Abstract

The resolution dependence of spatial-spectral volume holographic imaging systems on angular and spectral bandwidth of nonuniform gratings is investigated. Modeling techniques include a combination of the approximate coupled-wave analysis and the transfer-matrix method for holograms recorded in absorptive media. The effective thickness of the holograms is used as an estimator of the resolution of the imaging systems. The methodology, which assists in the design and optimization of volume holographic simulation results based on our approach, are confirmed with experiments and show proof of consistency and usefulness of the proposed models.

© 2011 Optical Society of America

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References

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  1. J. Fujimoto and D. Farkas, Biomedical Optical Imaging(Oxford University, 2009).
  2. A. Sinha, W. Sun, T. Shih, and G. Barbastathis, “Volume holographic imaging in transmission geometry,” Appl. Opt. 43, 1533–1551 (2004).
    [CrossRef] [PubMed]
  3. Z. Li, D. Psaltis, W. Liu, W. R. Johnson, and G. Bearman, “Volume holographic spectral imaging,” Proc. SPIE 5694, 33–40 (2005).
    [CrossRef]
  4. A. Sinha and G. Barbastathis, “Broadband volume holographic imaging,” Appl. Opt. 43, 5214–5221 (2004).
    [CrossRef] [PubMed]
  5. Y. Luo, P. J. Gelsinger, G. Barbastathis, J. K. Barton, and R. K. Kostuk, “Optimization of multiplexed holographic gratings in PQ-PMMA for spectral-spatial filters,” Opt. Lett. 33, 566–568 (2008).
    [CrossRef] [PubMed]
  6. P. J. Gelsinger-Austin, Y. Luo, J. M. Watson, R. K. Kostuk, G. Barbastathis, J. K. Barton, and J. M. Castro, “Optical design for a spatial-spectral volume holographic imaging system,” Opt. Eng. 49, 043001 (2010).
    [CrossRef]
  7. Y. Lou, J. M. Castro, J. Barton, R. K. Kostuk, and G. Barbastathis, “Simulation and experiments of non-uniform multiplexed gratings in volume holographic imaging systems,” Opt. Express 18, 19273–19285 (2010).
    [CrossRef]
  8. A. Sato and R. K. Kostuk, “Holographic grating for dense wavelength division optical filters at 1550 nm using phenanthrenequinone doped poly(methylmethacrylate),” Proc. SPIE 5216, 44–52 (2003).
    [CrossRef]
  9. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
  10. L. Solymar and D. J. Cooke, Volume Holography and Volume Gratings (Academic, 1981).
  11. J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  12. D. Kermish, “Nonuniform sinusoidally modulated dielectric gratings,” J. Opt. Soc. Am. 59, 1409–1414 (1969).
    [CrossRef]
  13. R. Kowarschik, “Diffraction efficiency of attenuated sinusoidally modulated gratings in volume holograms,” J. Mod. Opt. 23, 1039–1051 (1976).
    [CrossRef]
  14. T. Kubota, “The diffraction efficiency of holograms gratings recorded in an absorptive medium,” Opt. Commun. 16, 347–349 (1976).
    [CrossRef]
  15. S. Gallego, M. Ortuño, C. Neipp, A. Marquez, A. Belendez, I. Pascual, J. V. Kelly, and J. T. Sheridan, “Physical and effective optical thickness of holographic diffraction gratings recorded in photopolymers,” Opt. Express 13, 1939–1947 (2005).
    [CrossRef] [PubMed]
  16. C. Neipp, J. T. Sheridan, S. Gallego, M. Ortuño, A. Marquez, I. Pascual, and A. Belenez, “Effect of depth attenuated refractive index profile in the angular responses of the efficiency of higher order in volume gratings recorded in a PVA/acrylamide photopolymer,” Opt. Commun. 233, 311–322 (2004).
    [CrossRef]
  17. V. V. Shelkovnikov, E. F. Pen, and V. I. Kovalevsky, “Optimum optical density of the absorbing holographic materials,” Opt. Mem. Neural Netw. 16, 75–83 (2007).
    [CrossRef]
  18. W. Liu, D. Psaltis, and G. Barbastathis, “Real-time spectral imaging in three spatial dimensions,” Opt. Lett. 27, 854–856 (2002).
    [CrossRef]
  19. S. B. Oh, J. M. Watson, and G. Barbastathis, “Theoretical analysis of curved Bragg diffraction images from plane reference volume holograms,” Appl. Opt. 48, 5984–5996(2009).
    [CrossRef] [PubMed]
  20. J. M. Castro, E. de Leon, J. K. Barton, and R. K. Kostuk, “Analysis of diffracted image patterns from volume holographic imaging systems and applications to imaging processing,” Appl. Opt. 50, 170–176(2011).
    [CrossRef] [PubMed]
  21. U. V. Mahilny, D. N. Marmysh, A. I. Stankevich, A. L. Tolstik, V. Matusevich, and R. Kowarschik, “Holographic volume absorption grating in glass-like polymer recording material,” Appl. Phys. B 82, 299–302 (2005).
    [CrossRef]
  22. R. K. Kostuk, W. Maeda, Ch. H.Chen, I. Djordjevic, and B. Vasic, “Cascaded holographic polymer reflection grating filters for optical–code-division multiple-access applications,” Appl. Opt. 44, 7581–7586 (2005).
    [CrossRef] [PubMed]
  23. M. G. Moharam and T. K. Gaylord, “Three-dimensional vector coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 73, 1105–1112 (1983).
    [CrossRef]
  24. M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
    [CrossRef]

