Abstract

Computational spectrometer based on a broadband diffractive optic was demonstrated with high spectral resolution over large bandwidth and high photon utilization efficiency. In this paper, we analyze such a spectrometer using singular value decomposition and propose a faster spectrum reconstruction algorithm with excellent accuracy by regularization. A new definition of spectral resolution based upon the Fourier analysis of singular vectors is described as well.

© 2014 Optical Society of America

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References

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  1. S. B. Utter, J. R. C. Lopez-Urrutia, P. Beiersdorfer, and E. Trabert, “Design and implementation of a high-resolution, high-efficiency optical spectrometer,” Rev. Sci. Instrum. 73(11), 3737–3741 (2002).
    [Crossref]
  2. C. P. Bacon, Y. Mattley, and R. DeFrece, “Miniature spectroscopic instrumentation: applications to biology and chemistry,” Rev. Sci. Instrum. 75(1), 1–16 (2004).
    [Crossref]
  3. G. Fortin and N. McCarthy, “Chirped holographic grating used as the dispersive element in an optical spectrometer,” Appl. Opt. 44(23), 4874–4883 (2005).
    [Crossref] [PubMed]
  4. K. Chaganti, I. Salakhutdinov, I. Avrutsky, and G. W. Auner, “A simple miniature optical spectrometer with a planar waveguide grating coupler in combination with a plano-convex lens,” Opt. Express 14(9), 4064–4072 (2006).
    [Crossref] [PubMed]
  5. R. F. Wolffenbuttel, “State-of-the-art in integrated optical microspectrometers,” IEEE Trans. Instrum. Meas. 53(1), 197–202 (2004).
    [Crossref]
  6. M. E. Gehm, S. T. McCain, N. P. Pitsianis, D. J. Brady, P. Potuluri, and M. E. Sullivan, “Static two-dimensional aperture coding for multimodal, multiplex spectroscopy,” Appl. Opt. 45(13), 2965–2974 (2006).
    [Crossref] [PubMed]
  7. S. D. Feller, H. Chen, D. J. Brady, M. E. Gehm, C. Hsieh, O. Momtahan, and A. Adibi, “Multiple order coded aperture spectrometer,” Opt. Express 15(9), 5625–5630 (2007).
    [Crossref] [PubMed]
  8. B. Redding and H. Cao, “Using a multimode fiber as a high-resolution, low-loss spectrometer,” Opt. Lett. 37(16), 3384–3386 (2012).
    [Crossref] [PubMed]
  9. B. Redding, S. M. Popoff, and H. Cao, “All-fiber spectrometer based on speckle pattern reconstruction,” Opt. Express 21(5), 6584–6600 (2013).
    [Crossref] [PubMed]
  10. B. Redding, S. F. Liew, R. Sarma, and H. Cao, “Compact spectrometer based on a disordered photonic chip,” Nat. Photonics 7(9), 746–751 (2013).
    [Crossref]
  11. T. W. Kohlgraf-Owens and A. Dogariu, “Transmission matrices of random media: means for spectral polarimetric measurements,” Opt. Lett. 35(13), 2236–2238 (2010).
    [Crossref] [PubMed]
  12. Z. Xu, Z. Wang, M. E. Sullivan, D. J. Brady, S. H. Foulger, and A. Adibi, “Multimodal multiplex spectroscopy using photonic crystals,” Opt. Express 11(18), 2126–2133 (2003).
    [Crossref] [PubMed]
  13. A. Nitkowski, L. Chen, and M. Lipson, “Cavity-enhanced on-chip absorption spectroscopy using microring resonators,” Opt. Express 16(16), 11930–11936 (2008).
    [Crossref] [PubMed]
  14. Z. Xia, A. A. Eftekhar, M. Soltani, B. Momeni, Q. Li, M. Chamanzar, S. Yegnanarayanan, and A. Adibi, “High resolution on-chip spectroscopy based on miniaturized microdonut resonators,” Opt. Express 19(13), 12356–12364 (2011).
    [Crossref] [PubMed]
  15. P. Wang and R. Menon, “Computational spectrometer based on a broadband diffractive optic,” Opt. Express 22(12), 14575–14587 (2014).
    [Crossref] [PubMed]
  16. P. C. Hansen, Discrete Inverse Problems: Insight and Algorithms (SIAM Press, 2010).
  17. Regularization tools for MATLAB developed by Prof. Hansen at the Technical University of Denmark: http://www.imm.dtu.dk/~pcha/Regutools/ .
  18. G. H. Golub and C. F. Van Loan, Matrix Computations (The Johns Hopkins University Press, 2013).
  19. G. Kim, J. A. Domínguez-Caballero, and R. Menon, “Design and analysis of multi-wavelength diffractive optics,” Opt. Express 20(3), 2814–2823 (2012).
    [Crossref] [PubMed]
  20. P. Wang, J. A. Dominguez-Caballero, D. J. Friedman, and R. Menon, “A new class of multi-bandgap high efficiency photovoltaics enabled by broadband diffractive optics,” Prog. Photovolt. Res. Appl.n/a (2014), doi:.
    [Crossref]
  21. P. Wang, C. G. Ebeling, J. Gerton, and R. Menon, “Hyper-spectral imaging in scanning-confocal-fluorescence microscopy using a novel broadband diffractive optic,” Opt. Commun. 324, 73–80 (2014).
    [Crossref]
  22. P. Wang and R. Menon, “Three-dimensional lithography via digital holography,” in Frontiers in Optics 2012/Laser Science XXVIII, OSA Technical Digest (online) (Optical Society of America, 2012), paper FTu3A.4. http://www.opticsinfobase.org/abstract.cfm?URI=FiO-2012-FTu3A.4
  23. P. Wang and R. Menon, “Optimization of periodic nanostructures for enhanced light-trapping in ultra-thin photovoltaics,” Opt. Express 21(5), 6274–6285 (2013).
    [Crossref] [PubMed]
  24. P. Wang and R. Menon, “Optimization of generalized dielectric nanostructures for enhanced light trapping in thin-film photovoltaics via boosting the local density of optical states,” Opt. Express 22(S1Suppl 1), A99–A110 (2014).
    [Crossref] [PubMed]

