Abstract

Recently, the fabrication of computer-generated holograms by diamond face turning with a nanometer-stroke fast tool servo (nFTS) has been demonstrated. Existing methods for the design of diamond-turned holograms account for their spiral-shaped surface topology and the fact that only the phase of a wave field can be modulated. Here we present an algorithm enabling the additional consideration of two important fabrication-related properties: the shape of the diamond tool used and the limited control frequency of the nFTS. Our method is based on the generalized projections method and enables the design of holograms for the reconstruction of arbitrary intensity distributions in the far field. Experimental results are presented, demonstrating the advantages of the method.

© 2010 Optical Society of America

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References

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  1. C. Falldorf, C. Dankwart, R. Gläbe, B. Lünemann, C. v. Kopylow, and R. B. Bergmann, “Holographic projection based on diamond-turned diffractive optical elements,” Appl. Opt. 48, 5782–5785 (2009).
    [CrossRef] [PubMed]
  2. B. Lünemann, R. Gläbe, E. Brinksmeier, C. Dankwart, C. Falldorf, and C. von Kopylow, “Open-loop nanometerstroke fast tool servo system for the generation of diffractive optical microstructures,” in Proceedings of the 11th International Conference and Exhibition on New Actuators and Drive Systems (HVG Hanseatische Veranstaltungs-GmbH, 2008), pp. 95–98.
  3. L. Li, A. Y. Yi, C. Huang, D. A. Grewell, A. Benatar, and Y. Chen, “Fabrication of diffractive optics by use of slow tool servo diamond turning process,” Opt. Eng. 45, 113401 (2006).
    [CrossRef]
  4. D. Michaelis, S. Kudaev, R. Steinkopf, A. Gebhardt, P. Schreiber, and A. Bräuer, “Incoherent beam shaping with freeform mirror,” Proc. SPIE 7059, 705905 (2008).
    [CrossRef]
  5. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Jena) 35, 237–246 (1972).
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    [CrossRef]
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    [CrossRef]
  8. D. C. Youla and H. Webb, “Image restoration by the method of projections onto convex sets. part i,” IEEE Trans. Med. Imaging 1, 81–91 (1982).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  13. A. Levi and H. Stark, “Image restoration by the method of generalized projections with application to restoration from magnitude,” J. Opt. Soc. Am. A 1, 932–943 (1984).
    [CrossRef]
  14. O. K. Ersoy, Diffraction, Fourier Optics and Imaging(Wiley, 2007).
    [CrossRef]
  15. M. Hayes, “The reconstruction of a multidimensional sequence from the phase of its Fourier transform,” IEEE Trans. Acoust. Speech Signal Process. 30, 140–154 (1982).
    [CrossRef]
  16. E. T. Whittaker, “On the functions which are represented by the expansions of the interpolation theory,” Proc. R. Soc. Edinburgh 35, 181–194 (1915).

2009

2008

D. Michaelis, S. Kudaev, R. Steinkopf, A. Gebhardt, P. Schreiber, and A. Bräuer, “Incoherent beam shaping with freeform mirror,” Proc. SPIE 7059, 705905 (2008).
[CrossRef]

2006

L. Li, A. Y. Yi, C. Huang, D. A. Grewell, A. Benatar, and Y. Chen, “Fabrication of diffractive optics by use of slow tool servo diamond turning process,” Opt. Eng. 45, 113401 (2006).
[CrossRef]

2000

P. Birch, R. Young, M. Farsari, C. Chatwin, and D. Budgett, “A comparison of the iterative Fourier transform method and evolutionary algorithms for the design of diffractive optical elements,” Opt. Lasers Eng. 33, 439–448(2000).
[CrossRef]

1990

1987

1984

1982

M. Hayes, “The reconstruction of a multidimensional sequence from the phase of its Fourier transform,” IEEE Trans. Acoust. Speech Signal Process. 30, 140–154 (1982).
[CrossRef]

D. C. Youla and H. Webb, “Image restoration by the method of projections onto convex sets. part i,” IEEE Trans. Med. Imaging 1, 81–91 (1982).
[CrossRef] [PubMed]

M. I. Sezan and H. Stark, “Image restoration by the method of projections onto convex sets. part ii,” IEEE Trans. Med. Imaging 1, 95–101 (1982).
[CrossRef] [PubMed]

1975

A. Papoulis, “A new algorithm in spectral analysis and bandlimited extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[CrossRef]

1974

R. Gerchberg, “Super resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

1972

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Jena) 35, 237–246 (1972).

