Abstract

The analysis of a slitless volume holographic spectrometer is presented in detail. The spectrometer is based on a spherical beam volume hologram followed by a Fourier-transforming lens and a CCD. It is shown that the spectrometer is not sensitive to the incident angle of the input beam for the practical range of applications. A holographic spectrometer based on the conventional implementation is also analyzed, and the results are used to compare the performance of the proposed method with the conventional one. The experimental results are consistent with the theoretical study. It is also shown that the slitless volume holographic spectrometer lumps three elements (the entrance slit, the collimator, and the diffractive element) of the conventional spectrometer into one spherical beam volume hologram. Based on the unique features of the slitless volume holographic spectrometer, we believe it is a good candidate for portable spectroscopy for environmental and biological applications.

© 2006 Optical Society of America

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References

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  1. R. G. Bingham, "Grating spectrometers and spectrographs reexamined," Q. J. R. Astron. Soc. 20, 395-421 (1979).
  2. S. Singh, "Diffraction gratings: aberrations and applications," Opt. Laser Technol. 31, 195-218 (1999).
    [CrossRef]
  3. C. Hsieh, O. Momtahan, A. Karbaschi, A. Adibi, M. E. Sullivan, and D. J. Brady, "A compact Fourier transform volume holographic spectrometer for diffuse source spectroscopy," Opt. Lett. 30, 836-838 (2005).
    [CrossRef] [PubMed]
  4. W. Cassarly, "Nonimaging optics: concentration and illumination," in OSA Handbook of Optics, 2nd ed. (McGraw-Hill, 2001), Vol. 3, Chap. 2.
  5. D. J. Brady, "Multiplex sensors and the constant radiance theorem," Opt. Lett. 27, 16-18 (2002).
    [CrossRef]
  6. A. Karbaschi, C. Hsieh, O. Momtahan, A. Adibi, M. E. Sullivan, and D. J. Brady, "Qualitative demonstration of spectral diversity filtering using spherical beam volume holograms," Opt. Express 12, 3018-3024 (2004).
    [CrossRef] [PubMed]
  7. O. Momtahan, C. Hsieh, A. Karbaschi, A. Adibi, M. E. Sullivan, and D. J. Brady, "Spherical beam volume holograms for spectroscopic applications: modeling and implementation," Appl. Opt. 43, 6557-6567 (2004).
    [CrossRef]
  8. C. Hsieh, O. Momtahan, A. Karbaschi, A. Adibi, M. E. Sullivan, and D. J. Brady, "Role of recording geometry in the performance of spectral diversity filters using spherical beam volume holograms," Opt. Lett. 30, 186-188 (2005).
    [CrossRef] [PubMed]
  9. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
  10. G. Barbastathis and D. Psaltis, "Volume holographic multiplexing methods," in Holographic Data Storage, H.J.Coufal, D.Psaltis, and G.T.Sincerbox, eds. (Springer, 2000), pp. 21-59.
  11. H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909-2947 (1969).
  12. T. K. Gaylord and M. G. Moharam, "Analysis and applications of optical diffraction by gratings," Proc. IEEE 73, 894-937 (1985).
    [CrossRef]
  13. R. T. Ingwall and D. Waldman, "Photopolymer systems," in Holographic Data Storage, H.J.Coufal, D.Psaltis, and G.T.Sincerbox, eds. (Springer, 2000), pp. 171-197. Also see www.aprilisinc.com.

2005

2004

2002

1999

S. Singh, "Diffraction gratings: aberrations and applications," Opt. Laser Technol. 31, 195-218 (1999).
[CrossRef]

1985

T. K. Gaylord and M. G. Moharam, "Analysis and applications of optical diffraction by gratings," Proc. IEEE 73, 894-937 (1985).
[CrossRef]

1979

R. G. Bingham, "Grating spectrometers and spectrographs reexamined," Q. J. R. Astron. Soc. 20, 395-421 (1979).

1969

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

Adibi, A.

Barbastathis, G.

G. Barbastathis and D. Psaltis, "Volume holographic multiplexing methods," in Holographic Data Storage, H.J.Coufal, D.Psaltis, and G.T.Sincerbox, eds. (Springer, 2000), pp. 21-59.

Bingham, R. G.

R. G. Bingham, "Grating spectrometers and spectrographs reexamined," Q. J. R. Astron. Soc. 20, 395-421 (1979).

Brady, D. J.

Cassarly, W.

W. Cassarly, "Nonimaging optics: concentration and illumination," in OSA Handbook of Optics, 2nd ed. (McGraw-Hill, 2001), Vol. 3, Chap. 2.

Gaylord, T. K.

T. K. Gaylord and M. G. Moharam, "Analysis and applications of optical diffraction by gratings," Proc. IEEE 73, 894-937 (1985).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Hsieh, C.

Ingwall, R. T.