2014 (3)

2013 (3)

2012 (2)

2011 (1)

2010 (1)

2008 (1)

2007 (1)

2006 (2)

2005 (1)

2004 (2)

R. F. Wolffenbuttel, “State-of-the-art in integrated optical microspectrometers,” IEEE Trans. Instrum. Meas. 53(1), 197–202 (2004).
[Crossref]

C. P. Bacon, Y. Mattley, and R. DeFrece, “Miniature spectroscopic instrumentation: applications to biology and chemistry,” Rev. Sci. Instrum. 75(1), 1–16 (2004).
[Crossref]

2003 (1)

2002 (1)

S. B. Utter, J. R. C. Lopez-Urrutia, P. Beiersdorfer, and E. Trabert, “Design and implementation of a high-resolution, high-efficiency optical spectrometer,” Rev. Sci. Instrum. 73(11), 3737–3741 (2002).
[Crossref]

Adibi, A.

Auner, G. W.

Avrutsky, I.

Bacon, C. P.

C. P. Bacon, Y. Mattley, and R. DeFrece, “Miniature spectroscopic instrumentation: applications to biology and chemistry,” Rev. Sci. Instrum. 75(1), 1–16 (2004).
[Crossref]

Beiersdorfer, P.

S. B. Utter, J. R. C. Lopez-Urrutia, P. Beiersdorfer, and E. Trabert, “Design and implementation of a high-resolution, high-efficiency optical spectrometer,” Rev. Sci. Instrum. 73(11), 3737–3741 (2002).
[Crossref]

Brady, D. J.

Cao, H.

Chaganti, K.

Chamanzar, M.

Chen, H.

Chen, L.

DeFrece, R.

C. P. Bacon, Y. Mattley, and R. DeFrece, “Miniature spectroscopic instrumentation: applications to biology and chemistry,” Rev. Sci. Instrum. 75(1), 1–16 (2004).
[Crossref]

Dogariu, A.

Dominguez-Caballero, J. A.

P. Wang, J. A. Dominguez-Caballero, D. J. Friedman, and R. Menon, “A new class of multi-bandgap high efficiency photovoltaics enabled by broadband diffractive optics,” Prog. Photovolt. Res. Appl.n/a (2014), doi:.
[Crossref]

Domínguez-Caballero, J. A.

