Abstract

The fabrication of computer-generated holograms (CGH) by e-beam or laser-writing machine specifically requires using polygon segments to approximate the continuously smooth fringe pattern of an ideal CGH. Wavefront phase errors introduced in this process depend on the size of the polygon segments and the shape of the fringes. In this paper, we propose a method for estimating the wavefront error and its spatial frequency, allowing optimization of the polygon sizes for required measurement accuracy. This method is validated with computer simulation and direct measurements from an interferometer.

© 2013 Optical Society of America

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References

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  1. D. Malacara, Optical Shop Testing, 3rd ed. (Wiley, 2007).
  2. P. Zhou and J. H. Burge, “Fabrication error analysis and experimental demonstration for computer-generated holograms,” Appl. Opt. 46, 657–663 (2007).
    [CrossRef]
  3. W. H. Lee, “Binary synthetic holograms,” Appl. Opt. 13, 1677–1682 (1974).
    [CrossRef]
  4. K. Creath and J. C. Wyant, Use of Computer-Generated Holograms in Optical Testing, Handbook of Optics, 2nd ed. (Optical Society America, 1995), Vol. 2, Chap. 31.
  5. S. M. Arnold, “Electron beam fabrication of computer-generated holograms,” Opt. Eng. 24, 245803 (1985).
    [CrossRef]
  6. J. H. Burge, “Applications of computer-generated holograms for interferometric measurement of large aspheric optics,” Proc. SPIE 2576, 258–269 (1995).
    [CrossRef]
  7. J. Fan, D. Zaleta, K. S. Urquhart, and S. H. Lee, “Efficient encoding algorithms for computer-aided design of diffractive optical elements by the use of electron-beam fabrication,” Appl. Opt. 34, 2522–2533 (1995).
    [CrossRef]
  8. I. Kallioniemi, J. Saarinen, K. Blomstedt, and J. Turunen, “Polygon approximation of the fringes of diffractive elements,” Appl. Opt. 36, 7217–7223 (1997).
    [CrossRef]
  9. C. Zhao and J. H. Burge, “Estimate of wavefront error introduced by encoding of computer generated holograms,” in Conference on Lasers and Electro-Optics/Pacific Rim (CLEOPR), Shanghai, China (2009), paper ThJ2_3.
  10. A. F. Fercher, “Computer-generated holograms for testing optical elements: error analysis and error compensation,” Opt. Acta 23, 347–365 (1976).
    [CrossRef]
  11. S. D. Peckham, “Profile, plan and streamline curvature: a simple derivation and applications,” 2011, http://www.geomorphometry.org .
  12. E. Sidick, “Power spectral density specification and analysis of large optical surfaces,” Proc. SPIE 7390, 73900L (2009).
    [CrossRef]
  13. P. Zhou and J. H. Burge, “Analysis of wavefront propagation using the Talbot effect,” Appl. Opt. 49, 5351–5359 (2010).
    [CrossRef]

2010

2009

E. Sidick, “Power spectral density specification and analysis of large optical surfaces,” Proc. SPIE 7390, 73900L (2009).
[CrossRef]

2007

1997

1995

1985

S. M. Arnold, “Electron beam fabrication of computer-generated holograms,” Opt. Eng. 24, 245803 (1985).
[CrossRef]

1976

A. F. Fercher, “Computer-generated holograms for testing optical elements: error analysis and error compensation,” Opt. Acta 23, 347–365 (1976).
[CrossRef]

1974

Arnold, S. M.

S. M. Arnold, “Electron beam fabrication of computer-generated holograms,” Opt. Eng. 24, 245803 (1985).
[CrossRef]

Blomstedt, K.

Burge, J. H.

P. Zhou and J. H. Burge, “Analysis of wavefront propagation using the Talbot effect,” Appl. Opt. 49, 5351–5359 (2010).
[CrossRef]

P. Zhou and J. H. Burge, “Fabrication error analysis and experimental demonstration for computer-generated holograms,” Appl. Opt. 46, 657–663 (2007).
[CrossRef]

J. H. Burge, “Applications of computer-generated holograms for interferometric measurement of large aspheric optics,” Proc. SPIE 2576, 258–269 (1995).
[CrossRef]

C. Zhao and J. H. Burge, “Estimate of wavefront error introduced by encoding of computer generated holograms,” in Conference on Lasers and Electro-Optics/Pacific Rim (CLEOPR), Shanghai, China (2009), paper ThJ2_3.

Creath, K.

K. Creath and J. C. Wyant, Use of Computer-Generated Holograms in Optical Testing, Handbook of Optics, 2nd ed. (Optical Society America, 1995), Vol. 2, Chap. 31.

Fan, J.

Fercher, A. F.

A. F. Fercher, “Computer-generated holograms for testing optical elements: error analysis and error compensation,” Opt. Acta 23, 347–365 (1976).
[CrossRef]

Kallioniemi, I.

Lee, S. H.

Lee, W. H.

Malacara, D.

D. Malacara, Optical Shop Testing, 3rd ed. (Wiley, 2007).

Saarinen, J.

Sidick, E.

E. Sidick, “Power spectral density specification and analysis of large optical surfaces,” Proc. SPIE 7390, 73900L (2009).
[CrossRef]

Turunen, J.

Urquhart, K. S.

Wyant, J. C.