2011 (1)

2010 (2)

P. J. Gelsinger-Austin, Y. Luo, J. M. Watson, R. K. Kostuk, G. Barbastathis, J. K. Barton, and J. M. Castro, “Optical design for a spatial-spectral volume holographic imaging system,” Opt. Eng. 49, 043001 (2010).
[CrossRef]

Y. Lou, J. M. Castro, J. Barton, R. K. Kostuk, and G. Barbastathis, “Simulation and experiments of non-uniform multiplexed gratings in volume holographic imaging systems,” Opt. Express 18, 19273–19285 (2010).
[CrossRef]

2009 (1)

2008 (1)

2007 (1)

V. V. Shelkovnikov, E. F. Pen, and V. I. Kovalevsky, “Optimum optical density of the absorbing holographic materials,” Opt. Mem. Neural Netw. 16, 75–83 (2007).
[CrossRef]

2005 (4)

2004 (3)

C. Neipp, J. T. Sheridan, S. Gallego, M. Ortuño, A. Marquez, I. Pascual, and A. Belenez, “Effect of depth attenuated refractive index profile in the angular responses of the efficiency of higher order in volume gratings recorded in a PVA/acrylamide photopolymer,” Opt. Commun. 233, 311–322 (2004).
[CrossRef]

A. Sinha and G. Barbastathis, “Broadband volume holographic imaging,” Appl. Opt. 43, 5214–5221 (2004).
[CrossRef] [PubMed]

A. Sinha, W. Sun, T. Shih, and G. Barbastathis, “Volume holographic imaging in transmission geometry,” Appl. Opt. 43, 1533–1551 (2004).
[CrossRef] [PubMed]

2003 (1)

A. Sato and R. K. Kostuk, “Holographic grating for dense wavelength division optical filters at 1550 nm using phenanthrenequinone doped poly(methylmethacrylate),” Proc. SPIE 5216, 44–52 (2003).
[CrossRef]

2002 (1)

1983 (1)

1981 (1)

1976 (2)

R. Kowarschik, “Diffraction efficiency of attenuated sinusoidally modulated gratings in volume holograms,” J. Mod. Opt. 23, 1039–1051 (1976).
[CrossRef]

T. Kubota, “The diffraction efficiency of holograms gratings recorded in an absorptive medium,” Opt. Commun. 16, 347–349 (1976).
[CrossRef]

1969 (2)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

D. Kermish, “Nonuniform sinusoidally modulated dielectric gratings,” J. Opt. Soc. Am. 59, 1409–1414 (1969).
[CrossRef]

Barbastathis, G.