Ebeling, C. G.

P. Wang, C. G. Ebeling, J. Gerton, and R. Menon, “Hyper-spectral imaging in scanning-confocal-fluorescence microscopy using a novel broadband diffractive optic,” Opt. Commun. 324, 73–80 (2014).
[Crossref]

Eftekhar, A. A.

Feller, S. D.

Fortin, G.

Foulger, S. H.

Friedman, D. J.

P. Wang, J. A. Dominguez-Caballero, D. J. Friedman, and R. Menon, “A new class of multi-bandgap high efficiency photovoltaics enabled by broadband diffractive optics,” Prog. Photovolt. Res. Appl.n/a (2014), doi:.
[Crossref]

Gehm, M. E.

Gerton, J.

P. Wang, C. G. Ebeling, J. Gerton, and R. Menon, “Hyper-spectral imaging in scanning-confocal-fluorescence microscopy using a novel broadband diffractive optic,” Opt. Commun. 324, 73–80 (2014).
[Crossref]

Hsieh, C.

Kim, G.

Kohlgraf-Owens, T. W.

Li, Q.

Liew, S. F.

B. Redding, S. F. Liew, R. Sarma, and H. Cao, “Compact spectrometer based on a disordered photonic chip,” Nat. Photonics 7(9), 746–751 (2013).
[Crossref]

Lipson, M.

Lopez-Urrutia, J. R. C.

S. B. Utter, J. R. C. Lopez-Urrutia, P. Beiersdorfer, and E. Trabert, “Design and implementation of a high-resolution, high-efficiency optical spectrometer,” Rev. Sci. Instrum. 73(11), 3737–3741 (2002).
[Crossref]

Mattley, Y.

C. P. Bacon, Y. Mattley, and R. DeFrece, “Miniature spectroscopic instrumentation: applications to biology and chemistry,” Rev. Sci. Instrum. 75(1), 1–16 (2004).
[Crossref]

McCain, S. T.

McCarthy, N.

Menon, R.

Momeni, B.

Momtahan, O.

Nitkowski, A.

Pitsianis, N. P.

Popoff, S. M.

Potuluri, P.

Redding, B.

Salakhutdinov, I.

Sarma, R.

B. Redding, S. F. Liew, R. Sarma, and H. Cao, “Compact spectrometer based on a disordered photonic chip,” Nat. Photonics 7(9), 746–751 (2013).
[Crossref]

Soltani, M.

Sullivan, M. E.

Trabert, E.

S. B. Utter, J. R. C. Lopez-Urrutia, P. Beiersdorfer, and E. Trabert, “Design and implementation of a high-resolution, high-efficiency optical spectrometer,” Rev. Sci. Instrum. 73(11), 3737–3741 (2002).
[Crossref]

Utter, S. B.

S. B. Utter, J. R. C. Lopez-Urrutia, P. Beiersdorfer, and E. Trabert, “Design and implementation of a high-resolution, high-efficiency optical spectrometer,” Rev. Sci. Instrum. 73(11), 3737–3741 (2002).
[Crossref]

Wang, P.

P. Wang, C. G. Ebeling, J. Gerton, and R. Menon, “Hyper-spectral imaging in scanning-confocal-fluorescence microscopy using a novel broadband diffractive optic,” Opt. Commun. 324, 73–80 (2014).
[Crossref]

P. Wang and R. Menon, “Optimization of generalized dielectric nanostructures for enhanced light trapping in thin-film photovoltaics via boosting the local density of optical states,” Opt. Express 22(S1Suppl 1), A99–A110 (2014).
[Crossref] [PubMed]

P. Wang and R. Menon, “Computational spectrometer based on a broadband diffractive optic,” Opt. Express 22(12), 14575–14587 (2014).
[Crossref] [PubMed]

P. Wang and R. Menon, “Optimization of periodic nanostructures for enhanced light-trapping in ultra-thin photovoltaics,” Opt. Express 21(5), 6274–6285 (2013).
[Crossref] [PubMed]

P. Wang, J. A. Dominguez-Caballero, D. J. Friedman, and R. Menon, “A new class of multi-bandgap high efficiency photovoltaics enabled by broadband diffractive optics,” Prog. Photovolt. Res. Appl.n/a (2014), doi:.
[Crossref]

Wang, Z.