R. T. Ingwall and D. Waldman, "Photopolymer systems," in Holographic Data Storage, H.J.Coufal, D.Psaltis, and G.T.Sincerbox, eds. (Springer, 2000), pp. 171-197. Also see www.aprilisinc.com.

Karbaschi, A.

Kogelnik, H.

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

Moharam, M. G.

T. K. Gaylord and M. G. Moharam, "Analysis and applications of optical diffraction by gratings," Proc. IEEE 73, 894-937 (1985).
[CrossRef]

Momtahan, O.

Psaltis, D.

G. Barbastathis and D. Psaltis, "Volume holographic multiplexing methods," in Holographic Data Storage, H.J.Coufal, D.Psaltis, and G.T.Sincerbox, eds. (Springer, 2000), pp. 21-59.

Singh, S.

S. Singh, "Diffraction gratings: aberrations and applications," Opt. Laser Technol. 31, 195-218 (1999).
[CrossRef]

Sullivan, M. E.

Waldman, D.

R. T. Ingwall and D. Waldman, "Photopolymer systems," in Holographic Data Storage, H.J.Coufal, D.Psaltis, and G.T.Sincerbox, eds. (Springer, 2000), pp. 171-197. Also see www.aprilisinc.com.

Appl. Opt.

Bell Syst. Tech. J.

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

Opt. Express

Opt. Laser Technol.

S. Singh, "Diffraction gratings: aberrations and applications," Opt. Laser Technol. 31, 195-218 (1999).
[CrossRef]

Opt. Lett.

Proc. IEEE

T. K. Gaylord and M. G. Moharam, "Analysis and applications of optical diffraction by gratings," Proc. IEEE 73, 894-937 (1985).
[CrossRef]

Q. J. R. Astron. Soc.

R. G. Bingham, "Grating spectrometers and spectrographs reexamined," Q. J. R. Astron. Soc. 20, 395-421 (1979).

Other

R. T. Ingwall and D. Waldman, "Photopolymer systems," in Holographic Data Storage, H.J.Coufal, D.Psaltis, and G.T.Sincerbox, eds. (Springer, 2000), pp. 171-197. Also see www.aprilisinc.com.

W. Cassarly, "Nonimaging optics: concentration and illumination," in OSA Handbook of Optics, 2nd ed. (McGraw-Hill, 2001), Vol. 3, Chap. 2.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

G. Barbastathis and D. Psaltis, "Volume holographic multiplexing methods," in Holographic Data Storage, H.J.Coufal, D.Psaltis, and G.T.Sincerbox, eds. (Springer, 2000), pp. 21-59.

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Figures (7)

Fig. 1
Fig. 1

(a) Recording geometry of a spherical beam volume hologram. The point source is located at (−a, 0, −d). The reference beam (plane-wave) incident angle is θ r. A line from the coordinate origin to the point source makes an angle θ s with the z axis. The thickness of the holographic material is L. (b) Slitless spectrometer configuration. The reading beam is the input to the spectrometer with an incident angle of θ si ′. The focal length of the lens is f. The CCD is located at the back focal plane of the lens.

Fig. 2
Fig. 2

Theoretical intensity distribution in the output of the slitless holographic spectrometer estimated for the region corresponding to the CCD area when the hologram is read with a spatially incoherent reading beam. The incident angle of the reading beam is assumed to be from −5° to 5° measured in the air in both the x and the y directions corresponding to the total solid angle of 0.03 sr. The hologram is assumed to be recorded by using the setup in Fig. 1(a) with d = 4 cm, L = 300 μm, θ r = 46°, and θ s = −9°. The reading wavelength is 532 nm, which is equal to the recording wavelength. The refractive index of the recording material is assumed to be 1.5.

Fig. 3
Fig. 3

Experimental arrangement of the slitless spectrometer. All the parameters are the same as those given in the caption of Fig. 1 (b).

Fig. 4
Fig. 4

Basic arrangement of a spectrometer that uses a plane-wave hologram as the diffractive element. The hologram dimensions are shown in the figure. The hologram height (the dimension in the y direction) is assumed to be L 2 (not shown in the figure). The focal length of both lenses is f. The input object is usually a slit in the yi direction.

Fig. 5
Fig. 5

Diffraction efficiency of a plane-wave hologram (a) as a function of normalized modulated permittivity (Δϵ∕ϵ) for a Bragg-matched reading beam and (b) as a function of the incident angle of the reading beam for Δϵ∕ϵ = 0.0062, calculated by using the Born approximation and the Kogelnik method. The hologram is assumed to be recorded by using two plane waves at a 532 nm wavelength. Each recording plane wave has an incident angle of 35° in air. The refractive index of the recording material is n = 1.5. The hologram thickness is assumed to be 100 μm. The polarization of the recording beams is TE. The diffraction efficiency in the modified Born approximation (η MB ) is calculated as η MB = sin 2 ( η Born ) , where ηBorn is the diffraction efficiency calculated by using the Born approximation as described in the text.