K. Creath and J. C. Wyant, Use of Computer-Generated Holograms in Optical Testing, Handbook of Optics, 2nd ed. (Optical Society America, 1995), Vol. 2, Chap. 31.

Zaleta, D.

Zhao, C.

C. Zhao and J. H. Burge, “Estimate of wavefront error introduced by encoding of computer generated holograms,” in Conference on Lasers and Electro-Optics/Pacific Rim (CLEOPR), Shanghai, China (2009), paper ThJ2_3.

Zhou, P.

Appl. Opt.

Opt. Acta

A. F. Fercher, “Computer-generated holograms for testing optical elements: error analysis and error compensation,” Opt. Acta 23, 347–365 (1976).
[CrossRef]

Opt. Eng.

S. M. Arnold, “Electron beam fabrication of computer-generated holograms,” Opt. Eng. 24, 245803 (1985).
[CrossRef]

Proc. SPIE

J. H. Burge, “Applications of computer-generated holograms for interferometric measurement of large aspheric optics,” Proc. SPIE 2576, 258–269 (1995).
[CrossRef]

E. Sidick, “Power spectral density specification and analysis of large optical surfaces,” Proc. SPIE 7390, 73900L (2009).
[CrossRef]

Other

D. Malacara, Optical Shop Testing, 3rd ed. (Wiley, 2007).

K. Creath and J. C. Wyant, Use of Computer-Generated Holograms in Optical Testing, Handbook of Optics, 2nd ed. (Optical Society America, 1995), Vol. 2, Chap. 31.

S. D. Peckham, “Profile, plan and streamline curvature: a simple derivation and applications,” 2011, http://www.geomorphometry.org .

C. Zhao and J. H. Burge, “Estimate of wavefront error introduced by encoding of computer generated holograms,” in Conference on Lasers and Electro-Optics/Pacific Rim (CLEOPR), Shanghai, China (2009), paper ThJ2_3.

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Figures (10)

Fig. 1.
Fig. 1.

Process of segmenting a smooth fringe to polygons.

Fig. 2.
Fig. 2.

Segmenting the same fringe central line with different error boundaries ( σ 1 = 2 σ 2 , ε 1 = 2 ε 2 , L 1 = 2 L 2 . P-V phase error and segment length decreased as the boundary decreased).

Fig. 3.
Fig. 3.

Gradient, aspect, and tangential vector of a given phase function at f ( 0 , 0 ) .

Fig. 4.
Fig. 4.

Encoding-induced phase error map of a FZP and its zoomed-in characters. (color bar unit: nanometer)

Fig. 5.
Fig. 5.

1D PSD of a local area on a phase map when individual fringe is resolved. It has a shape close to a triangle.

Fig. 6.
Fig. 6.

PSD parametric model to calculate the actual RMS phase error using the ratio of the shaded area to the whole area of the triangle. ϒ is the error frequency from Eq. (11). S / λ is the system cut-off frequency. S / λ is the spatial frequency related to the fringe spacing, where S is the local slope magnitude of the wavefront.

Fig. 7.
Fig. 7.

(a) Fringe map (FZP). (b) Encoding-induced RMS phase error map (FZP).

Fig. 8.
Fig. 8.

(a) Fringe map (spherical aberration with tilt). (b) Encoding induced RMS phase error map (spherical aberration with tilt).

Fig. 9.
Fig. 9.

Interferometric measurement for the phase error due to encoding the CGH pattern of a FZP.

Fig. 10.
Fig. 10.

Comparison results from measurements and parametric model for the local RMS phase error.

Tables (3)

Tables Icon

Table 1. Comparison of the Measurement Results and Simulation Results at Different Levels of Encoding Residual

Tables Icon

Table 2. Comparison of Peak Frequencies among Analytical, Simulated, and Measured Values

Tables Icon

Table 3. Comparison of RMS Phase Errors between Simulated and Measured Values

Equations (18)

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Δ ϕ ( x , y ) = m λ σ ( x , y ) d ( x , y ) ,
ε = σ d .
L = 4 R · σ = 4 ε d / c ,
ϒ = 1 L = 1 4 c ε d .
f = ( f x , f y ) ,
S ( x , y ) = | f | = f x 2 + f y 2 .
θ ( x , y ) = arctan ( f y f x ) .
g = ( f y , f x ) ,
κ c ( x , y ) = θ ( x , y ) · ( g ( x , y ) / S ( x , y ) ) ,
κ c = ( f y 2 f x x 2 f x f y f x y + f x 2 f y y ) / S 3 .
ϒ ( x , y ) = 1 4 c ε d = 1 4 | κ c | ε λ / S = 1 4 S | f y 2 f x x 2 f x f y f x y + f x 2 f y y | / ε λ .
f FZP ( x , y ) = x 2 + y 2 2 F ,
ϒ FZP = 1 4 ε λ F .
PSD ( u , v ) = 1 A | F { h ( x , y ) } | 2 ,
PSD 1 D ( ρ ) = 0 2 π PSD ( ρ , θ ) d θ ,
σ 2 = PSD ( u , v ) d u d v .
σ RMS ( x , y ) = ε λ 2 1 [ S ( x , y ) / λ f cutoff S ( x , y ) / λ γ ( x , y ) ] 2 ,
σ RMS = 1 M N x = 1 M y = 1 N σ RMS ( x , y ) .

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