Abstract

A binary diffraction model is introduced to study the sensitivity of the wavefront phase of binary computer-generated holograms on groove depth and duty-cycle variations. Analytical solutions to diffraction efficiency, diffracted wavefront phase functions, and wavefront sensitivity functions are derived. The derivation of these relationships is obtained by using the Fourier method. Results from experimental data confirm the analysis. Several phase anomalies were discovered, and a simple graphical model of the complex fields is applied to explain these phenomena.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. Reichelt, C. Pruß, and H. J. Tiziani, "New design techniques and calibration methods for CGH-null testing of aspheric surfaces," in Proc. SPIE 4778, 158-168 (2002).
    [CrossRef]
  2. T. Herrmann, "Testing aspheric surfaces with CGH of different accuracy in industrial manufacturing environ," in Proc. SPIE 4440, 120-126 (2001).
    [CrossRef]
  3. C. Pruss, S. Reichelt, H. J. Tiziani, and W. Osten, "Computer-generated holograms in interferometric testing," Opt. Eng. 43, 2534-3540 (2004).
    [CrossRef]
  4. S. Reichelt, C. Pruß, and H. J. Tiziani, "Specification and characterization of CGHs for interferometrical optical testing," in Proc. SPIE 4778, 206-216 (2002).
    [CrossRef]
  5. G. J. Swanson, "Binary optics technology: theoretical limits on the diffraction efficiency of multilevel diffractive optical elements," technical report (Lincoln Library, Massachusetts Institute of Technology, 1991).
  6. J. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, 1978).
  7. M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, 1980).
  8. Y. C. Chang and J. H. Burge, "Errors analysis for CGH optical testing," in Optical Manufacturing and Testing III, Proc. SPIE 3782, 358-366 (1999).
    [CrossRef]

2004

C. Pruss, S. Reichelt, H. J. Tiziani, and W. Osten, "Computer-generated holograms in interferometric testing," Opt. Eng. 43, 2534-3540 (2004).
[CrossRef]

2002

S. Reichelt, C. Pruß, and H. J. Tiziani, "Specification and characterization of CGHs for interferometrical optical testing," in Proc. SPIE 4778, 206-216 (2002).
[CrossRef]

S. Reichelt, C. Pruß, and H. J. Tiziani, "New design techniques and calibration methods for CGH-null testing of aspheric surfaces," in Proc. SPIE 4778, 158-168 (2002).
[CrossRef]

2001

T. Herrmann, "Testing aspheric surfaces with CGH of different accuracy in industrial manufacturing environ," in Proc. SPIE 4440, 120-126 (2001).
[CrossRef]

1999

Y. C. Chang and J. H. Burge, "Errors analysis for CGH optical testing," in Optical Manufacturing and Testing III, Proc. SPIE 3782, 358-366 (1999).
[CrossRef]

1991

G. J. Swanson, "Binary optics technology: theoretical limits on the diffraction efficiency of multilevel diffractive optical elements," technical report (Lincoln Library, Massachusetts Institute of Technology, 1991).

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, 1980).

Burge, J. H.

Y. C. Chang and J. H. Burge, "Errors analysis for CGH optical testing," in Optical Manufacturing and Testing III, Proc. SPIE 3782, 358-366 (1999).
[CrossRef]

Chang, Y. C.

Y. C. Chang and J. H. Burge, "Errors analysis for CGH optical testing," in Optical Manufacturing and Testing III, Proc. SPIE 3782, 358-366 (1999).
[CrossRef]

Gaskill, J.

J. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, 1978).

Herrmann, T.

T. Herrmann, "Testing aspheric surfaces with CGH of different accuracy in industrial manufacturing environ," in Proc. SPIE 4440, 120-126 (2001).
[CrossRef]

Osten, W.

C. Pruss, S. Reichelt, H. J. Tiziani, and W. Osten, "Computer-generated holograms in interferometric testing," Opt. Eng. 43, 2534-3540 (2004).
[CrossRef]

Pruss, C.

C. Pruss, S. Reichelt, H. J. Tiziani, and W. Osten, "Computer-generated holograms in interferometric testing," Opt. Eng. 43, 2534-3540 (2004).
[CrossRef]

Pruß, C.

S. Reichelt, C. Pruß, and H. J. Tiziani, "Specification and characterization of CGHs for interferometrical optical testing," in Proc. SPIE 4778, 206-216 (2002).
[CrossRef]

S. Reichelt, C. Pruß, and H. J. Tiziani, "New design techniques and calibration methods for CGH-null testing of aspheric surfaces," in Proc. SPIE 4778, 158-168 (2002).
[CrossRef]

Reichelt, S.

C. Pruss, S. Reichelt, H. J. Tiziani, and W. Osten, "Computer-generated holograms in interferometric testing," Opt. Eng. 43, 2534-3540 (2004).
[CrossRef]

S. Reichelt, C. Pruß, and H. J. Tiziani, "Specification and characterization of CGHs for interferometrical optical testing," in Proc. SPIE 4778, 206-216 (2002).
[CrossRef]

S. Reichelt, C. Pruß, and H. J. Tiziani, "New design techniques and calibration methods for CGH-null testing of aspheric surfaces," in Proc. SPIE 4778, 158-168 (2002).
[CrossRef]

Swanson, G. J.