1915

E. T. Whittaker, “On the functions which are represented by the expansions of the interpolation theory,” Proc. R. Soc. Edinburgh 35, 181–194 (1915).

Allebach, J. P.

Benatar, A.

L. Li, A. Y. Yi, C. Huang, D. A. Grewell, A. Benatar, and Y. Chen, “Fabrication of diffractive optics by use of slow tool servo diamond turning process,” Opt. Eng. 45, 113401 (2006).
[CrossRef]

Bergmann, R. B.

Birch, P.

P. Birch, R. Young, M. Farsari, C. Chatwin, and D. Budgett, “A comparison of the iterative Fourier transform method and evolutionary algorithms for the design of diffractive optical elements,” Opt. Lasers Eng. 33, 439–448(2000).
[CrossRef]

Bräuer, A.

D. Michaelis, S. Kudaev, R. Steinkopf, A. Gebhardt, P. Schreiber, and A. Bräuer, “Incoherent beam shaping with freeform mirror,” Proc. SPIE 7059, 705905 (2008).
[CrossRef]

Brinksmeier, E.

B. Lünemann, R. Gläbe, E. Brinksmeier, C. Dankwart, C. Falldorf, and C. von Kopylow, “Open-loop nanometerstroke fast tool servo system for the generation of diffractive optical microstructures,” in Proceedings of the 11th International Conference and Exhibition on New Actuators and Drive Systems (HVG Hanseatische Veranstaltungs-GmbH, 2008), pp. 95–98.

Budgett, D.

P. Birch, R. Young, M. Farsari, C. Chatwin, and D. Budgett, “A comparison of the iterative Fourier transform method and evolutionary algorithms for the design of diffractive optical elements,” Opt. Lasers Eng. 33, 439–448(2000).
[CrossRef]

Chatwin, C.

P. Birch, R. Young, M. Farsari, C. Chatwin, and D. Budgett, “A comparison of the iterative Fourier transform method and evolutionary algorithms for the design of diffractive optical elements,” Opt. Lasers Eng. 33, 439–448(2000).
[CrossRef]

Chen, Y.

L. Li, A. Y. Yi, C. Huang, D. A. Grewell, A. Benatar, and Y. Chen, “Fabrication of diffractive optics by use of slow tool servo diamond turning process,” Opt. Eng. 45, 113401 (2006).
[CrossRef]

Dankwart, C.

C. Falldorf, C. Dankwart, R. Gläbe, B. Lünemann, C. v. Kopylow, and R. B. Bergmann, “Holographic projection based on diamond-turned diffractive optical elements,” Appl. Opt. 48, 5782–5785 (2009).
[CrossRef] [PubMed]

B. Lünemann, R. Gläbe, E. Brinksmeier, C. Dankwart, C. Falldorf, and C. von Kopylow, “Open-loop nanometerstroke fast tool servo system for the generation of diffractive optical microstructures,” in Proceedings of the 11th International Conference and Exhibition on New Actuators and Drive Systems (HVG Hanseatische Veranstaltungs-GmbH, 2008), pp. 95–98.

Ersoy, O. K.

O. K. Ersoy, Diffraction, Fourier Optics and Imaging(Wiley, 2007).
[CrossRef]

Falldorf, C.

C. Falldorf, C. Dankwart, R. Gläbe, B. Lünemann, C. v. Kopylow, and R. B. Bergmann, “Holographic projection based on diamond-turned diffractive optical elements,” Appl. Opt. 48, 5782–5785 (2009).
[CrossRef] [PubMed]

B. Lünemann, R. Gläbe, E. Brinksmeier, C. Dankwart, C. Falldorf, and C. von Kopylow, “Open-loop nanometerstroke fast tool servo system for the generation of diffractive optical microstructures,” in Proceedings of the 11th International Conference and Exhibition on New Actuators and Drive Systems (HVG Hanseatische Veranstaltungs-GmbH, 2008), pp. 95–98.

Farsari, M.

