Abstract

Multivariate optical elements (MOEs) are multilayer optical interference coatings with arbitrary spectral profiles that are used in multivariate pattern recognition to perform the task of projecting magnitudes of special basis functions (regression vectors) out of optical spectra. Because MOEs depend on optical interference effects, their performance is sensitive to the angle of incidence of incident light. This angle dependence complicates their use in imaging applications. We report a method for the design of angle-insensitive MOEs based on modification of a previously described nonlinear optimization algorithm. This algorithm operates when the effects of deviant angles of incidence are simulated prior to optimization, which treats the angular deviation as an interferent in the measurement. To demonstrate the algorithm, a 13-layer imaging MOE (IMOE, with alternating layers of high-index Nb2O5 and low-index SiO2) for the determination of Bismarck Brown dye in mixtures of Bismarck Brown and Crystal Violet, was designed and its performance simulated. For angles of incidence that range from 42° to 48°, the IMOE has an average standard error of prediction (SEP) of 0.55 µM for Bismarck Brown. This compares with a SEP of 2.8 µM for a MOE designed by a fixed-angle algorithm.

© 2002 Optical Society of America

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References

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  1. P. Geladi, H. Grahn, Multivariate Image Analysis (Wiley, New York, 1996).
  2. S. E. Bialkowski, “A scheme for species discrimination and quantitative estimation using incoherent lidar optical signal processing,” Anal. Chem. 58, 2561–2563 (1986).
    [Crossref]
  3. A. Kasparov, V. Ryabenko, “Numerical study of a pattern recognition multispectral system with optimal spectral splitting,” Pattern Recogn. Image Anal. 1, 347–354 (1991).
  4. M. P. Nelson, J. F. Aust, J. A. Dobrowolski, P. G. Verly, M. L. Myrick, “Multivariate optical computation for predictive spectroscopy,” Anal. Chem. 70, 73–82 (1998).
    [Crossref] [PubMed]
  5. K. S. Booksh, A. M. C. Prakash, C. M. Stellman, “Optical regression: a method for improving quantitative precision of multivariate predictions with single channel spectrometers,” Chemom. Intell. Lab. Syst. 46, 265–274 (1999).
    [Crossref]
  6. O. Soyemi, D. Eastwood, L. Zhang, H. Li, J. Karunamuni, P. Gemperline, R. A. Synowicki, M. L. Myrick, “Design and testing of a multivariate optical element: the first demonstration of multivariate optical computing for predictive spectroscopy,” Anal. Chem. 73, 1069–1079 (2001).
    [Crossref]
  7. O. Soyemi, F. G. Haibach, A. E. Greer, M. V. Shiza, P. Gemperline, M. L. Myrick, “Nonlinear optimization algorithm for multivariate optical element design,” Appl. Spectrosc. 56, 396–398 (2002).
    [Crossref]
  8. J. E. Dennis, R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, N.J., 1983).

2002 (1)

O. Soyemi, F. G. Haibach, A. E. Greer, M. V. Shiza, P. Gemperline, M. L. Myrick, “Nonlinear optimization algorithm for multivariate optical element design,” Appl. Spectrosc. 56, 396–398 (2002).
[Crossref]

2001 (1)

O. Soyemi, D. Eastwood, L. Zhang, H. Li, J. Karunamuni, P. Gemperline, R. A. Synowicki, M. L. Myrick, “Design and testing of a multivariate optical element: the first demonstration of multivariate optical computing for predictive spectroscopy,” Anal. Chem. 73, 1069–1079 (2001).
[Crossref]

1999 (1)

K. S. Booksh, A. M. C. Prakash, C. M. Stellman, “Optical regression: a method for improving quantitative precision of multivariate predictions with single channel spectrometers,” Chemom. Intell. Lab. Syst. 46, 265–274 (1999).
[Crossref]

1998 (1)

M. P. Nelson, J. F. Aust, J. A. Dobrowolski, P. G. Verly, M. L. Myrick, “Multivariate optical computation for predictive spectroscopy,” Anal. Chem. 70, 73–82 (1998).
[Crossref] [PubMed]

1991 (1)

A. Kasparov, V. Ryabenko, “Numerical study of a pattern recognition multispectral system with optimal spectral splitting,” Pattern Recogn. Image Anal. 1, 347–354 (1991).

