Abstract

We present a matrix formalism to describe the near-field diffraction pattern, at fractions of a Talbot distance, of a grating whose unit cell is composed of a discrete substructure. We show that this formalism is useful for designing Lohmann array illuminators.

© 1995 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. T. Winthrop, C. R. Worthingon, J. Opt. Soc. Am. 55, 373 (1965).
    [CrossRef]
  2. J. P. Guigay, Opt. Acta 18, 677 (1971).
    [CrossRef]
  3. R. E. Ioseliani, Opt. Spectrosc. (USSR) 55, 544 (1983).
  4. V. Arrizón, J. Ojeda-Castañeda, Appl. Opt. 33, 5925 (1994).
    [CrossRef] [PubMed]
  5. J. R. Leger, G. J. Swanson, Opt. Lett. 15,288 (1990).
    [CrossRef] [PubMed]
  6. A. W. Lohmann, Optik 79, 41 (1988).

1994 (1)

1990 (1)

1988 (1)

A. W. Lohmann, Optik 79, 41 (1988).

1983 (1)

R. E. Ioseliani, Opt. Spectrosc. (USSR) 55, 544 (1983).

1971 (1)

J. P. Guigay, Opt. Acta 18, 677 (1971).
[CrossRef]

1965 (1)

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

Opt. Acta (1)

J. P. Guigay, Opt. Acta 18, 677 (1971).
[CrossRef]

Opt. Lett. (1)

Opt. Spectrosc. (1)

R. E. Ioseliani, Opt. Spectrosc. (USSR) 55, 544 (1983).

Optik (1)

A. W. Lohmann, Optik 79, 41 (1988).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1
Fig. 1

One period (a) of a substructured phase grating G(x) with Q subintervals and (b) of the binary grating B(x) with opening ratio 1/Q.

Fig. 2
Fig. 2

Generation of an array illuminator with a compression ratio of 8: (a) four-level phase distribution ψ(x) of grating G(x) at z = 0, (b) irradiance distribution at the plane z = Zt/16.

Tables (2)

Tables Icon

Table 1 Values of |C(L, N)| and Phase Shift ϕ0

Tables Icon

Table 2 Phases θn,m for the Components of Matrix M (Case N = 16)

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

u ( x ; z = Z t / N ) = L = 0 N - 1 C ( L , N ) G ( x - L d / N ) ,
C ( L , N ) = 1 N q = 0 N - 1 exp [ i ( 2 π / N ) q ( L - q ) ] .
C ( L , N ) = C ( L , N ) exp [ i ϕ ( L , N ) ] ,
ϕ ( L , N ) = ( L 2 / 2 N ) π - ϕ 0 ,
u ( x ; z = Z t / N ) = n = 0 Q - 1 c n G ( x - n d / Q ) ,
c n = { C ( n , N ) for N odd C ( 2 n , N ) for N / 2 even .
B ( x ) = q = - rect ( x - q d d / Q ) ,
G ( x ) = m = 0 Q - 1 α m B ( x - m d / Q ) .
u ( x ; z = Z t / N ) = n = 0 Q - 1 m = 0 Q - 1 α m c n B [ x - ( n + m ) d / Q ] .
u ( x ; Z t / N ) = p = 0 Q - 1 β p B ( x - p d / Q ) ,
β p = m = 0 p α m c p - m + m = p + 1 Q - 1 α m c p - m + Q .
β = M α ,
α = [ α 0 α 1 α Q - 1 ] ,             β = [ β 0 β 1 β Q - 1 ] ,             M = [ c 0 c Q - 1 c 2 c 1 c 1 c 0 c 3 c 2 c Q - 2 c Q - 3 c 0 c Q - 1 c Q - 1 c Q - 2 c 1 c 0 ] .
u ( x ; L Z t / N ) = p = 0 Q - 1 γ p B ( x - p d / Q ) ,
γ = M L α .
α = M - 1 β .
M n , m = ( 1 / 8 ) exp ( i θ n , m ) ,
α = [ exp ( i π / 4 ) exp ( i π / 8 ) exp ( - i π / 4 ) exp ( - i 7 π / 8 ) exp ( i π / 4 ) exp ( - i 7 π / 8 ) exp ( - i π / 4 ) exp ( i π / 8 ) ] ,             β = [ 8 0 0 0 0 0 0 0 ] .
α = [ 1 1 1 1 - 1 - 1 - 1 - 1 ] ,             β = 2 [ 0 exp ( - i π / 8 ) exp ( - i π / 8 ) 0 0 - exp ( - i π / 8 ) - exp ( - i π / 8 ) 0 ] .

Metrics