Wolffenbuttel, R. F.

R. F. Wolffenbuttel, “State-of-the-art in integrated optical microspectrometers,” IEEE Trans. Instrum. Meas. 53(1), 197–202 (2004).
[Crossref]

Xia, Z.

Xu, Z.

Yegnanarayanan, S.

Appl. Opt. (2)

IEEE Trans. Instrum. Meas. (1)

R. F. Wolffenbuttel, “State-of-the-art in integrated optical microspectrometers,” IEEE Trans. Instrum. Meas. 53(1), 197–202 (2004).
[Crossref]

Nat. Photonics (1)

B. Redding, S. F. Liew, R. Sarma, and H. Cao, “Compact spectrometer based on a disordered photonic chip,” Nat. Photonics 7(9), 746–751 (2013).
[Crossref]

Opt. Commun. (1)

P. Wang, C. G. Ebeling, J. Gerton, and R. Menon, “Hyper-spectral imaging in scanning-confocal-fluorescence microscopy using a novel broadband diffractive optic,” Opt. Commun. 324, 73–80 (2014).
[Crossref]

Opt. Express (10)

P. Wang and R. Menon, “Optimization of periodic nanostructures for enhanced light-trapping in ultra-thin photovoltaics,” Opt. Express 21(5), 6274–6285 (2013).
[Crossref] [PubMed]

P. Wang and R. Menon, “Optimization of generalized dielectric nanostructures for enhanced light trapping in thin-film photovoltaics via boosting the local density of optical states,” Opt. Express 22(S1Suppl 1), A99–A110 (2014).
[Crossref] [PubMed]

G. Kim, J. A. Domínguez-Caballero, and R. Menon, “Design and analysis of multi-wavelength diffractive optics,” Opt. Express 20(3), 2814–2823 (2012).
[Crossref] [PubMed]

Z. Xu, Z. Wang, M. E. Sullivan, D. J. Brady, S. H. Foulger, and A. Adibi, “Multimodal multiplex spectroscopy using photonic crystals,” Opt. Express 11(18), 2126–2133 (2003).
[Crossref] [PubMed]

A. Nitkowski, L. Chen, and M. Lipson, “Cavity-enhanced on-chip absorption spectroscopy using microring resonators,” Opt. Express 16(16), 11930–11936 (2008).
[Crossref] [PubMed]

Z. Xia, A. A. Eftekhar, M. Soltani, B. Momeni, Q. Li, M. Chamanzar, S. Yegnanarayanan, and A. Adibi, “High resolution on-chip spectroscopy based on miniaturized microdonut resonators,” Opt. Express 19(13), 12356–12364 (2011).
[Crossref] [PubMed]

P. Wang and R. Menon, “Computational spectrometer based on a broadband diffractive optic,” Opt. Express 22(12), 14575–14587 (2014).
[Crossref] [PubMed]

B. Redding, S. M. Popoff, and H. Cao, “All-fiber spectrometer based on speckle pattern reconstruction,” Opt. Express 21(5), 6584–6600 (2013).
[Crossref] [PubMed]

S. D. Feller, H. Chen, D. J. Brady, M. E. Gehm, C. Hsieh, O. Momtahan, and A. Adibi, “Multiple order coded aperture spectrometer,” Opt. Express 15(9), 5625–5630 (2007).
[Crossref] [PubMed]

K. Chaganti, I. Salakhutdinov, I. Avrutsky, and G. W. Auner, “A simple miniature optical spectrometer with a planar waveguide grating coupler in combination with a plano-convex lens,” Opt. Express 14(9), 4064–4072 (2006).
[Crossref] [PubMed]

Opt. Lett. (2)

Rev. Sci. Instrum. (2)

S. B. Utter, J. R. C. Lopez-Urrutia, P. Beiersdorfer, and E. Trabert, “Design and implementation of a high-resolution, high-efficiency optical spectrometer,” Rev. Sci. Instrum. 73(11), 3737–3741 (2002).
[Crossref]

C. P. Bacon, Y. Mattley, and R. DeFrece, “Miniature spectroscopic instrumentation: applications to biology and chemistry,” Rev. Sci. Instrum. 75(1), 1–16 (2004).
[Crossref]

Other (5)

P. Wang, J. A. Dominguez-Caballero, D. J. Friedman, and R. Menon, “A new class of multi-bandgap high efficiency photovoltaics enabled by broadband diffractive optics,” Prog. Photovolt. Res. Appl.n/a (2014), doi:.
[Crossref]

P. C. Hansen, Discrete Inverse Problems: Insight and Algorithms (SIAM Press, 2010).

Regularization tools for MATLAB developed by Prof. Hansen at the Technical University of Denmark: http://www.imm.dtu.dk/~pcha/Regutools/ .