Barton, J.

Barton, J. K.

Bearman, G.

Z. Li, D. Psaltis, W. Liu, W. R. Johnson, and G. Bearman, “Volume holographic spectral imaging,” Proc. SPIE 5694, 33–40 (2005).
[CrossRef]

Belendez, A.

Belenez, A.

C. Neipp, J. T. Sheridan, S. Gallego, M. Ortuño, A. Marquez, I. Pascual, and A. Belenez, “Effect of depth attenuated refractive index profile in the angular responses of the efficiency of higher order in volume gratings recorded in a PVA/acrylamide photopolymer,” Opt. Commun. 233, 311–322 (2004).
[CrossRef]

Castro, J. M.

Chen, Ch. H.

Cooke, D. J.

L. Solymar and D. J. Cooke, Volume Holography and Volume Gratings (Academic, 1981).

de Leon, E.

Djordjevic, I.

Farkas, D.

J. Fujimoto and D. Farkas, Biomedical Optical Imaging(Oxford University, 2009).

Fujimoto, J.

J. Fujimoto and D. Farkas, Biomedical Optical Imaging(Oxford University, 2009).

Gallego, S.

S. Gallego, M. Ortuño, C. Neipp, A. Marquez, A. Belendez, I. Pascual, J. V. Kelly, and J. T. Sheridan, “Physical and effective optical thickness of holographic diffraction gratings recorded in photopolymers,” Opt. Express 13, 1939–1947 (2005).
[CrossRef] [PubMed]

C. Neipp, J. T. Sheridan, S. Gallego, M. Ortuño, A. Marquez, I. Pascual, and A. Belenez, “Effect of depth attenuated refractive index profile in the angular responses of the efficiency of higher order in volume gratings recorded in a PVA/acrylamide photopolymer,” Opt. Commun. 233, 311–322 (2004).
[CrossRef]

Gaylord, T. K.

Gelsinger, P. J.

Gelsinger-Austin, P. J.

P. J. Gelsinger-Austin, Y. Luo, J. M. Watson, R. K. Kostuk, G. Barbastathis, J. K. Barton, and J. M. Castro, “Optical design for a spatial-spectral volume holographic imaging system,” Opt. Eng. 49, 043001 (2010).
[CrossRef]

Goodman, J.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Johnson, W. R.

Z. Li, D. Psaltis, W. Liu, W. R. Johnson, and G. Bearman, “Volume holographic spectral imaging,” Proc. SPIE 5694, 33–40 (2005).
[CrossRef]

Kelly, J. V.

Kermish, D.

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Kostuk, R. K.

Kovalevsky, V. I.

V. V. Shelkovnikov, E. F. Pen, and V. I. Kovalevsky, “Optimum optical density of the absorbing holographic materials,” Opt. Mem. Neural Netw. 16, 75–83 (2007).
[CrossRef]

Kowarschik, R.

U. V. Mahilny, D. N. Marmysh, A. I. Stankevich, A. L. Tolstik, V. Matusevich, and R. Kowarschik, “Holographic volume absorption grating in glass-like polymer recording material,” Appl. Phys. B 82, 299–302 (2005).
[CrossRef]

R. Kowarschik, “Diffraction efficiency of attenuated sinusoidally modulated gratings in volume holograms,” J. Mod. Opt. 23, 1039–1051 (1976).
[CrossRef]

Kubota, T.

T. Kubota, “The diffraction efficiency of holograms gratings recorded in an absorptive medium,” Opt. Commun. 16, 347–349 (1976).
[CrossRef]

Li, Z.

Z. Li, D. Psaltis, W. Liu, W. R. Johnson, and G. Bearman, “Volume holographic spectral imaging,” Proc. SPIE 5694, 33–40 (2005).
[CrossRef]

Liu, W.