P. Birch, R. Young, M. Farsari, C. Chatwin, and D. Budgett, “A comparison of the iterative Fourier transform method and evolutionary algorithms for the design of diffractive optical elements,” Opt. Lasers Eng. 33, 439–448(2000).
[CrossRef]

Gebhardt, A.

D. Michaelis, S. Kudaev, R. Steinkopf, A. Gebhardt, P. Schreiber, and A. Bräuer, “Incoherent beam shaping with freeform mirror,” Proc. SPIE 7059, 705905 (2008).
[CrossRef]

Gerchberg, R.

R. Gerchberg, “Super resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Jena) 35, 237–246 (1972).

Gläbe, R.

C. Falldorf, C. Dankwart, R. Gläbe, B. Lünemann, C. v. Kopylow, and R. B. Bergmann, “Holographic projection based on diamond-turned diffractive optical elements,” Appl. Opt. 48, 5782–5785 (2009).
[CrossRef] [PubMed]

B. Lünemann, R. Gläbe, E. Brinksmeier, C. Dankwart, C. Falldorf, and C. von Kopylow, “Open-loop nanometerstroke fast tool servo system for the generation of diffractive optical microstructures,” in Proceedings of the 11th International Conference and Exhibition on New Actuators and Drive Systems (HVG Hanseatische Veranstaltungs-GmbH, 2008), pp. 95–98.

Grewell, D. A.

L. Li, A. Y. Yi, C. Huang, D. A. Grewell, A. Benatar, and Y. Chen, “Fabrication of diffractive optics by use of slow tool servo diamond turning process,” Opt. Eng. 45, 113401 (2006).
[CrossRef]

Guest, C.

Hayes, M.

M. Hayes, “The reconstruction of a multidimensional sequence from the phase of its Fourier transform,” IEEE Trans. Acoust. Speech Signal Process. 30, 140–154 (1982).
[CrossRef]

Huang, C.

L. Li, A. Y. Yi, C. Huang, D. A. Grewell, A. Benatar, and Y. Chen, “Fabrication of diffractive optics by use of slow tool servo diamond turning process,” Opt. Eng. 45, 113401 (2006).
[CrossRef]

Kim, M.

Kopylow, C. v.

Kudaev, S.

D. Michaelis, S. Kudaev, R. Steinkopf, A. Gebhardt, P. Schreiber, and A. Bräuer, “Incoherent beam shaping with freeform mirror,” Proc. SPIE 7059, 705905 (2008).
[CrossRef]

Levi, A.

Li, L.

L. Li, A. Y. Yi, C. Huang, D. A. Grewell, A. Benatar, and Y. Chen, “Fabrication of diffractive optics by use of slow tool servo diamond turning process,” Opt. Eng. 45, 113401 (2006).
[CrossRef]

Lünemann, B.

C. Falldorf, C. Dankwart, R. Gläbe, B. Lünemann, C. v. Kopylow, and R. B. Bergmann, “Holographic projection based on diamond-turned diffractive optical elements,” Appl. Opt. 48, 5782–5785 (2009).
[CrossRef] [PubMed]

B. Lünemann, R. Gläbe, E. Brinksmeier, C. Dankwart, C. Falldorf, and C. von Kopylow, “Open-loop nanometerstroke fast tool servo system for the generation of diffractive optical microstructures,” in Proceedings of the 11th International Conference and Exhibition on New Actuators and Drive Systems (HVG Hanseatische Veranstaltungs-GmbH, 2008), pp. 95–98.

Michaelis, D.

D. Michaelis, S. Kudaev, R. Steinkopf, A. Gebhardt, P. Schreiber, and A. Bräuer, “Incoherent beam shaping with freeform mirror,” Proc. SPIE 7059, 705905 (2008).
[CrossRef]

Papoulis, A.

A. Papoulis, “A new algorithm in spectral analysis and bandlimited extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[CrossRef]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Jena) 35, 237–246 (1972).

Schreiber, P.

D. Michaelis, S. Kudaev, R. Steinkopf, A. Gebhardt, P. Schreiber, and A. Bräuer, “Incoherent beam shaping with freeform mirror,” Proc. SPIE 7059, 705905 (2008).
[CrossRef]

Seldowitz, M. A.

Sezan, M. I.