G. H. Golub and C. F. Van Loan, Matrix Computations (The Johns Hopkins University Press, 2013).

P. Wang and R. Menon, “Three-dimensional lithography via digital holography,” in Frontiers in Optics 2012/Laser Science XXVIII, OSA Technical Digest (online) (Optical Society of America, 2012), paper FTu3A.4. http://www.opticsinfobase.org/abstract.cfm?URI=FiO-2012-FTu3A.4

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Figures (6)

Fig. 1
Fig. 1

Schematic explaining the principle of the diffraction-based computational spectrometer. The polychromat is a pixelated micro-optic with discretized heights. Each wavelength generates a specific intensity pattern after propagation through the polychromat. The image plane is at distance d away from the polychromat.

Fig. 2
Fig. 2

(a) AFM measurement of a segment (70μm × 30μm) of the fabricated polychromat with maximum height of 1.2μm and groove width of 3μm. (b) Schematic of the setup for characterizing the SS-PSF of the polychromat. The beam from super-continuum source is collimated and expanded. The fiber-tip detector, connected to a conventional spectrometer, is scanned at the image plane.

Fig. 3
Fig. 3

Singular value decomposition analysis of the experimental data. (a) Singular values σi of the measured SS-PSFs at different propagation distances. (inset: the spectral resolution estimated by Fourier analysis of singular vectors versus propagation distance, and the spectral resolution predicted by a correlation function) (b) Measured SS-PSF of the polychromat at d = 49mm. (c) Measured SS-PSF of the polychromat at d = 450mm. (d) Each column i represents the Fourier transform of the singular vector vi of the SS-PSF at d = 450mm. The low- and high-frequency regimes are separated by i = 420. (e) The Fourier transform of the singular vector v420. Peak frequency f0 = 0.53nm−1 indicates a spectral resolution of 0.95nm.

Fig. 4
Fig. 4

L-curve for selecting the regularization parameter ω to reconstruct the unknown spectrum of a yellow LED by the Tikhonov regularization method. Spectrum reconstruction results by using ω = 0.001 (top left inset), ω = 0.1 (top right inset), ω = 1.857 (middle right inset), ω = 10 (bottom left inset) and ω = 1000 (bottom right inset). The best result is obtained by ω = 1.857. The reconstructed plots are in red and the reference measurements are in black. They are normalized. The captured grey-scale camera image is shown as the middle inset.

Fig. 5
Fig. 5

Spectrum reconstruction results of a solid-state green laser with 532nm central wavelength (a) and a super-continuum source (b) by the Tikhonov regularization. The optimal parameters are ω = 0.938 and ω = 4.004, respectively. Red curves are the reconstructed results and black ones are the reference measurements. (Top insets: L-curves for selecting the best parameters; Bottom insets: photographs of the images projected on white paper at d = 450mm).

Fig. 6
Fig. 6

Explanation of the achievable spectral resolution. (a) A perfect harmonic signal in the spectral domain with period of Λ = 50nm from 500nm to 600nm. (b) Normalized correlation function of the harmonic signal, indicating a minimum at δλ = 25nm, which is half the period.

Equations (10)

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

g(x)= K(x,λ)f(λ)dλ .
b=Ax,
S=PSFΨ.
A=UΣV.
min{ Axb 2 2 + ω 2 x 2 2 }.
x ω = i=1 n ϕ i [ω] u i T b σ i v i ,
ϕ i [ω] = σ i 2 σ i 2 + ω 2 .
f max = 1 2 1 ( BW m ) .
resolution= 1 2 f 0 .
C(δλ)= < v i (λ) v i (λ+δλ)> < v i (λ)>< v i (λ+δλ)> .

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