ν_{y} = c {(δ ν)}^{2} sinc [c δ ν (x + x_{0})] Σ Σ (n, m) W_{n m} sinc (y W_{n m} δ ν) E {δ ν [(x + x_{0}) (n + P_{n m}) + y m]} .

rect (x / Δ x) rect (y / Δ x) \iint H E d ν_{x} d ν_{y} = h (x, y); h (x, y) = const u (x, y) .

\begin{array}{l} u (x, y) = \iint \tilde{u} (ν_{x}, ν_{y}) E (x ν_{x} + y ν_{y}) d ν_{x} d ν_{y} \\ = rect (x / Δ x) rect (y / Δ x) \sum_{(n, m)} \sum \tilde{u} (n δ ν), m δ ν) E [δ ν (x n + y m)] . \end{array}

\begin{array}{l} c {(δ ν)}^{2} W_{n m} E [x_{0} δ ν (n + P_{n m})] & \approx const \tilde{u} (n δ ν, m δ ν); \\ const \tilde{u} (n δ ν, m δ ν) & = c {(δ ν)}^{2} A_{n m} E (φ_{n m} / 2 π); \\ W_{n m} & \approx A_{n m}; P_{n m} + n \approx φ_{n m} / 2 π x_{0} δ ν . \end{array}

P_{n m} \approx φ_{n m} / 2 π M .

E [x_{0} δ ν (n + P_{n m})] = E (M P_{n m}); φ_{n m} = 2 π M P_{n m}; \bar{φ} = 2 π M \bar{P} .

\begin{array}{l} (a) sinc [c δ ν (x + x_{0})] & \approx const, in ∣ x ∣ \leq Δ x / 2; \\ (b) sinc (y W_{n m} δ ν) & \approx 1 in ∣ y ∣ \leq Δ x / 2; \\ (c) E (x P_{n m} δ ν) & \approx 1 in ∣ x ∣ \leq Δ x / 2. \end{array}

∣ φ ∣ \leq π; φ = 2 π M P; ∣ P ∣ \leq \frac{1}{2 M} .

sinc [c δ ν (x + x_{0})] \approx const, in ∣ x ∣ \leq Δ x / 2.

sinc (y W δ ν) \approx 1,

E (x P_{n m} δ ν) \approx 1 x in ∣ x ∣ \leq Δ x / 2.

u (x, y) \neq 0 only in ∣ x ∣, ∣ y ∣ \leq ξ Δ x / 2; ξ \leq 1.

u (x, y) = v (x, y) sinc [c δ ν (x + x_{0})] .

h (x, y) / sinc [c δ ν (x + x_{0})] = const v (x, y)

\begin{array}{l} v (x, y) = u (x, y) / sinc [c δ ν (x + x_{0})] \\ = rect (x / Δ x) rect (y / Δ x) \sum_{(n, m)} \sum {\tilde{v}}_{n m} E [δ ν (x n + y m)]; \end{array}

{\tilde{v}}_{n m} = {(1 / Δ x)}^{2} \iint u / sinc E [- δ ν (x n + y m)] d x d y;

h (x, y) / sinc [c δ ν (x + x_{0})] = rect (x / Δ x) rect (y / Δ x) c {(δ ν)}^{2} Σ Σ (n, m) W_{n m} sinc (y W_{n m} δ ν) E [M P_{n m} + (x n + y m) δ ν + x P_{n m} δ ν] .

const {\tilde{v}}_{j k} = c {(δ ν / Δ x)}^{2} \sum_{(n, m)} \sum W_{n m} \times \int \int_{(- Δ x / 2)}^{(+ Δ x / 2)} sinc (y W_{n m} δ ν) E [∊] d x d y;

∊ = M P_{n m} + (n - j) x δ ν + (m - k) y δ ν + x P_{n m} δ ν .

const {\tilde{v}}_{j k} = c {(δ ν)}^{2} \sum_{(m, n)} \sum W_{n m} E (M P_{n m}) σ (n, j; m, k);

σ (n, j; m, k) = σ_{x} (n, j; m) σ_{y} (m, k; n);

σ_{x} = (1 / Δ x) \int_{- Δ x / 2}^{+ Δ x / 2} E [x δ ν (n - j + P_{n m})] d x = sinc (n - j + P_{n m});

σ_{y} = (1 / Δ x) \int_{- Δ x / 2}^{+ Δ x / 2} sinc (y W_{n m} δ ν) E [y δ ν (m - k)] d y = [Si (m - k + W_{n m} / 2) - Si (m - k - W_{n m} / 2) / W_{n m} .

Si (z) = \int_{0}^{z} sinc (z^{'}) d z^{'} .

sinc (y W δ ν) = (1 / W δ ν) \int_{- \infty}^{+ \infty} rect (μ / W δ ν) E (μ y) d μ .

const {\tilde{v}}_{j k} / c {(δ ν)}^{2} = A_{j k} E (φ_{j k} / 2 π);

\sum_{(n, m)} \sum W_{n m} E (M P_{n m}) σ (n, j; m, k) = W_{j k} E (M P_{j k}) σ (j, j; k, k) + R_{j k} (W_{n m}, P_{n m}) .