G. J. Swanson, "Binary optics technology: theoretical limits on the diffraction efficiency of multilevel diffractive optical elements," technical report (Lincoln Library, Massachusetts Institute of Technology, 1991).

Tiziani, H. J.

C. Pruss, S. Reichelt, H. J. Tiziani, and W. Osten, "Computer-generated holograms in interferometric testing," Opt. Eng. 43, 2534-3540 (2004).
[CrossRef]

S. Reichelt, C. Pruß, and H. J. Tiziani, "Specification and characterization of CGHs for interferometrical optical testing," in Proc. SPIE 4778, 206-216 (2002).
[CrossRef]

S. Reichelt, C. Pruß, and H. J. Tiziani, "New design techniques and calibration methods for CGH-null testing of aspheric surfaces," in Proc. SPIE 4778, 158-168 (2002).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, 1980).

Opt. Eng.

C. Pruss, S. Reichelt, H. J. Tiziani, and W. Osten, "Computer-generated holograms in interferometric testing," Opt. Eng. 43, 2534-3540 (2004).
[CrossRef]

Proc. SPIE

S. Reichelt, C. Pruß, and H. J. Tiziani, "Specification and characterization of CGHs for interferometrical optical testing," in Proc. SPIE 4778, 206-216 (2002).
[CrossRef]

S. Reichelt, C. Pruß, and H. J. Tiziani, "New design techniques and calibration methods for CGH-null testing of aspheric surfaces," in Proc. SPIE 4778, 158-168 (2002).
[CrossRef]

T. Herrmann, "Testing aspheric surfaces with CGH of different accuracy in industrial manufacturing environ," in Proc. SPIE 4440, 120-126 (2001).
[CrossRef]

Y. C. Chang and J. H. Burge, "Errors analysis for CGH optical testing," in Optical Manufacturing and Testing III, Proc. SPIE 3782, 358-366 (1999).
[CrossRef]

Other

G. J. Swanson, "Binary optics technology: theoretical limits on the diffraction efficiency of multilevel diffractive optical elements," technical report (Lincoln Library, Massachusetts Institute of Technology, 1991).

J. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, 1978).

M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, 1980).

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 (19)

Fig. 1
Fig. 1

Output complex wavefront immediately after the diffraction grating.

Fig. 2
Fig. 2

Diffraction efficiency of a phase grating as a function of the duty cycle and phase depth for the zero-order diffraction beam.

Fig. 3
Fig. 3

Diffraction efficiency of phase grating as a function of the duty cycle for the non-zero-order diffraction beams with 0.5λ phase depth.

Fig. 4
Fig. 4

Diffraction wavefront phase of phase grating as a function of the phase depth and duty cycle for the zero-order diffraction beam.

Fig. 5
Fig. 5

Wavefront phases versus phase depth for first- and second-order diffraction beams for the phase grating.

Fig. 6
Fig. 6

Wavefront phase versus duty cycle for nonzero diffraction orders at 0.5λ phase depth from phase grating.

Fig. 7
Fig. 7

Analytical result for the wavefront phase sensitivity of phase grating to phase depth variation for the zero-order beam.

Fig. 8
Fig. 8

Analytical result for the wavefront phase sensitivity of phase grating to the duty cycle for the zero-order beam.

Fig. 9
Fig. 9

Experimental setup for diffraction wavefronts measurements of phase grating.

Fig. 10
Fig. 10

Design layout of the sample phase grating.

Fig. 11
Fig. 11

Diffraction wavefront phase sensitivities to phase depth for the zero-order beam. Experimental data (vertical bar) versus theoretical results (solid line).

Fig. 12
Fig. 12

Diffraction wavefront phase deviations per 1% duty-cycle variation for the zero-order beam. Experimental data (vertical bar) versus theoretical results (solid curve).

Fig. 13
Fig. 13

Wavefront phase sensitivity functions per phase depth variations for the first-order diffraction wavefront. Experimental data (vertical bars) versus theoretical data (solid curve).

Fig. 14
Fig. 14

Wavefront phase sensitivity functions per 1% duty-cycle variations for the first-order diffraction wavefront of phase grating. Experimental data (vertical bars) versus theoretical data (solid curve).

Fig. 15
Fig. 15

Interferogram indicating diffraction efficiency distribution and wavefront phase as a function of the duty cycle at different diffraction orders. The top chart shows the corresponding theoretical values.

Fig. 16
Fig. 16

Graphical representation of the complex diffraction wavefront produced by a phase grating with 40% duty cycle for the zero-order beam.

Fig. 17
Fig. 17

Graphical representation of the complex diffraction fields at five different diffraction orders for a phase grating with a 40% duty cycle.

Fig. 18
Fig. 18

Graphical representation of the complex diffraction fields in zero diffraction order produced by a phase grating with a 50% duty cycle.

Fig. 19
Fig. 19

Graphical representation of the complex diffraction fields in zero diffraction order produced by a phase grating with 45% and 55% duty cycles.

