Abstract

A three dimensional (3D) pupil is an optical element, most commonly implemented on a volume hologram, that processes the incident optical field on a 3D fashion. Here we analyze the diffraction properties of a 3D pupil with finite lateral aperture in the 4-f imaging system configuration, using the Wigner Distribution Function (WDF) formulation. Since 3D imaging pupil is finite in both lateral and longitudinal directions, the WDF of the volume holographic 4-f imager theoretically predicts distinct Bragg diffraction patterns in phase space. These result in asymmetric profiles of diffracted coherent point spread function between degenerate diffraction and Bragg diffraction, elucidating the fundamental performance of volume holographic imaging. Experimental measurements are also presented, confirming the theoretical predictions.

© 2015 Optical Society of America

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References

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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  5. Y. Luo, P. J. Gelsinger, J. K. Barton, G. Barbastathis, and R. K. Kostuk, “Optimization of multiplexed holographic gratings in PQ-PMMA for spectral-spatial imaging filters,” Opt. Lett. 33(6), 566–568 (2008).
    [Crossref] [PubMed]
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    [Crossref]
  7. H.-T. Hsieh, W. Liu, F. Havermeyer, C. Moser, and D. Psaltis, “Beam-width-dependent filtering properties of strong volume holographic gratings,” Appl. Opt. 45(16), 3774–3780 (2006).
    [Crossref] [PubMed]
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    [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  15. Y. Luo, V. R. Singh, D. Bhattacharya, E. Y. S. Yew, J.-C. Tsai, S.-L. Yu, H.-H. Chen, J.-M. Wong, P. Matsudaira, P. T. C. So, and G. Barbastathis, “Talbot holographic illumination nonscanning (THIN) fluorescence microscopy,” Laser and Photonics Reviews 8(5), L71–L75 (2014).
    [Crossref]
  16. S. B. Oh, “Volume holographic pupils in ray, wave, statistical optics, and Wigner space,” Ph.D. dissertation (Massachusetts Institute of Technology, 2009).
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    [Crossref] [PubMed]
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    [Crossref]

2014 (1)

Y. Luo, V. R. Singh, D. Bhattacharya, E. Y. S. Yew, J.-C. Tsai, S.-L. Yu, H.-H. Chen, J.-M. Wong, P. Matsudaira, P. T. C. So, and G. Barbastathis, “Talbot holographic illumination nonscanning (THIN) fluorescence microscopy,” Laser and Photonics Reviews 8(5), L71–L75 (2014).
[Crossref]

2013 (1)

2010 (2)

2009 (1)

2008 (3)

2007 (1)

2006 (1)

2003 (2)

2002 (1)

1984 (1)

K. H. Brenner and J. Ojeda-Castañeda, “Ambiguity Function and Wigner Distribution Function Applied to Partially Coherent Imagery,” Optica Acta: International Journal of Optics 31(2), 213–223 (1984).
[Crossref]

1979 (1)

1969 (1)

H. W. Kogelnik, “Coupled Wave Theory for Thick Hologram Gratings,” Bell Syst. Tech. J. 48(9), 2909–2947 (1969).
[Crossref]

Barbastathis, G.

Y. Luo, V. R. Singh, D. Bhattacharya, E. Y. S. Yew, J.-C. Tsai, S.-L. Yu, H.-H. Chen, J.-M. Wong, P. Matsudaira, P. T. C. So, and G. Barbastathis, “Talbot holographic illumination nonscanning (THIN) fluorescence microscopy,” Laser and Photonics Reviews 8(5), L71–L75 (2014).
[Crossref]

S. B. Oh, Z.-Q. J. Lu, J.-C. Tsai, H.-H. Chen, G. Barbastathis, and Y. Luo, “Phase-coded volume holographic gratings for spatial-spectral imaging filters,” Opt. Lett. 38(4), 477–479 (2013).
[Crossref] [PubMed]

Y. Luo, J. Castro, J. K. Barton, R. K. Kostuk, and G. Barbastathis, “Simulations and experiments of aperiodic and multiplexed gratings in volume holographic imaging systems,” Opt. Express 18(18), 19273–19285 (2010).
[Crossref] [PubMed]

Y. Luo, S. B. Oh, and G. Barbastathis, “Wavelength-coded multifocal microscopy,” Opt. Lett. 35(5), 781–783 (2010).
[PubMed]

S. B. Oh and G. Barbastathis, “Wigner distribution function of volume holograms,” Opt. Lett. 34(17), 2584–2586 (2009).
[Crossref] [PubMed]

Y. Luo, P. J. Gelsinger, J. K. Barton, G. Barbastathis, and R. K. Kostuk, “Optimization of multiplexed holographic gratings in PQ-PMMA for spectral-spatial imaging filters,” Opt. Lett. 33(6), 566–568 (2008).
[Crossref] [PubMed]

