Abstract

Diffraction gratings can be used to form multiple images in optical systems utilizing incoherent, monochromatic light. The properties of two-dimensional phase gratings having periodic rectangular, sinusoidal, or triangular surface corrugation have been investigated. It is shown that these gratings can be used to form an array of equally intense images by an optical cascading process, each step in the cascade increasing the number of images by a factor of 9. These simple gratings would be particularly useful for forming relatively small numbers of large images. The efficiency per step would be between 70% and 80% and the distortion less than 2%. If spatially incoherent laser light is used, the resolution and general image quality would be limited mainly by the lenses used. These conclusions are corroborated by experimental results obtained with rectangular-type phase gratings.

© 1972 Optical Society of America

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References

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  1. P. A. Newman, V. E. Rible, Appl. Opt. 5, 1225 (1966).
    [CrossRef] [PubMed]
  2. R. G. Olsson, “Generation of Micro Circuit Pattern Arrays by Diffraction Imaging,” presented at the Conference on the Use of Optics in Microelectronics, Optical Society of America, January 1971, Las Vegas, Nevada.
  3. S. Laventhal, A. Werts, N. Rembault, Compt. Rend. B267, 120 (1968).
  4. Sun Lu, Proc. IEEE 56, 116 (1968).
    [CrossRef]
  5. G. Groh, Appl. Opt. 8, 967 (1969).
    [CrossRef] [PubMed]
  6. R. Bartolini, W. Hannan, D. Karlsons, M. Lurie, Appl. Opt. 9, 2283 (1970).
    [CrossRef] [PubMed]
  7. H. J. Gerritsen, W. J. Hannan, E. G. Ramberg, Appl. Opt. 7, 2301 (1968).
    [CrossRef] [PubMed]
  8. H. Dammann, K. Görtler, Opt. Commun. 3, 312 (1971).
    [CrossRef]
  9. W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966), p. 333.
  10. M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1964), pp. 401, 402.
  11. M. Abramowitz, I. A. Stegren, Handbook of Mathematical Functions, A.M.S. 55 (U.S. Govt. Printing Office, Washington, D.C., 1964), pp. 887, 916.

1971 (1)

H. Dammann, K. Görtler, Opt. Commun. 3, 312 (1971).
[CrossRef]

1970 (1)

1969 (1)

1968 (3)

S. Laventhal, A. Werts, N. Rembault, Compt. Rend. B267, 120 (1968).

Sun Lu, Proc. IEEE 56, 116 (1968).
[CrossRef]

H. J. Gerritsen, W. J. Hannan, E. G. Ramberg, Appl. Opt. 7, 2301 (1968).
[CrossRef] [PubMed]

1966 (1)

Abramowitz, M.

M. Abramowitz, I. A. Stegren, Handbook of Mathematical Functions, A.M.S. 55 (U.S. Govt. Printing Office, Washington, D.C., 1964), pp. 887, 916.

Bartolini, R.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1964), pp. 401, 402.

Dammann, H.

H. Dammann, K. Görtler, Opt. Commun. 3, 312 (1971).
[CrossRef]

Gerritsen, H. J.

Görtler, K.

H. Dammann, K. Görtler, Opt. Commun. 3, 312 (1971).
[CrossRef]

Groh, G.

Hannan, W.

Hannan, W. J.

Karlsons, D.

Laventhal, S.

S. Laventhal, A. Werts, N. Rembault, Compt. Rend. B267, 120 (1968).

Lu, Sun

Sun Lu, Proc. IEEE 56, 116 (1968).
[CrossRef]

Lurie, M.

Newman, P. A.

Olsson, R. G.

R. G. Olsson, “Generation of Micro Circuit Pattern Arrays by Diffraction Imaging,” presented at the Conference on the Use of Optics in Microelectronics, Optical Society of America, January 1971, Las Vegas, Nevada.

Ramberg, E. G.

Rembault, N.

S. Laventhal, A. Werts, N. Rembault, Compt. Rend. B267, 120 (1968).

Rible, V. E.

Smith, W. J.

W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966), p. 333.

Stegren, I. A.

M. Abramowitz, I. A. Stegren, Handbook of Mathematical Functions, A.M.S. 55 (U.S. Govt. Printing Office, Washington, D.C., 1964), pp. 887, 916.

Werts, A.

S. Laventhal, A. Werts, N. Rembault, Compt. Rend. B267, 120 (1968).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1964), pp. 401, 402.