M. I. Sezan and H. Stark, “Image restoration by the method of projections onto convex sets. part ii,” IEEE Trans. Med. Imaging 1, 95–101 (1982).
[CrossRef] [PubMed]

Stark, H.

A. Levi and H. Stark, “Image restoration by the method of generalized projections with application to restoration from magnitude,” J. Opt. Soc. Am. A 1, 932–943 (1984).
[CrossRef]

M. I. Sezan and H. Stark, “Image restoration by the method of projections onto convex sets. part ii,” IEEE Trans. Med. Imaging 1, 95–101 (1982).
[CrossRef] [PubMed]

Steinkopf, R.

D. Michaelis, S. Kudaev, R. Steinkopf, A. Gebhardt, P. Schreiber, and A. Bräuer, “Incoherent beam shaping with freeform mirror,” Proc. SPIE 7059, 705905 (2008).
[CrossRef]

Sweeney, D. W.

von Kopylow, C.

B. Lünemann, R. Gläbe, E. Brinksmeier, C. Dankwart, C. Falldorf, and C. von Kopylow, “Open-loop nanometerstroke fast tool servo system for the generation of diffractive optical microstructures,” in Proceedings of the 11th International Conference and Exhibition on New Actuators and Drive Systems (HVG Hanseatische Veranstaltungs-GmbH, 2008), pp. 95–98.

Webb, H.

D. C. Youla and H. Webb, “Image restoration by the method of projections onto convex sets. part i,” IEEE Trans. Med. Imaging 1, 81–91 (1982).
[CrossRef] [PubMed]

Whittaker, E. T.

E. T. Whittaker, “On the functions which are represented by the expansions of the interpolation theory,” Proc. R. Soc. Edinburgh 35, 181–194 (1915).

Yi, A. Y.

L. Li, A. Y. Yi, C. Huang, D. A. Grewell, A. Benatar, and Y. Chen, “Fabrication of diffractive optics by use of slow tool servo diamond turning process,” Opt. Eng. 45, 113401 (2006).
[CrossRef]

Youla, D. C.

D. C. Youla and H. Webb, “Image restoration by the method of projections onto convex sets. part i,” IEEE Trans. Med. Imaging 1, 81–91 (1982).
[CrossRef] [PubMed]

Young, R.

P. Birch, R. Young, M. Farsari, C. Chatwin, and D. Budgett, “A comparison of the iterative Fourier transform method and evolutionary algorithms for the design of diffractive optical elements,” Opt. Lasers Eng. 33, 439–448(2000).
[CrossRef]

Appl. Opt.

IEEE Trans. Acoust. Speech Signal Process.

M. Hayes, “The reconstruction of a multidimensional sequence from the phase of its Fourier transform,” IEEE Trans. Acoust. Speech Signal Process. 30, 140–154 (1982).
[CrossRef]

IEEE Trans. Circuits Syst.

A. Papoulis, “A new algorithm in spectral analysis and bandlimited extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[CrossRef]

IEEE Trans. Med. Imaging

D. C. Youla and H. Webb, “Image restoration by the method of projections onto convex sets. part i,” IEEE Trans. Med. Imaging 1, 81–91 (1982).
[CrossRef] [PubMed]

M. I. Sezan and H. Stark, “Image restoration by the method of projections onto convex sets. part ii,” IEEE Trans. Med. Imaging 1, 95–101 (1982).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A

Opt. Acta

R. Gerchberg, “Super resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

Opt. Eng.

L. Li, A. Y. Yi, C. Huang, D. A. Grewell, A. Benatar, and Y. Chen, “Fabrication of diffractive optics by use of slow tool servo diamond turning process,” Opt. Eng. 45, 113401 (2006).
[CrossRef]

Opt. Lasers Eng.

P. Birch, R. Young, M. Farsari, C. Chatwin, and D. Budgett, “A comparison of the iterative Fourier transform method and evolutionary algorithms for the design of diffractive optical elements,” Opt. Lasers Eng. 33, 439–448(2000).
[CrossRef]

Optik (Jena)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Jena) 35, 237–246 (1972).

Proc. R. Soc. Edinburgh

E. T. Whittaker, “On the functions which are represented by the expansions of the interpolation theory,” Proc. R. Soc. Edinburgh 35, 181–194 (1915).

