Abstract

Photonic crystal spectrometers possess significant size and cost advantages over traditional grating-based spectrometers. In a previous work [Pervez, et al, Opt. Express 18, 8277 (2010)] we demonstrated a proof of this concept by implementing a 9-element array photonic crystal spectrometer with a resolution of 20nm. Here we demonstrate a photonic crystal spectrometer with improved performance. The dependence of the spectral recovery resolution on the number of photonic crystal arrays and the width of the response function from each photonic crystal is investigated. A mathematical treatment, regularization based on known information of the spectrum, is utilized in order to stabilize the spectral estimation inverse problem and achieve improved spectral recovery. Colorimetry applications, the measurement of CIE 1931 chromaticities and the color rendering index, are demonstrated with the improved spectrometer.

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  13. S. H. Kim, H. S. Park, J. H. Choi, J. W. Shim, and S. M. Yang, “Integration of colloidal photonic crystals toward miniaturized spectrometers,” Adv. Mater.22, 946–950 (2010).
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    [CrossRef] [PubMed]
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  16. P. C. Hansen, Discrete Inverse Problems: Insight and Algorithms (Society for Industrial and Applied Mathematics, 2010).
    [CrossRef]
  17. J. C. Santamarina and D. Fratta, Discrete Signals and Inverse Problems (Wiley, 2005).
    [CrossRef]
  18. G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd Ed. (Johns Hopkins University,1996)

2011 (1)

U. Kurokawa, B. I. Choi, and C-. C. Chang, “Filter-based miniature spectrometers: spectrum reconstruction using adaptive regularization,” IEEE Sensors J.11, 1556–1563 (2011)
[CrossRef]

2010 (2)

S. H. Kim, H. S. Park, J. H. Choi, J. W. Shim, and S. M. Yang, “Integration of colloidal photonic crystals toward miniaturized spectrometers,” Adv. Mater.22, 946–950 (2010).
[CrossRef] [PubMed]

N. K. Pervez, W. Cheng, Z. Jia, M. P. Cox, H. M. Edrees, and I. Kymissis, “Photonic crystal spectrometer,” Opt. Express18, 8277–8285 (2010).
[CrossRef] [PubMed]

2009 (3)

B. Momeni, E. S. Hosseini, and A. Adibi, “Planar photonic crystal microspectrometers in silicon-nitride for the visible range,” Opt. Express17, 17060–17069 (2009).
[CrossRef] [PubMed]

W. M. Johnston, “Color measurement in dentistry,” J. Dent.37, e2–e6 (2009).
[CrossRef] [PubMed]

B. Momeni, E. S. Hosseini, M. Askari, M. Soltani, and A. Adibi, “Integrated photonic crystal spectrometers for sensing applications,” Opt. Commun.282, 3168–3171 (2009).
[CrossRef]

2007 (1)

2002 (1)

J. Y. Hardeberg, F. Schmitt, and H. Brettel, “Multispectral color image capture using a liquid crystal tunable filter,” Opt. Eng.41, 2532–2548 (2002).
[CrossRef]

2000 (1)

P. G. Herzog and F. Koenig, “Spectral scanner in the quality control of fabrics manufacturing,” Proc. SPIE4300, 25–32 (2000).
[CrossRef]

1995 (1)

S. S. Murtaza and J. C. Campbell, “Effects of variations in layer thickness on the reflectivity spectra of semiconductor Bragg mirrors,” J. Appl. Phys.77, 3641–3644 (1995).
[CrossRef]

Adibi, A.

B. Momeni, E. S. Hosseini, and A. Adibi, “Planar photonic crystal microspectrometers in silicon-nitride for the visible range,” Opt. Express17, 17060–17069 (2009).
[CrossRef] [PubMed]

B. Momeni, E. S. Hosseini, M. Askari, M. Soltani, and A. Adibi, “Integrated photonic crystal spectrometers for sensing applications,” Opt. Commun.282, 3168–3171 (2009).
[CrossRef]

Ahuja, S.

S. Ahuja and S. Scypinski, Handbook of Modern Pharmaceutical Analysis (Academic, 2010).

Askari, M.

B. Momeni, E. S. Hosseini, M. Askari, M. Soltani, and A. Adibi, “Integrated photonic crystal spectrometers for sensing applications,” Opt. Commun.282, 3168–3171 (2009).
[CrossRef]

Bassler, M.

Brettel, H.

J. Y. Hardeberg, F. Schmitt, and H. Brettel, “Multispectral color image capture using a liquid crystal tunable filter,” Opt. Eng.41, 2532–2548 (2002).
[CrossRef]

Campbell, J. C.