P. Wissmann, S. B. Oh, and G. Barbastathis, “Simulation and optimization of volume holographic imaging systems in Zemax®,” Opt. Express 16(10), 7516–7524 (2008).
[Crossref] [PubMed]

A. Sinha and G. Barbastathis, “Volume holographic imaging for surface metrology at long working distances,” Opt. Express 11(24), 3202–3209 (2003).
[Crossref] [PubMed]

W. Liu, D. Psaltis, and G. Barbastathis, “Real-time spectral imaging in three spatial dimensions,” Opt. Lett. 27(10), 854–856 (2002).
[Crossref] [PubMed]

Barton, J. K.

Bastiaans, M. J.

Bhattacharya, D.

Y. Luo, V. R. Singh, D. Bhattacharya, E. Y. S. Yew, J.-C. Tsai, S.-L. Yu, H.-H. Chen, J.-M. Wong, P. Matsudaira, P. T. C. So, and G. Barbastathis, “Talbot holographic illumination nonscanning (THIN) fluorescence microscopy,” Laser and Photonics Reviews 8(5), L71–L75 (2014).
[Crossref]

Brenner, K. H.

K. H. Brenner and J. Ojeda-Castañeda, “Ambiguity Function and Wigner Distribution Function Applied to Partially Coherent Imagery,” Optica Acta: International Journal of Optics 31(2), 213–223 (1984).
[Crossref]

Castro, J.

Chen, H.-H.

Y. Luo, V. R. Singh, D. Bhattacharya, E. Y. S. Yew, J.-C. Tsai, S.-L. Yu, H.-H. Chen, J.-M. Wong, P. Matsudaira, P. T. C. So, and G. Barbastathis, “Talbot holographic illumination nonscanning (THIN) fluorescence microscopy,” Laser and Photonics Reviews 8(5), L71–L75 (2014).
[Crossref]

S. B. Oh, Z.-Q. J. Lu, J.-C. Tsai, H.-H. Chen, G. Barbastathis, and Y. Luo, “Phase-coded volume holographic gratings for spatial-spectral imaging filters,” Opt. Lett. 38(4), 477–479 (2013).
[Crossref] [PubMed]

Gelsinger, P. J.

Havermeyer, F.

Hsieh, H.-T.

Kogelnik, H. W.

H. W. Kogelnik, “Coupled Wave Theory for Thick Hologram Gratings,” Bell Syst. Tech. J. 48(9), 2909–2947 (1969).
[Crossref]

Kostuk, R. K.

Liu, W.

Lu, Z.-Q. J.

Luo, Y.

Matsudaira, P.

Y. Luo, V. R. Singh, D. Bhattacharya, E. Y. S. Yew, J.-C. Tsai, S.-L. Yu, H.-H. Chen, J.-M. Wong, P. Matsudaira, P. T. C. So, and G. Barbastathis, “Talbot holographic illumination nonscanning (THIN) fluorescence microscopy,” Laser and Photonics Reviews 8(5), L71–L75 (2014).
[Crossref]

Moser, C.

Oh, S. B.

Ojeda-Castañeda, J.

K. H. Brenner and J. Ojeda-Castañeda, “Ambiguity Function and Wigner Distribution Function Applied to Partially Coherent Imagery,” Optica Acta: International Journal of Optics 31(2), 213–223 (1984).
[Crossref]

Pan, W.

Psaltis, D.

Sheridan, J. T.

Singh, V. R.

Y. Luo, V. R. Singh, D. Bhattacharya, E. Y. S. Yew, J.-C. Tsai, S.-L. Yu, H.-H. Chen, J.-M. Wong, P. Matsudaira, P. T. C. So, and G. Barbastathis, “Talbot holographic illumination nonscanning (THIN) fluorescence microscopy,” Laser and Photonics Reviews 8(5), L71–L75 (2014).
[Crossref]

Sinha, A.

Situ, G.

So, P. T. C.

Y. Luo, V. R. Singh, D. Bhattacharya, E. Y. S. Yew, J.-C. Tsai, S.-L. Yu, H.-H. Chen, J.-M. Wong, P. Matsudaira, P. T. C. So, and G. Barbastathis, “Talbot holographic illumination nonscanning (THIN) fluorescence microscopy,” Laser and Photonics Reviews 8(5), L71–L75 (2014).
[Crossref]

Sun, C.-C.

C.-C. Sun, “Simplified model for diffraction analysis of volume holograms,” Opt. Eng. 42(5), 1184–1185 (2003).
[Crossref]

Tsai, J.-C.