Proc. SPIE

D. Michaelis, S. Kudaev, R. Steinkopf, A. Gebhardt, P. Schreiber, and A. Bräuer, “Incoherent beam shaping with freeform mirror,” Proc. SPIE 7059, 705905 (2008).
[CrossRef]

Other

B. Lünemann, R. Gläbe, E. Brinksmeier, C. Dankwart, C. Falldorf, and C. von Kopylow, “Open-loop nanometerstroke fast tool servo system for the generation of diffractive optical microstructures,” in Proceedings of the 11th International Conference and Exhibition on New Actuators and Drive Systems (HVG Hanseatische Veranstaltungs-GmbH, 2008), pp. 95–98.

O. K. Ersoy, Diffraction, Fourier Optics and Imaging(Wiley, 2007).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Fabrication process and resulting surface geometry. (a) Scheme for the machining process for DTHs. (b) Schematic structure in the feed direction and imprinting of the diamond tool shape. (c) Section of the surface of a hologram. (d) Detailed view of the same part of the surface. The highlighted area A n corresponds to one single cutting depth value d n in the control signal of the nFTS. The maximum height difference displayed equals 700 nm .

Fig. 2
Fig. 2

Structure of the proposed algorithm. The gray squares indicate which steps in the algorithm can be assigned to which projection operator.

Fig. 3
Fig. 3

Comparison of far-field intensities. (a) Simulation of the intensity generated by a hologram designed with a combination of the IFTA with a sinc-interpolation, as in Ref. [1]. (b) Corresponding measurement. (c) Magnification of the measured region marked by the white square. (d) Simulation of the intensity generated by a hologram designed with the method presented in this paper. (e) Corresponding measurement. (f) Magnification of the measured region marked by the white square. Significant improvements can be observed regarding homogeneity and background noise.

Equations (20)

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τ d ( r , ϕ ) = τ c ( r , ϕ ) n = 1 N δ ( r r n , ϕ ϕ n ) = n = 1 N exp ( i 2 k d n ) δ ( r r n , ϕ ϕ n ) ,
τ s ( r , ϕ ) = exp [ i 2 k s ( r ) ] rect ( r Δ r ) rect ( ϕ Δ ϕ ) .
τ ( r , ϕ ) = S { τ d } = 0 0 2 π τ d ( r , ϕ ) τ s ( r r , ϕ ϕ ) d ϕ d r .
E ( ρ , Φ ) = C ( ρ ) F { τ } ( ρ λ z , Φ ) ,
C ( ρ ) = E 0 1 i λ z exp [ i k ( z + ρ 2 2 z ) ] ,
I ( ρ , Φ ) = I 0 λ 2 z 2 | F { S { τ d } } | 2 ( ρ λ z , Φ ) ,
P q f C q ,
P q f f = min y f y C q ,
f m + 1 = T 1 T 2 T Q f m ,
T q = λ q 1 + λ q P q with     0 λ q a i .
P 1 E 1 = F 1 { E ^ 1 , P ( ρ , Φ ) } ,
E ^ 1 , P ( ρ , Φ ) = { E ^ 1 ( ρ , Φ ) ( ρ , Φ ) a arg [ E ^ 1 ( ρ , Φ ) ] I opt ( ρ , Φ ) ( ρ , Φ ) a ,
P 2 E 1 = S τ d ,
d n = 1 2 k arg ( E 1 , s , n ) .
E 1 , s , n = A n E 1 ( r , ϕ ) exp ( i 2 k s ( r r n ) ) d r d ϕ .
P 2 E 1 E 1 = E 0 τ E 1 .
P 2 E 1 E 1 2 A | E 0 | 2 + | E 1 | 2 2 E 0 τ E 1 d r d ϕ ,
P 2 E 1 E 1 2 = C 1 2 n = 1 N A n E 0 exp [ i 2 k d n + i 2 k s ( r r n ) ] · E 1 d r d ϕ .
P 2 E 1 E 1 2 = C 1 2 n = 1 N A n E 0 exp ( i 2 k d n ) · E 1 exp [ i 2 k s ( r r n ) ] d r d ϕ .
P 2 E 1 E 1 2 = C 1 2 n = 1 N E 0 exp ( i 2 k d n ) · E 1 , s , n .

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