S. S. Murtaza and J. C. Campbell, “Effects of variations in layer thickness on the reflectivity spectra of semiconductor Bragg mirrors,” J. Appl. Phys.77, 3641–3644 (1995).
[CrossRef]

Celikiz, G.

G. Celikiz and R. G. Kuehni, Color Technology in the Textile Industry (American Association of Textile Chemists and Colorists: 1983).

Chang, C-. C.

U. Kurokawa, B. I. Choi, and C-. C. Chang, “Filter-based miniature spectrometers: spectrum reconstruction using adaptive regularization,” IEEE Sensors J.11, 1556–1563 (2011)
[CrossRef]

Cheng, W.

Choi, B. I.

U. Kurokawa, B. I. Choi, and C-. C. Chang, “Filter-based miniature spectrometers: spectrum reconstruction using adaptive regularization,” IEEE Sensors J.11, 1556–1563 (2011)
[CrossRef]

Choi, J. H.

S. H. Kim, H. S. Park, J. H. Choi, J. W. Shim, and S. M. Yang, “Integration of colloidal photonic crystals toward miniaturized spectrometers,” Adv. Mater.22, 946–950 (2010).
[CrossRef] [PubMed]

Cox, M. P.

Edrees, H. M.

Fratta, D.

J. C. Santamarina and D. Fratta, Discrete Signals and Inverse Problems (Wiley, 2005).
[CrossRef]

Gaurav, S.

S. Gaurav, Digital Color Imaging Handbook (CRC, 2003).

Golub, G. H.

G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd Ed. (Johns Hopkins University,1996)

Hansen, P. C.

P. C. Hansen, Discrete Inverse Problems: Insight and Algorithms (Society for Industrial and Applied Mathematics, 2010).
[CrossRef]

Hardeberg, J. Y.

J. Y. Hardeberg, F. Schmitt, and H. Brettel, “Multispectral color image capture using a liquid crystal tunable filter,” Opt. Eng.41, 2532–2548 (2002).
[CrossRef]

Herzog, P. G.

P. G. Herzog and F. Koenig, “Spectral scanner in the quality control of fabrics manufacturing,” Proc. SPIE4300, 25–32 (2000).
[CrossRef]

Hosseini, E. S.

B. Momeni, E. S. Hosseini, M. Askari, M. Soltani, and A. Adibi, “Integrated photonic crystal spectrometers for sensing applications,” Opt. Commun.282, 3168–3171 (2009).
[CrossRef]

B. Momeni, E. S. Hosseini, and A. Adibi, “Planar photonic crystal microspectrometers in silicon-nitride for the visible range,” Opt. Express17, 17060–17069 (2009).
[CrossRef] [PubMed]

Jia, Z.

Johnston, W. M.

W. M. Johnston, “Color measurement in dentistry,” J. Dent.37, e2–e6 (2009).
[CrossRef] [PubMed]

Kiesel, P.

Kim, S. H.

S. H. Kim, H. S. Park, J. H. Choi, J. W. Shim, and S. M. Yang, “Integration of colloidal photonic crystals toward miniaturized spectrometers,” Adv. Mater.22, 946–950 (2010).
[CrossRef] [PubMed]

Kipphan, H.

H. Kipphan, Handbook of Print Media: Technologies and Production Methods (Springer, 2001).

Koenig, F.

P. G. Herzog and F. Koenig, “Spectral scanner in the quality control of fabrics manufacturing,” Proc. SPIE4300, 25–32 (2000).
[CrossRef]

Kuehni, R. G.

G. Celikiz and R. G. Kuehni, Color Technology in the Textile Industry (American Association of Textile Chemists and Colorists: 1983).

Kurokawa, U.

U. Kurokawa, B. I. Choi, and C-. C. Chang, “Filter-based miniature spectrometers: spectrum reconstruction using adaptive regularization,” IEEE Sensors J.11, 1556–1563 (2011)
[CrossRef]

Kymissis, I.

Momeni, B.

B. Momeni, E. S. Hosseini, and A. Adibi, “Planar photonic crystal microspectrometers in silicon-nitride for the visible range,” Opt. Express17, 17060–17069 (2009).
[CrossRef] [PubMed]

B. Momeni, E. S. Hosseini, M. Askari, M. Soltani, and A. Adibi, “Integrated photonic crystal spectrometers for sensing applications,” Opt. Commun.282, 3168–3171 (2009).
[CrossRef]

Murtaza, S. S.