Appl. Opt. (4)

Compt. Rend. (1)

S. Laventhal, A. Werts, N. Rembault, Compt. Rend. B267, 120 (1968).

Opt. Commun. (1)

H. Dammann, K. Görtler, Opt. Commun. 3, 312 (1971).
[CrossRef]

Proc. IEEE (1)

Sun Lu, Proc. IEEE 56, 116 (1968).
[CrossRef]

Other (4)

R. G. Olsson, “Generation of Micro Circuit Pattern Arrays by Diffraction Imaging,” presented at the Conference on the Use of Optics in Microelectronics, Optical Society of America, January 1971, Las Vegas, Nevada.

W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966), p. 333.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1964), pp. 401, 402.

M. Abramowitz, I. A. Stegren, Handbook of Mathematical Functions, A.M.S. 55 (U.S. Govt. Printing Office, Washington, D.C., 1964), pp. 887, 916.

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Figures (13)

Fig. 1
Fig. 1

(a) Three-dimensional view of a portion of a two-dimensional rectangular-type phase grating. (b) Profiles of the periodic rectangular, sinusoidal, and triangular phase gratings, along either the x or y direction.

Fig. 2
Fig. 2

Schematic diagram of an optical system using phase gratings to form multiple images.

Fig. 3
Fig. 3

Diffraction envelopes along u or v axis corresponding to the two-dimensional phase gratings: (a) rectangular (α = 1.004), (b) sinusoidal (α = 1.435), (c) triangular (α = 1.822).

Fig. 4
Fig. 4

Normalized interference term along u or v axis corresponding to a diffraction grating having twenty elements along x and y directions.

Fig. 5
Fig. 5

Normalized diffraction patterns along u or v axis corresponding to the two-dimensional phase gratings, for N = 20: (a) rectangular (α = 1.004), (b) sinusoidal (α = 1.435), (c) triangular (α = 1.822).

Fig. 6
Fig. 6

Normalized diffraction pattern along u or v axis corresponding to a square aperture of the same dimension as the grating boundary, and with sides parallel to the x and y elements of the grating.

Fig. 7
Fig. 7

Diagram explaining the symbols used in the calculation of the distortion introduced by a diffraction grating.

Fig. 8
Fig. 8

Photomicrographs (640×) of the surface of a rectangular-type phase grating having an element spacing of 25 μ: (a) after the first etch, (b) after the second etch.

Fig. 9
Fig. 9

Diffraction pattern formed by a rectangular-type phase grating used in conjunction with a lens, when the limiting aperture is circular.

Fig. 10
Fig. 10

Multiple imaging with phase gratings: images formed in plane P1 (Fig. 2): (a) without the grating G1, (b) with the grating G1. Shown at approximately four times the actual size.

Fig. 11
Fig. 11

Influence of the grating on the quality of the images shown in Fig. 10. (a) Detail of the images formed without the grating [Fig. 10(a)]. (b) Same detail, with the grating, for the image at the center of the array [Fig. 10(b)]. (c) Same as (b), but for the image at one of the corners of the array. The finest resolved line is about 5 μ for the three cases. The magnification is about 270×.

Fig. 12
Fig. 12

Images formed in plane P2 (Fig. 2) after a two-stage image multiplication by means of gratings G1 and G2. Shown at about five times the actual size.

Fig. 13
Fig. 13

Influence of refraction in determining the phase modulation in the z = 0 plane introduced by a corrugation-type phase grating.

Tables (1)

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Table I Characteristics of Three Phase Gratings (Two Dimensional) That Can Be Used to Form Nine Images of Equal Intensity

Equations (31)