Y. Luo, V. R. Singh, D. Bhattacharya, E. Y. S. Yew, J.-C. Tsai, S.-L. Yu, H.-H. Chen, J.-M. Wong, P. Matsudaira, P. T. C. So, and G. Barbastathis, “Talbot holographic illumination nonscanning (THIN) fluorescence microscopy,” Laser and Photonics Reviews 8(5), L71–L75 (2014).
[Crossref]

S. B. Oh, Z.-Q. J. Lu, J.-C. Tsai, H.-H. Chen, G. Barbastathis, and Y. Luo, “Phase-coded volume holographic gratings for spatial-spectral imaging filters,” Opt. Lett. 38(4), 477–479 (2013).
[Crossref] [PubMed]

Wissmann, P.

Wong, J.-M.

Y. Luo, V. R. Singh, D. Bhattacharya, E. Y. S. Yew, J.-C. Tsai, S.-L. Yu, H.-H. Chen, J.-M. Wong, P. Matsudaira, P. T. C. So, and G. Barbastathis, “Talbot holographic illumination nonscanning (THIN) fluorescence microscopy,” Laser and Photonics Reviews 8(5), L71–L75 (2014).
[Crossref]

Yew, E. Y. S.

Y. Luo, V. R. Singh, D. Bhattacharya, E. Y. S. Yew, J.-C. Tsai, S.-L. Yu, H.-H. Chen, J.-M. Wong, P. Matsudaira, P. T. C. So, and G. Barbastathis, “Talbot holographic illumination nonscanning (THIN) fluorescence microscopy,” Laser and Photonics Reviews 8(5), L71–L75 (2014).
[Crossref]

Yu, S.-L.

Y. Luo, V. R. Singh, D. Bhattacharya, E. Y. S. Yew, J.-C. Tsai, S.-L. Yu, H.-H. Chen, J.-M. Wong, P. Matsudaira, P. T. C. So, and G. Barbastathis, “Talbot holographic illumination nonscanning (THIN) fluorescence microscopy,” Laser and Photonics Reviews 8(5), L71–L75 (2014).
[Crossref]

Appl. Opt. (2)

Bell Syst. Tech. J. (1)

H. W. Kogelnik, “Coupled Wave Theory for Thick Hologram Gratings,” Bell Syst. Tech. J. 48(9), 2909–2947 (1969).
[Crossref]

J. Opt. Soc. Am. (1)

Laser and Photonics Reviews (1)

Y. Luo, V. R. Singh, D. Bhattacharya, E. Y. S. Yew, J.-C. Tsai, S.-L. Yu, H.-H. Chen, J.-M. Wong, P. Matsudaira, P. T. C. So, and G. Barbastathis, “Talbot holographic illumination nonscanning (THIN) fluorescence microscopy,” Laser and Photonics Reviews 8(5), L71–L75 (2014).
[Crossref]

Opt. Eng. (1)

C.-C. Sun, “Simplified model for diffraction analysis of volume holograms,” Opt. Eng. 42(5), 1184–1185 (2003).
[Crossref]

Opt. Express (3)

Opt. Lett. (6)

Optica Acta: International Journal of Optics (1)

K. H. Brenner and J. Ojeda-Castañeda, “Ambiguity Function and Wigner Distribution Function Applied to Partially Coherent Imagery,” Optica Acta: International Journal of Optics 31(2), 213–223 (1984).
[Crossref]

Other (3)

W. Sun, “Profilometry with volume holographic imaging,” Ph.D. dissertation (Massachusetts Institute of Technology, 2006).

G. Barbastathis, “The Transfer Function of Volume Holographic Optical Systems,” in Photorefractive Materials and Their Applications 3, P. Günter, and J.-P. Huignard, eds., pp. 51–76 (Springer, 2007).

S. B. Oh, “Volume holographic pupils in ray, wave, statistical optics, and Wigner space,” Ph.D. dissertation (Massachusetts Institute of Technology, 2009).

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Figures (8)

Fig. 1
Fig. 1

. Illustration of (a) volume holographic recording geometry, (b) k-sphere diagram probed and recorded using wavevectors kp and kd, respectively, for the recorded grating vector K ; (c) 4-f imaging system geometry with 3D pupil, where f is the focal length, a and L are the lateral and axial apertures, respectively, and red dash-line denotes the Fourier plane. To obtain a clear pupil, the VH would be removed from the system and be replaced by a simple aperture of the same size, located at the Fourier plane.

Fig. 2
Fig. 2

Wigner function of input corresponding to different output location of diffracted beam. The output location ( x 4 ) is (i) 0, (ii) 0.25,(iii), 0.5,(iv) 0.8mm with same diffracted angle u 4 = θ s /λ , where a=1mm , L=1mm , λ=500nm and θ s = 30 Red dash frame denotes the edge of the recorded region inside the holographic material.