Z. Li, D. Psaltis, W. Liu, W. R. Johnson, and G. Bearman, “Volume holographic spectral imaging,” Proc. SPIE 5694, 33–40 (2005).
[CrossRef]

W. Liu, D. Psaltis, and G. Barbastathis, “Real-time spectral imaging in three spatial dimensions,” Opt. Lett. 27, 854–856 (2002).
[CrossRef]

Lou, Y.

Luo, Y.

P. J. Gelsinger-Austin, Y. Luo, J. M. Watson, R. K. Kostuk, G. Barbastathis, J. K. Barton, and J. M. Castro, “Optical design for a spatial-spectral volume holographic imaging system,” Opt. Eng. 49, 043001 (2010).
[CrossRef]

Y. Luo, P. J. Gelsinger, G. Barbastathis, J. K. Barton, and R. K. Kostuk, “Optimization of multiplexed holographic gratings in PQ-PMMA for spectral-spatial filters,” Opt. Lett. 33, 566–568 (2008).
[CrossRef] [PubMed]

Maeda, W.

Mahilny, U. V.

U. V. Mahilny, D. N. Marmysh, A. I. Stankevich, A. L. Tolstik, V. Matusevich, and R. Kowarschik, “Holographic volume absorption grating in glass-like polymer recording material,” Appl. Phys. B 82, 299–302 (2005).
[CrossRef]

Marmysh, D. N.

U. V. Mahilny, D. N. Marmysh, A. I. Stankevich, A. L. Tolstik, V. Matusevich, and R. Kowarschik, “Holographic volume absorption grating in glass-like polymer recording material,” Appl. Phys. B 82, 299–302 (2005).
[CrossRef]

Marquez, A.

S. Gallego, M. Ortuño, C. Neipp, A. Marquez, A. Belendez, I. Pascual, J. V. Kelly, and J. T. Sheridan, “Physical and effective optical thickness of holographic diffraction gratings recorded in photopolymers,” Opt. Express 13, 1939–1947 (2005).
[CrossRef] [PubMed]

C. Neipp, J. T. Sheridan, S. Gallego, M. Ortuño, A. Marquez, I. Pascual, and A. Belenez, “Effect of depth attenuated refractive index profile in the angular responses of the efficiency of higher order in volume gratings recorded in a PVA/acrylamide photopolymer,” Opt. Commun. 233, 311–322 (2004).
[CrossRef]

Matusevich, V.

U. V. Mahilny, D. N. Marmysh, A. I. Stankevich, A. L. Tolstik, V. Matusevich, and R. Kowarschik, “Holographic volume absorption grating in glass-like polymer recording material,” Appl. Phys. B 82, 299–302 (2005).
[CrossRef]

Moharam, M. G.

Neipp, C.

S. Gallego, M. Ortuño, C. Neipp, A. Marquez, A. Belendez, I. Pascual, J. V. Kelly, and J. T. Sheridan, “Physical and effective optical thickness of holographic diffraction gratings recorded in photopolymers,” Opt. Express 13, 1939–1947 (2005).
[CrossRef] [PubMed]

C. Neipp, J. T. Sheridan, S. Gallego, M. Ortuño, A. Marquez, I. Pascual, and A. Belenez, “Effect of depth attenuated refractive index profile in the angular responses of the efficiency of higher order in volume gratings recorded in a PVA/acrylamide photopolymer,” Opt. Commun. 233, 311–322 (2004).
[CrossRef]

Oh, S. B.

Ortuño, M.

S. Gallego, M. Ortuño, C. Neipp, A. Marquez, A. Belendez, I. Pascual, J. V. Kelly, and J. T. Sheridan, “Physical and effective optical thickness of holographic diffraction gratings recorded in photopolymers,” Opt. Express 13, 1939–1947 (2005).
[CrossRef] [PubMed]

C. Neipp, J. T. Sheridan, S. Gallego, M. Ortuño, A. Marquez, I. Pascual, and A. Belenez, “Effect of depth attenuated refractive index profile in the angular responses of the efficiency of higher order in volume gratings recorded in a PVA/acrylamide photopolymer,” Opt. Commun. 233, 311–322 (2004).
[CrossRef]

Pascual, I.