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Φ ( x , y ) = ϕ ( x ) + ϕ ( y ) .
A ( p , q ) = exp [ - i Φ ( x , y ) ] exp [ ( - i 2 π / λ ) ( p x + q y ) ] d x d y ,
I ( u , v ) = E ( u , v ) I F ( u , v ) = E 1 ( u ) E 1 ( v ) I 1 ( u ) I 1 ( v ) ,
E 1 ( u ) = | 0 d exp [ - i ϕ ( x ) ] exp ( - i u x / d ) d x | 2 , E 1 ( v ) = | 0 d exp [ - i ϕ ( y ) exp ( - i v y / d ) d y | 2 ,
I 1 ( u ) = sin 2 2 N u / sin 2 2 u ,             I 1 ( v ) = sin 2 2 N v / sin 2 2 v ,
u = π d sin θ / 2 λ ,             v = π d sin β / 2 λ .
Rectangular ( Rec ) ϕ ( x ) = 2 π ( n - 1 ) Δ / λ 0 < x < d / 2 , ϕ ( x ) = 0 d / 2 < x < d , Sinusoidal ( Sin ) ϕ ( x ) = [ π ( n - 1 ) Δ / λ ] × [ ( 1 + cos ( 2 π x / d ) ] 0 < x < d , Triangular ( Tri ) ϕ ( x ) = 4 π ( n - 1 ) Δ x / λ d 0 < x / d / 2 , ϕ ( x ) = 4 π ( n - 1 ) Δ ( d - x ) / λ d d / 2 < x < d ,
Rec : E 1 ( u , α ) = sin 2 u cos 2 ( u + α ) / u 2 ; Sin : when u = m π / 2 , m = 0 , ± 1 , ± 2 , ( i . e . , at the principal maxima ) :
E 1 ( m π / 2 , α ) = J 2 m ( α ) ,
A = - π + π cos [ α cos x + ( 2 u x / π ) ] d x , B = - π + π sin [ α cos x + ( 2 u x / π ) ] d x .
E 1 ( u , α ) = [ sin 2 ( u + α ) / ( u + α ) 2 ] + [ sin 2 ( u - α ) / ( u - α ) 2 ] - [ sin 2 ( u - α ) / ( u 2 - α 2 ) ] - [ sin 2 ( u + α ) / ( u 2 - α 2 ) ] + [ sin 2 2 u / ( u 2 - α 2 ) ] .
Rec { cos 2 ( α ) m = 0 sin 2 ( α ) / ( m π / 2 ) 2 m = odd integer 0 m = even integer Sin J 2 m ( α ) all m Tri { sin 2 ( α ) / α 2 m = 0 α 2 cos 2 ( α ) / [ ( m 2 π 2 / 4 ) - α 2 ] 2 m = odd α 2 sin 2 ( α ) / [ m 2 π 2 / 4 ) - α 2 ] 2 m = even
Rec I 1 / m 2 , m = odd Sin I ( e α ) 2 m / m ( 2 m ) 2 m , Tri I 1 / m 4 .
Rec : α = tan - 1 π / 2 α = 1.004 , Sin : J o ( α ) = J 1 ( α ) α = 1.435 , Tri : tan α = α 2 / [ ( π 2 / 4 ) - α 2 ] α = 1.822.
e = 9 E 1 2 ( 0 , α ) / E 1 2 ( 0 , 0 ) .
R d R / d α Rec ( 2 tan α ) / π ( 2 sec 2 α ) / π Sin J 1 ( α ) / J o ( α ) 1 + [ J 1 2 ( α ) / J o 2 ( α ) ] - [ J 1 ( α ) / α J o ( α ) ] Tri α 2 cot α / ( π 2 / 4 - α 2 ) π 2 α cot α / 2 ( π 2 / 4 - α 2 ) 2 - α 2 csc 2 α / ( π 2 / 4 - α 2 )
sin θ = sin γ + ( m λ / d ) ,             sin β = sin ω + ( n λ / d ) ,             m , n = 0 , ± 1 , ± 2 , .
sin θ - sin γ = m λ / d .
D = [ ( d s / d r ) θ - ( d s / d r ) 0 ( d s / d r ) 0 ] γ = 0
D = ( 1 - cos 3 θ ) / cos 3 θ ,
Δ θ m Δ λ / d cos θ             and             Δ β n Δ λ / d cos β .
Δ s / s = Δ t / t Δ λ / λ .
z = Δ ( 1 + { [ cos ( 2 π x / d ) + cos ( 2 π y / d ) ] / 2 } ) .
n z ( x , y ) - z ( x , y ) .
n z ( x + δ x , y + δ y ) - [ z ( x + δ x , y + δ y ) / cos ( θ - θ ) ] ,
tan θ = ( π Δ / d ) [ sin 2 ( 2 π x / d ) + sin 2 ( 2 π y / d ) ] 1 2 ,
( θ - θ ) ( n - 1 ) θ [ ( n - 1 ) π Δ / d ] × [ sin 2 ( 2 π x / d ) + sin 2 ( 2 π y / d ) ] 1 2 .
e = ( n - 1 ) [ ( z / x ) δ x + ( z / y ) δ y ] - [ ( θ - θ ) 2 z / 2 ] .
e = - 3 Δ ( n . - 1 ) 2 ( π Δ / d ) 2 sin 2 ( 2 π x / d ) [ 1 + cos ( 2 π x / d ) ] .
3 cos 2 ( 2 π x / d ) + 2 cos ( 2 π x / d ) - 1 = 0
e max = 4 Δ ( n - 1 ) 2 ( π Δ / d ) 2 .

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