Equations (122)

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

u ( x ) = A 0 + [ A 1 exp ( i ϕ ) A 0 ] rect ( x b ) 1 S   comb ( x S ) ,
A 0
A 1
A 0
A 1
U ( ξ ) = { u ( x ) } = A 0 δ ( ξ ) + [ A 1 exp ( i ϕ ) A 0 ] b   sinc ( b ξ )   comb ( S ξ ) = { A 0 δ ( ξ ) + [ A 1 cos ( ϕ ) A 0 ] D   sinc ( D S ξ ) m = δ ( ξ m S ) } + i { A 1 sin ( ϕ ) D   sinc ( D S ξ ) m = δ ( ξ m S ) } ;
= { { A 0 + [ A 1 cos ( ϕ ) A 0 ] D } + i { A 1 sin ( ϕ ) D } ,                     m = 0 { [ A 1 cos ( ϕ ) A 0 ] D   sinc ( m D ) } + i { A 1 sin ( ϕ ) D  sinc ( m D ) } , m = ± 1 , ± 2 , ,
ξ = m λ / S
D = b / S
U ( ξ )
1 / S
η = | U ( ξ ) | 2 | U 0 ( ξ ) | 2 .
| U 0 ( ξ ) | 2
( m = 0 )
η | m = 0 = A 0     2 ( 1 D ) 2 + A 1     2 D 2 + 2 A 0 A 1 D ( 1 D ) cos ( ϕ ) .
( m 0 )
η | m 0 = [ A 0     2 + A 1     2 - 2 A 0 A 1 cos ( ϕ ) ] D 2 sinc 2 ( m D ) .
U ( ξ )
tan ( Ψ ) = Im { U ( ξ ) } Re { U ( ξ ) } .
m = 0 : tan ( Ψ ) m = 0 = D A 1 sin ( ϕ ) A 0 ( 1 D ) + A 1 D cos ( ϕ ) ,
m = ± 1 , ± 2 , : tan ( Ψ ) m 0 = A 1 sin ( ϕ ) sinc ( m D ) [ A 0 + A 1 cos ( ϕ ) ] sinc ( m D ) .
W = Ψ 2 π .
sinc ( m D )
U ( ξ )
m = 0 : Ψ m = 0 D = 1 1 + [ tan ( Ψ ) m = 0 ] 2 tan ( Ψ ) m = 0 D = A 0 A 1 sin ϕ A 1     2 D 2 + A 0     2 ( 1 D ) 2 + 2 A 0 A 1 D ( 1 D ) cos ϕ ,
m = ± 1 , ± 2 , : Ψ m 0 D = 1 1 + [ tan ( Ψ ) m 0 ] 2 tan ( Ψ ) m 0 D = { ,       for   sinc ( m D ) = 0 , 0 , otherwise .
sinc ( m D )
sinc ( m D )
m = 0 : Ψ m = 0 ϕ = 1 1 + [ tan ( Ψ ) m = 0 ] 2 tan ( Ψ ) m = 0 ϕ = A 1     2 D 2 + A 0 A 1 D ( 1 D ) cos ϕ A 1     2 D 2 + A 0     2 ( 1 D ) 2 + 2 A 0 A 1 D ( 1 D ) cos ϕ ,
m = ± 1 , ± 2 , : Ψ m 0 ϕ = 1 1 + [ tan ( Ψ ) m 0 ] 2 tan ( Ψ ) m 0 ϕ = A 1     2 A 0 A 1 cos ϕ A 1     2 + A 0     2 2 A 0 A 1 cos ϕ .
Δ W D = 1 2 π Ψ D Δ D = 1 2 π Ψ D ( Δ D D ) D ,
Δ W ϕ = Ψ ϕ Δ ϕ = Ψ ϕ ( Δ ϕ ϕ ) ϕ ,
Δ D
Δ W D
Δ ϕ
Δ W ϕ
A 0 = A 1 = 1
λ / 2
50 %
( m 0 )
λ / 2
50 %
50 %
λ / 2
λ / 2
50 %
λ / 2
( m 0 )
λ / 2
λ / 2
50 %
λ / 2
33.3 %
66.6 %
sinc ( m D )
± π   rad
0 %
100 %
0.1 λ
0.075 λ
30 %
0.5 λ
+ 0.03 λ
30 %
0.1 λ
50 %
0.5 λ
1 %
1 %
0.004
30 %
λ / 20
( λ = 632.8   nm )
0.633   μm
5 × 11
2 %
44 %   to   60 %
0 %
100 %
0.22 λ
0.24 λ
0.25 λ
0.27 λ
0.29 λ
1 %
0 %
100 %
10 %
sinc ( m D ) = 0
λ / 2
40 %
( r )
( r )
( r )
( Ψ )
ϕ = 0.2 λ
40 %
η = 0.82 2 = 67 %
Ψ = 28 ° / 360 ° = 0.077
50 %
λ / 2
50 %
0.45 λ
0.55 λ
81 °
81 °
162 °
( r )
λ / 2
+ 90 °
9 0 °
0.5 λ
45 %
55 %
0 °
180 °
50 %
0.5 λ
0 °
180 °
50 %
50 %

Metrics