A_{j k} E (φ_{j k} / 2 π) = W_{j k} E (M P_{j k}) σ (j, j; k, k) + R_{j k};

σ (j, j; k, k) = sinc (P_{j k}) Si (W_{j k} / 2) / (W_{j k} / 2) .

R_{j k} = 0; σ (j, j; k, k) = 1.

\begin{array}{l} A_{j k} E (φ_{j k} / 2 π) = {W_{j k}}^{(0)} E (M {P_{j k}}^{(0)}); \\ {W_{j k}}^{(0)} = A_{j k}; {P_{j k}}^{(0)} = φ_{j k} / 2 π M . \end{array}

A_{j k} E (φ_{j k} / 2 π) = {W_{j k}}^{(1)} E (M {P_{j k}}^{(1)}) σ ({W_{j k}}^{(0)}, {P_{j k}}^{(0)});

{P_{j k}}^{(1)} = φ_{j k} / 2 π M = {P_{j k}}^{(0)};

{W_{j k}}^{(1)} = A_{j k} / σ ({W_{j k}}^{(0)}, {P_{j k}}^{(0)}) = {W_{j k}}^{(0)} / σ ({W_{j k}}^{(0)}, {P_{j k}}^{(0)}) .

A_{j k} E (φ_{j k} / 2 π) = {W_{j k}}^{(2)} E (M {P_{j k}}^{(2)}) σ_{j k} ({W_{j k}}^{(1)}, {P_{j k}}^{(1)}) + R_{j k} ({W_{n m}}^{(1)}, {P_{n m}}^{(1)});

A_{j k} E (φ_{j k} / 2 π) - R_{j k} ({W_{n m}}^{(1)}, {P_{n m}}^{(1)}) = B_{j k} E (β_{j k} / 2 π);

{W_{j k}}^{(2)} = B_{j k} / σ_{j k} ({W_{j k}}^{(1)}, {P_{j k}}^{(1)}); {P_{j k}}^{(2)} = β_{j k} / 2 π M .

σ (n, j; m, k) = σ_{x} (n, j; m) σ_{y} (m, k; n); σ_{x} (n \neq j; m m) \approx {(- 1)}^{n - j} sin (π P_{n m}) / π (n - j);

σ_{y} (m \neq k, n) \approx {(- 1)}^{m - k - 1} {[W_{n m} / 2 (m - k)]}^{2} / 3.

\begin{matrix} \tilde{u} (ν_{x}, ν_{y}) \approx 0 for ∣ ν_{x} ∣, ∣ ν_{y} ∣ \leq Δ ν / 2 = 1 / 2 δ x; \\ Δ x_{H} = Δ y_{H} = λ f Δ ν . \end{matrix}

u (n δ x, m δ x) = u_{n m} .

\begin{array}{l} v_{n m} = u_{n m} / sinc [c δ ν (x_{0} + n δ x)] = u_{n m} / sinc [c (M + n / N)]; \\ δ ν = 1 / Δ x; x_{0} δ ν = M . \end{array}

v_{n m} \to \tilde{v} (j δ ν, k δ ν) = ∣ {\tilde{v}}_{j k} ∣ E (φ_{j k} / 2 π)

const ∣ \tilde{v} ∣_{max} = c {(δ ν)}^{2} W_{max} .

const ∣ ν_{j k} ∣ / c {(δ ν)}^{2} = A_{j k}; A_{j k} \leq W_{max} .

∣ n - j ∣ {(m - j)}^{2} \leq {n_{0}}^{3}

u (x) \to u (x) Σ (n) δ (x - n δ x) = v (x) .

\tilde{u} (ν) = \int u (x) E (- v x) d x;

\tilde{v} (ν) = Σ (n) \int u (x) δ (x - n δ x) E (- ν x) d x = Σ (n) u (n δ x) E (- ν n δ x) .

Σ (n) δ (x - n δ x) = Σ (m) E (m x / δ x);

\tilde{v} (ν) = Σ (m) \int u (x) E [x (- ν + m / δ x)] d x = Σ (m) \tilde{u} (ν - m / δ x) .

δ x \leq 1 / Δ ν_{0} .

\tilde{v} (ν_{x}, ν_{y}) = Σ Σ (n, m) \tilde{u} (ν_{x} - n / δ_{x}, ν_{y} - m / δ x) .

Abstract

References

1966 (5)

1965 (2)

1964 (1)

1961 (1)

1958 (1)

1957 (1)

Brown, B. R.

Cooley, J. W.

Hauk, D.

Kelly, D. L.

Kozma, A.

Leith, E. N.

Lipson, H.

Lohmann, A.

Lohmann, A. W.

Marquet, M.

Paris, D. P.

Taylor, C. A.

Tsujiuchi, J.

Tukey, J. W.

Upatnieks, J.

Waters, J. P.

Appl. Opt. (2)

Appl. Phys. Letters (1)

IEEE J. Quantum Electron. (1)

J. Opt. Soc. Am. (3)

Math. Comp. (1)

Opt. Acta (2)

Optik (1)

Other (1)

Cited By

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Equations (65)

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