S. S. Murtaza and J. C. Campbell, “Effects of variations in layer thickness on the reflectivity spectra of semiconductor Bragg mirrors,” J. Appl. Phys.77, 3641–3644 (1995).
[CrossRef]

Park, H. S.

S. H. Kim, H. S. Park, J. H. Choi, J. W. Shim, and S. M. Yang, “Integration of colloidal photonic crystals toward miniaturized spectrometers,” Adv. Mater.22, 946–950 (2010).
[CrossRef] [PubMed]

Pervez, N. K.

Santamarina, J. C.

J. C. Santamarina and D. Fratta, Discrete Signals and Inverse Problems (Wiley, 2005).
[CrossRef]

Schmidt, O.

Schmitt, F.

J. Y. Hardeberg, F. Schmitt, and H. Brettel, “Multispectral color image capture using a liquid crystal tunable filter,” Opt. Eng.41, 2532–2548 (2002).
[CrossRef]

Scypinski, S.

S. Ahuja and S. Scypinski, Handbook of Modern Pharmaceutical Analysis (Academic, 2010).

Shim, J. W.

S. H. Kim, H. S. Park, J. H. Choi, J. W. Shim, and S. M. Yang, “Integration of colloidal photonic crystals toward miniaturized spectrometers,” Adv. Mater.22, 946–950 (2010).
[CrossRef] [PubMed]

Soltani, M.

B. Momeni, E. S. Hosseini, M. Askari, M. Soltani, and A. Adibi, “Integrated photonic crystal spectrometers for sensing applications,” Opt. Commun.282, 3168–3171 (2009).
[CrossRef]

Stiles, W. S.

G. Wyszecki and W. S. Stiles, Color Science, 2nd Ed. (Wiley, 1982).

Van Loan, C. F.

G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd Ed. (Johns Hopkins University,1996)

Wyszecki, G.

G. Wyszecki and W. S. Stiles, Color Science, 2nd Ed. (Wiley, 1982).

Yang, S. M.

S. H. Kim, H. S. Park, J. H. Choi, J. W. Shim, and S. M. Yang, “Integration of colloidal photonic crystals toward miniaturized spectrometers,” Adv. Mater.22, 946–950 (2010).
[CrossRef] [PubMed]

Adv. Mater. (1)

S. H. Kim, H. S. Park, J. H. Choi, J. W. Shim, and S. M. Yang, “Integration of colloidal photonic crystals toward miniaturized spectrometers,” Adv. Mater.22, 946–950 (2010).
[CrossRef] [PubMed]

IEEE Sensors J. (1)

U. Kurokawa, B. I. Choi, and C-. C. Chang, “Filter-based miniature spectrometers: spectrum reconstruction using adaptive regularization,” IEEE Sensors J.11, 1556–1563 (2011)
[CrossRef]

J. Appl. Phys. (1)

S. S. Murtaza and J. C. Campbell, “Effects of variations in layer thickness on the reflectivity spectra of semiconductor Bragg mirrors,” J. Appl. Phys.77, 3641–3644 (1995).
[CrossRef]

J. Dent. (1)

W. M. Johnston, “Color measurement in dentistry,” J. Dent.37, e2–e6 (2009).
[CrossRef] [PubMed]

Opt. Commun. (1)

B. Momeni, E. S. Hosseini, M. Askari, M. Soltani, and A. Adibi, “Integrated photonic crystal spectrometers for sensing applications,” Opt. Commun.282, 3168–3171 (2009).
[CrossRef]

Opt. Eng. (1)

J. Y. Hardeberg, F. Schmitt, and H. Brettel, “Multispectral color image capture using a liquid crystal tunable filter,” Opt. Eng.41, 2532–2548 (2002).
[CrossRef]

Opt. Express (3)

Proc. SPIE (1)

P. G. Herzog and F. Koenig, “Spectral scanner in the quality control of fabrics manufacturing,” Proc. SPIE4300, 25–32 (2000).
[CrossRef]

Other (8)

G. Celikiz and R. G. Kuehni, Color Technology in the Textile Industry (American Association of Textile Chemists and Colorists: 1983).