Fig. 3
Fig. 3

Wigner function of clear pupil, where a=1mm , L=0mm , λ=500nm Note that the spatial coordinates is defined as x 3 = x 4 owing to δ( x 3 x 4 ) in Eq. (7).

Fig. 4
Fig. 4

The illustration of Bragg match condition (a), (b) is the mapping against x 3 and x 4 , where u 3 =0 and u 4 =θ/λ as well as Bragg match. (c) is the projection on x 3 (orange curve) and x 4 (blue curve). The grid white lines in (b) denote the locations of peak value mapping against x 3 and x 4 .

Fig. 5
Fig. 5

The phase to phase representation as the beam illuminates on (a) 2D clear pupil, (b) vicinity of center of 3D pupil, where x 3 =0.25mm and x 4 =0mm , and (c) edge of 3D pupil, where x 3 =0.375mm and x 4 =0.8mm . Here, a=1mm , λ=500nm and θ s = 30 , and the thickness is (a) L=0mm for the clear pupil, (b, c) L=1mm for the volume hologram.

Fig. 6
Fig. 6

(a) Wigner function of x 4 u 4 . (b) I 4 for a 4-f imager with 3D pupil (marked in red line), and with 2D clear pupil (marked in blue line). (c) PSF of the 4-f imager with 3D (marked in red line), and 2D pupil (marked in blue line), with a=1mm , L=1mm , λ=500nm and θ s = 30 .

Fig. 7
Fig. 7

(a) Wigner function of x 4 u 4 . (b) I 4 for a 4-f imager with 3D pupil (marked in red line), and with 2D clear pupil (marked in blue line). (c) PSF of the 4-f imager with 3D (marked in red line), and 2D pupil (marked in blue line), with a=1mm , L=1mm , λ=500nm and θ s = 68 .

Fig. 8
Fig. 8

The measurement (a, c) and simulation (b, d) of PSF of VHIS, (a) and (b) is PSF for a=0.57mm ; (c) and (d) is PSF for a=1mm . (i, ii) intensity profile passing through the peak of the main lobe along x 2 and y 2 for a=0.57mm , and (iii, iv) intensity profile passing through the peak of the main lobe along x 2 and y 2 for a=1mm .

Equations (9)

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W 3 ( x 3 , u 3 )= W 1 (λf u 3 , L 2f u 3 + x 3 λf ), W 4 ( x 4 , u 4 )= W 2 (λf u 4 , L 2f u 4 x 4 λf ),
W 4 ( x 4 , u 4 )= K 3D ( x 4 , u 4 ; x 3 , u 3 ) W 3 ( x 3 , u 3 )d x 3 d u 3 .
K 3D ( x 4 , u 4 ; x 3 , u 3 )= K VHIS (λf u 4 , L 2f u 4 x 4 λf ;λf u 3 , L 2f u 3 + x 3 λf ),
K VHIS ( x 2 , u 2 ; x 1 , u 1 )= d x 2 d x 1 exp(i2π( u 2 x 2 u 1 x 1 )) × h VHIS ( x 2 + x 2 2 ; x 1 + x 1 2 ) h VHIS ( x 2 x 2 2 ; x 1 x 1 2 ),
h VHIS ( x 2 ; x 1 )=sinc( a λf ( x 1 + x 2 f θ s ) )×sinc( L 2λ f 2 ( x 1 2 x 2 2 + f 2 θ s 2 ) ),
K 3D ( x 4 , u 4 ; x 3 , u 3 )= d u 3 d u 4 exp{ i2π[ u 4 ( Lλ 2 u 4 x 4 )+ u 3 ( Lλ 2 u 3 + x 3 ) ] }× sinc{ a[ u 3 + u 4 + 1 2 ( u 3 + u 4 ) θ s λ ] }sinc{ a[ u 3 + u 4 1 2 ( u 3 + u 4 ) θ s λ ] }× sinc{ Lλ 2 [ ( u 3 + u 3 2 ) 2 ( u 4 + u 4 2 ) 2 + ( θ s λ ) 2 ] }×sinc{ Lλ 2 [ ( u 3 + u 3 2 ) 2 ( u 4 u 4 2 ) 2 + ( θ s λ ) 2 ] }.
K 2D ( x 4 , u 4 ; x 3 , u 3 )= a 2 δ( x 3 x 4 )Λ( x 4 a/2 )sinc{ (2a4| x 4 |)( u 3 u 4 ) },
I 4 ( x 4 )= W 4 ( x 4 , u 4 )d u 4 , | PSF( x 2 ) | 2 = W 4 ( L 2f x 2 λf u 2 , x 2 λf ) d u 2 ,
C M,S = i=1 n ( M i M )( S i S ) i=1 n ( M i M ) 2 i=1 n ( S i S ) 2 ,

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