Fig. 6
Fig. 6

Output of the slitless spectrometer for an input beam with wavelength components at 492, 532, and 562 nm obtained from (a) experiment and (b) theory. The SBVH was recorded by using the parameters in Fig. 1 (a) with d = 4 cm, θ r = 46° (in air), θ s = −9° (in air), L = 300 μm, and f = 10 cm. The recording wavelength was 532 nm. The pixel size of the CCD camera was 9 μm × 9 μm. Note that the sidelobes in the experimental results look stronger than those in the theoretical results. We believe this is because of the high diffraction efficiency of the SBVH in the experimental case that is not precisely modeled using the Born approximation.

Fig. 7
Fig. 7

Distribution of the output intensity of the conventional spectrometer shown in Fig. 4 obtained from both theory and experiment for (a) xo direction and (b) yo direction. The hologram dimensions were L 1 = 1 cm, L 2 = 1 cm, and L 3 = 100 μm. The focal length of both lenses was 6.5 cm. The hologram was recorded at 532 nm by using two plane waves, each with an incident angle of 35° measured in air. The hologram was read by a beam at 532 nm obtained by passing white light through a monochromator. The FWHM of the output spectrum of the monochromator at a 532 nm wavelength was 7.5 nm. The beam was collimated and passed through a square opening with dimensions of 140 μm × 140 μm (the object in Fig. 4). The square shape was selected to show the difference in the output for different input directions. The output was monitored by using a commercial CCD camera with a pixel size of 9.8 μm × 9.8 μm. Note that only the range of CCD pixels corresponding to a significant output signal is shown.

Equations (16)

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E i ( k i x , k i y , k i z ) = A i exp [ j ( k i x x + k i y y + k i z z ) + φ i ] .
E i d ( x , y , z ) = exp [ j ( k r x + k i x ) x ] exp ( j k i y y ) 4 π 2 × E ˜ i d ( k x , k y , z ) exp [ j ( k x x + k y y ) ] d k x d k y ,
E ˜ i d ( k x , k y , z ) = j 2 π 2 Δϵ k 2 L A i exp ( j φ i ) ϵ k i d z exp ( j k i d z z ) × sinc [ L 2 π ( K g z + k i z k i d z ) ] .
K g z = k r z ( k 2 k x     2 k y     2 ) 1 / 2 ,
k i d z = [ k 2 ( K g x + k i x ) 2 ( K g y + k i y ) 2 ] 1 / 2 ,
E i o ( u , v , z = 2 f ) = A i j λ f F { E i d ( x , y , L / 2 ) } | f x = u / ( λ f )   and   f y = v / ( λ f ) ,
P ˜ ( 2 π f x , 2 π f y , z ) = F { p ( x , y , z ) } = p ( x , y , z ) ×   exp [ j 2 π ( f x x + f y y ) ] d x d y .
E i o ( u , v , 2 f ) = A i j λ f F { exp [ j ( k r x + k i x ) x ] exp ( j k i y y ) F 1 { E ˜ d i ( k x , k y , 2 f ) } x x y y } f x = u / ( λ f ) and   f y = v / ( λ f )     = A i j λ f E ˜ i d [ ( 2 π f x k r x k i x ) , ( 2 π f y k i y ) , 2 f ] | f x = u / ( λ f )  and   f y = v / ( λ f )     = A i j λ f E ˜ i d ( k u / f + k r x + k i x , k v / f + k i y , 2 f ) .
H ( u , v , z = 2 f , λ ) = E i o ( u , v , 2 f ) / [ A i exp ( j φ i ) ] = j 2 π 2 Δϵ k 2 L ϵ 0 [ k 2 ( k u / f ) 2 ( k v / f ) 2 ] 1 / 2 × exp { j 2 f [ k 2 ( k u / f ) 2 ( k v / f ) 2 ] 1 / 2 } sinc ( L / 2 π { k r z [ k 2 ( k u / f k r x k i x ) 2 ( k v / f k i y ) 2 ] 1 / 2 + k i z [ k 2 ( k u / f ) 2 ( k v / f ) 2 ] 1 / 2 } ) .
I o ( u , v , 2 f ) = A i     2 ( k i x , k i y ) | H ( u , v , z = 2 f , λ ) | 2 d k x d k y
= | E i o ( u , v , 2 f ) | 2 d k x d k y ,
h ( x i , y i ; x o , y o ; λ ) = C   sinc ( L 1 2 π k 1 ) ×  sinc ( L 2 2 π k 2 ) sinc ( L 3 2 π k 3 ) ,
k 1 = 2 π λ [ ( x i f + 1 ) cos ( α ) ( x o f 1 ) sin ( α ) ] + K g ,
k 2 = 2 π λ ( y i f + y o f ) ,
k 3 = n 2 π λ [ ( x i f + 1 ) sin ( α ) ( x o f 1 ) cos ( α ) ] ,
rect ( u ) = { 1 | u | < 1 / 2 0   otherwise .

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