S. Ahuja and S. Scypinski, Handbook of Modern Pharmaceutical Analysis (Academic, 2010).

S. Gaurav, Digital Color Imaging Handbook (CRC, 2003).

H. Kipphan, Handbook of Print Media: Technologies and Production Methods (Springer, 2001).

G. Wyszecki and W. S. Stiles, Color Science, 2nd Ed. (Wiley, 1982).

P. C. Hansen, Discrete Inverse Problems: Insight and Algorithms (Society for Industrial and Applied Mathematics, 2010).
[CrossRef]

J. C. Santamarina and D. Fratta, Discrete Signals and Inverse Problems (Wiley, 2005).
[CrossRef]

G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd Ed. (Johns Hopkins University,1996)

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Figures (7)

Fig. 1
Fig. 1

(a) Working principle for the photonic crystal spectrometer; (b) a microscopic photo of the 17-channel photonic crystal spectrometer when illuminated with a white LED. Each square channel is 30 μm by 30 μm.

Fig. 2
Fig. 2

Reconstruction from noiseless data; discrete first derivative regularization (left), discrete second derivative (middle), and Γ = I (right). True spectra are dashed black curves, recovered are solid colored curves.

Fig. 3
Fig. 3

Reconstruction from actual channel data; discrete first derivative regularization (left), discrete second derivative (middle), and Γ = I (right). True spectra are dashed black curves, recovered are solid red curves.

Fig. 4
Fig. 4

Reconstruction of various LED spectra; aqua (top left), blue (top middle), orange (top right), red (bottom left), and yellow (bottom right). In each case the true spectrum is shown as the dashed black curve.

Fig. 5
Fig. 5

A computational experiment showing the dependence of spectral recovery resolution on the number of photonic crystal channels and the width of their response functions.

Fig. 6
Fig. 6

LEDs’ (triangles) and photonic crystal spectrometer’s (circles) chromaticities plotted on the CIE 1931 (x,y) color space. The center cross (x) is the chromaticities of a white LED based photonic crystal spectrometer measurement. Please refer to text for further details.

Fig. 7
Fig. 7

(a) the coordinates of the 8 given samples under the reference light source, and the tested light source (a white LED in this case) are calculated and plotted in CIE 1976 (L*, u*, v*) space. (b) Comparison of the CIE 1931 coordinates, CCTs, and CRIs

Tables (1)

Tables Icon

Table 1 CIE 1931 chromaticities of a variety of LEDs

Equations (22)

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b i = λ min λ max A i ( λ ) x ( λ ) d λ
Ax = b
x rec = argmin x n ( Ax b meas 2 2 + λ Γ x 2 2 )
( A T A + λ 2 Γ T Γ ) x rec = A T b meas .
Γ = [ 1 1 0 0 0 0 1 1 0 0 0 0 0 1 1 0 0 0 0 1 ] or Γ = [ 2 1 0 0 0 1 2 1 0 0 0 0 1 2 1 0 0 0 1 2 ]
[ A λ Γ ] x = [ b meas 0 ] .
W = λ min λ max w ( λ ) x ( λ ) d λ
W = w T x .
x rec = V S λ U T b
W = w T V S λ U T b = c λ T b
c λ = U ( S λ ) T V T w .
c 0 = U ( S 0 ) T V T w ,
W * = w T V S 0 U T b * ,
W λ = w T V S λ U T ( b * + e ) ,
W λ W * = w ˜ T ( S λ S 0 ) b ˜ + w ˜ T S λ U T e .
E ( ( W λ W * ) 2 ) = ( w ˜ T ( S λ S 0 ) b ˜ ) 2 + w ˜ T S λ U T CU ( S λ ) T w ˜ .
E ( ( W λ W * ) 2 ) = ( w ˜ T ( S λ S 0 ) b ˜ ) 2 + σ 2 w ˜ T S λ 2 2 .
E ( ( w ˜ T ( S λ S 0 ) b ˜ meas ) 2 ) = ( w ˜ T ( S λ S 0 ) b ˜ ) 2 + σ 2 w ˜ T ( S λ S 0 ) 2 2 .
E ( ( W λ W * ) 2 ) = E ( ( w ˜ T ( S λ S 0 ) b ˜ meas ) 2 ) σ 2 w ˜ T ( S λ S 0 ) 2 2 + σ 2 w ˜ T S λ 2 2 .
Q ( λ ) = ( w ˜ T ( S λ S 0 ) b ˜ meas ) 2 σ 2 w ˜ T ( S λ S 0 ) 2 2 + σ 2 w ˜ T S λ 2 2
Q ( λ ) = ( w ˜ T ( S λ S 0 ) b ˜ meas ) 2 σ 2 w ˜ T ( S λ S 0 ) 2 2 + 1 1 + σ 2 w ˜ T S λ U T B 2 2
E ( ( w T V S λ U T e ) 2 ) = σ 2 w ˜ T S λ 2 2 .

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