S. Gallego, M. Ortuño, C. Neipp, A. Marquez, A. Belendez, I. Pascual, J. V. Kelly, and J. T. Sheridan, “Physical and effective optical thickness of holographic diffraction gratings recorded in photopolymers,” Opt. Express 13, 1939–1947 (2005).
[CrossRef] [PubMed]

C. Neipp, J. T. Sheridan, S. Gallego, M. Ortuño, A. Marquez, I. Pascual, and A. Belenez, “Effect of depth attenuated refractive index profile in the angular responses of the efficiency of higher order in volume gratings recorded in a PVA/acrylamide photopolymer,” Opt. Commun. 233, 311–322 (2004).
[CrossRef]

Pen, E. F.

V. V. Shelkovnikov, E. F. Pen, and V. I. Kovalevsky, “Optimum optical density of the absorbing holographic materials,” Opt. Mem. Neural Netw. 16, 75–83 (2007).
[CrossRef]

Psaltis, D.

Z. Li, D. Psaltis, W. Liu, W. R. Johnson, and G. Bearman, “Volume holographic spectral imaging,” Proc. SPIE 5694, 33–40 (2005).
[CrossRef]

W. Liu, D. Psaltis, and G. Barbastathis, “Real-time spectral imaging in three spatial dimensions,” Opt. Lett. 27, 854–856 (2002).
[CrossRef]

Sato, A.

A. Sato and R. K. Kostuk, “Holographic grating for dense wavelength division optical filters at 1550 nm using phenanthrenequinone doped poly(methylmethacrylate),” Proc. SPIE 5216, 44–52 (2003).
[CrossRef]

Shelkovnikov, V. V.

V. V. Shelkovnikov, E. F. Pen, and V. I. Kovalevsky, “Optimum optical density of the absorbing holographic materials,” Opt. Mem. Neural Netw. 16, 75–83 (2007).
[CrossRef]

Sheridan, J. T.

S. Gallego, M. Ortuño, C. Neipp, A. Marquez, A. Belendez, I. Pascual, J. V. Kelly, and J. T. Sheridan, “Physical and effective optical thickness of holographic diffraction gratings recorded in photopolymers,” Opt. Express 13, 1939–1947 (2005).
[CrossRef] [PubMed]

C. Neipp, J. T. Sheridan, S. Gallego, M. Ortuño, A. Marquez, I. Pascual, and A. Belenez, “Effect of depth attenuated refractive index profile in the angular responses of the efficiency of higher order in volume gratings recorded in a PVA/acrylamide photopolymer,” Opt. Commun. 233, 311–322 (2004).
[CrossRef]

Shih, T.

Sinha, A.

Solymar, L.

L. Solymar and D. J. Cooke, Volume Holography and Volume Gratings (Academic, 1981).

Stankevich, A. I.

U. V. Mahilny, D. N. Marmysh, A. I. Stankevich, A. L. Tolstik, V. Matusevich, and R. Kowarschik, “Holographic volume absorption grating in glass-like polymer recording material,” Appl. Phys. B 82, 299–302 (2005).
[CrossRef]

Sun, W.

Tolstik, A. L.

U. V. Mahilny, D. N. Marmysh, A. I. Stankevich, A. L. Tolstik, V. Matusevich, and R. Kowarschik, “Holographic volume absorption grating in glass-like polymer recording material,” Appl. Phys. B 82, 299–302 (2005).
[CrossRef]

Vasic, B.

Watson, J. M.

P. J. Gelsinger-Austin, Y. Luo, J. M. Watson, R. K. Kostuk, G. Barbastathis, J. K. Barton, and J. M. Castro, “Optical design for a spatial-spectral volume holographic imaging system,” Opt. Eng. 49, 043001 (2010).
[CrossRef]

S. B. Oh, J. M. Watson, and G. Barbastathis, “Theoretical analysis of curved Bragg diffraction images from plane reference volume holograms,” Appl. Opt. 48, 5984–5996(2009).
[CrossRef] [PubMed]

Appl. Opt. (5)

Appl. Phys. B (1)

U. V. Mahilny, D. N. Marmysh, A. I. Stankevich, A. L. Tolstik, V. Matusevich, and R. Kowarschik, “Holographic volume absorption grating in glass-like polymer recording material,” Appl. Phys. B 82, 299–302 (2005).
[CrossRef]

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

J. Mod. Opt. (1)

R. Kowarschik, “Diffraction efficiency of attenuated sinusoidally modulated gratings in volume holograms,” J. Mod. Opt. 23, 1039–1051 (1976).
[CrossRef]

J. Opt. Soc. Am. (3)

Opt. Commun. (2)

T. Kubota, “The diffraction efficiency of holograms gratings recorded in an absorptive medium,” Opt. Commun. 16, 347–349 (1976).
[CrossRef]

C. Neipp, J. T. Sheridan, S. Gallego, M. Ortuño, A. Marquez, I. Pascual, and A. Belenez, “Effect of depth attenuated refractive index profile in the angular responses of the efficiency of higher order in volume gratings recorded in a PVA/acrylamide photopolymer,” Opt. Commun. 233, 311–322 (2004).
[CrossRef]

Opt. Eng. (1)

P. J. Gelsinger-Austin, Y. Luo, J. M. Watson, R. K. Kostuk, G. Barbastathis, J. K. Barton, and J. M. Castro, “Optical design for a spatial-spectral volume holographic imaging system,” Opt. Eng. 49, 043001 (2010).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Opt. Mem. Neural Netw. (1)

V. V. Shelkovnikov, E. F. Pen, and V. I. Kovalevsky, “Optimum optical density of the absorbing holographic materials,” Opt. Mem. Neural Netw. 16, 75–83 (2007).
[CrossRef]

Proc. SPIE (2)

Z. Li, D. Psaltis, W. Liu, W. R. Johnson, and G. Bearman, “Volume holographic spectral imaging,” Proc. SPIE 5694, 33–40 (2005).
[CrossRef]

A. Sato and R. K. Kostuk, “Holographic grating for dense wavelength division optical filters at 1550 nm using phenanthrenequinone doped poly(methylmethacrylate),” Proc. SPIE 5216, 44–52 (2003).
[CrossRef]

Other (3)

L. Solymar and D. J. Cooke, Volume Holography and Volume Gratings (Academic, 1981).

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

J. Fujimoto and D. Farkas, Biomedical Optical Imaging(Oxford University, 2009).

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Figures (12)

Fig. 1
Fig. 1

Layout and basic description of the S 2 -VHIS operation.

Fig. 2
Fig. 2

Relationship between the angular and spectral DE for angles and wavelengths close to the Bragg condition.

Fig. 3
Fig. 3

PQ-PMMA absorption for exposed and unexposed material.

Fig. 4
Fig. 4

Index modulation profile (left) and DE (right) of two gratings with identical t eff 1 .

Fig. 5
Fig. 5

HOE chromatic dispersive properties produce degradation in the x-axis image resolution. A broadband point source in object space forms a blurred spot along the dispersive axis of the S 2 -VHIS.

Fig. 6
Fig. 6

Relationship between spectral DE and angular dispersion.

Fig. 7
Fig. 7

Top, measured DE (square marks) and modeled DE (solid gray curve) for the HOE constructed at 457 nm (HOE 1). Bottom, estimated index modulation profile. Effective thickness = 1.32 mm using the 5% peak criterion.

Fig. 8
Fig. 8

Top, measured DE (square marks) and modeled DE (solid gray curve) for the HOE constructed at 488 nm (HOE 2). Bottom, estimated index modulation profile. Effective thickness = 1.55 mm using the 5% peak criterion.

Fig. 9
Fig. 9

Top, measured DE (square marks) and modeled (DE) (solid gray curve) for the HOE constructed at 514 nm (HOE 3). Bottom, estimated index modulation profile. Effective thickness = 1.8 mm using the 5% peak criterion.

Fig. 10
Fig. 10

x-axis MTF for the three holograms: (triangular marks) HOE 1 ( 457 nm ); (square marks) HOE 2 ( 488 nm ); and (solid gray curve) HOE 3 ( 514.5 nm ).

Fig. 11
Fig. 11

x-axis contrast measurements for the three holograms: (triangular marks) HOE 1 ( 457 nm ; (square marks) HOE 2 ( 488 nm ); and (solid gray curve) HOE 3 ( 514.5 nm ).

Fig. 12
Fig. 12

Image of the Air Force Bar Chart 1951, Group 7 reconstructed at 505 nm . Image produced using (a) HOE 1 ( 457 nm ), (b) HOE 2 ( 488 nm ), and (c) HOE 3 ( 514.5 nm ).

Equations (26)

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

u d = ( u x i K x ) x + ( u y i K y ) y + u z d z , u d = u x d x + u y d y + u z d z ,
u z d = k 2 2 | u x d | 2 | u y d | 2 , k 2 = 2 π n 2 λ ,
K = 2 π n 2 λ c [ r m s m ] ,
DE ( θ , λ ) = sin 2 ( v ( θ , λ ) 2 + ξ ( θ , λ ) 2 ) 1 + ( ξ ( θ , λ ) v ( θ , λ ) ) 2 ,
ν ( θ , λ ) = π Δ n t H λ c r ( θ ) c s ( θ ) ,
ϑ ( θ , λ ) = K cos ( ϕ θ ) K 2 4 π n 2 λ ,
ξ ( θ , λ ) = ϑ ( θ , λ ) t H 2 c s ( θ ) ,
DE a ( θ ) = DE ( θ , λ = λ r ) ,
DE s ( λ ) = DE ( θ = θ r , λ ) ,
ϑ ( Δ θ , Δ λ ) = K sin ( φ θ b ) Δ θ + K 2 4 π n 2 Δ λ .
v ( θ , λ ) = v ( θ b , λ b ) = c o n s t . , c s ( θ ) = c s ( θ b ) = c o n s t . , c r ( θ ) = c r ( θ b ) = c o n s t .
DE ( θ , λ ) = DE ( θ b + Δ θ , λ b + Δ λ ) = DE ° ( Δ θ , Δ λ ) .
DE s ( Δ λ ) = DE a ( Δ θ = ( K 4 π n 2 sin ( ϕ θ b ) ) Δ λ ) .
dn ( z ) = Δ n f ( z ) ,
dn i ( z ) = Δ n f ( i t H N ) ,
T 1 ( θ , λ ) = j DE ( θ , λ ) ,
T 0 ( θ , λ ) = cos ( v ( θ , λ ) 2 + ξ ( θ , λ ) 2 ) + ξ ( θ , λ ) v ( θ , λ ) T 1 ( θ , λ ) .
M i ( θ , λ ) = | T 0 ( θ , λ ) T 1 ( θ , λ ) T 1 ( θ , λ ) ( T 0 ( θ , λ ) ) * | ,
M ( θ , λ ) = i = 1 N M i ( θ , λ ) .
t eff 1 = 0 t H f ( z ) d z .
t eff 1 = 1 e α t H α .
t H = ln ( 1 α t eff 1 ) α ,
N x = B λ Δ λ .
x = f c tan ( θ ) = f c tan ( θ b + Δ θ ) x o + f c Δ θ ,
PSF x ( x ) DE s ( Δ θ m ) = DE s ( x / m f ) h ( x ) ,
MTF x ( f x ) I { DE s ( x / m f ) } ,

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