1986 (1)

S. E. Bialkowski, “A scheme for species discrimination and quantitative estimation using incoherent lidar optical signal processing,” Anal. Chem. 58, 2561–2563 (1986).
[Crossref]

Aust, J. F.

M. P. Nelson, J. F. Aust, J. A. Dobrowolski, P. G. Verly, M. L. Myrick, “Multivariate optical computation for predictive spectroscopy,” Anal. Chem. 70, 73–82 (1998).
[Crossref] [PubMed]

Bialkowski, S. E.

S. E. Bialkowski, “A scheme for species discrimination and quantitative estimation using incoherent lidar optical signal processing,” Anal. Chem. 58, 2561–2563 (1986).
[Crossref]

Booksh, K. S.

K. S. Booksh, A. M. C. Prakash, C. M. Stellman, “Optical regression: a method for improving quantitative precision of multivariate predictions with single channel spectrometers,” Chemom. Intell. Lab. Syst. 46, 265–274 (1999).
[Crossref]

Dennis, J. E.

J. E. Dennis, R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, N.J., 1983).

Dobrowolski, J. A.

M. P. Nelson, J. F. Aust, J. A. Dobrowolski, P. G. Verly, M. L. Myrick, “Multivariate optical computation for predictive spectroscopy,” Anal. Chem. 70, 73–82 (1998).
[Crossref] [PubMed]

Eastwood, D.

O. Soyemi, D. Eastwood, L. Zhang, H. Li, J. Karunamuni, P. Gemperline, R. A. Synowicki, M. L. Myrick, “Design and testing of a multivariate optical element: the first demonstration of multivariate optical computing for predictive spectroscopy,” Anal. Chem. 73, 1069–1079 (2001).
[Crossref]

Geladi, P.

P. Geladi, H. Grahn, Multivariate Image Analysis (Wiley, New York, 1996).

Gemperline, P.

O. Soyemi, F. G. Haibach, A. E. Greer, M. V. Shiza, P. Gemperline, M. L. Myrick, “Nonlinear optimization algorithm for multivariate optical element design,” Appl. Spectrosc. 56, 396–398 (2002).
[Crossref]

O. Soyemi, D. Eastwood, L. Zhang, H. Li, J. Karunamuni, P. Gemperline, R. A. Synowicki, M. L. Myrick, “Design and testing of a multivariate optical element: the first demonstration of multivariate optical computing for predictive spectroscopy,” Anal. Chem. 73, 1069–1079 (2001).
[Crossref]

Grahn, H.

P. Geladi, H. Grahn, Multivariate Image Analysis (Wiley, New York, 1996).

Greer, A. E.

O. Soyemi, F. G. Haibach, A. E. Greer, M. V. Shiza, P. Gemperline, M. L. Myrick, “Nonlinear optimization algorithm for multivariate optical element design,” Appl. Spectrosc. 56, 396–398 (2002).
[Crossref]

Haibach, F. G.

O. Soyemi, F. G. Haibach, A. E. Greer, M. V. Shiza, P. Gemperline, M. L. Myrick, “Nonlinear optimization algorithm for multivariate optical element design,” Appl. Spectrosc. 56, 396–398 (2002).
[Crossref]

Karunamuni, J.

O. Soyemi, D. Eastwood, L. Zhang, H. Li, J. Karunamuni, P. Gemperline, R. A. Synowicki, M. L. Myrick, “Design and testing of a multivariate optical element: the first demonstration of multivariate optical computing for predictive spectroscopy,” Anal. Chem. 73, 1069–1079 (2001).
[Crossref]

Kasparov, A.

A. Kasparov, V. Ryabenko, “Numerical study of a pattern recognition multispectral system with optimal spectral splitting,” Pattern Recogn. Image Anal. 1, 347–354 (1991).

Li, H.

O. Soyemi, D. Eastwood, L. Zhang, H. Li, J. Karunamuni, P. Gemperline, R. A. Synowicki, M. L. Myrick, “Design and testing of a multivariate optical element: the first demonstration of multivariate optical computing for predictive spectroscopy,” Anal. Chem. 73, 1069–1079 (2001).
[Crossref]

Myrick, M. L.

O. Soyemi, F. G. Haibach, A. E. Greer, M. V. Shiza, P. Gemperline, M. L. Myrick, “Nonlinear optimization algorithm for multivariate optical element design,” Appl. Spectrosc. 56, 396–398 (2002).
[Crossref]

O. Soyemi, D. Eastwood, L. Zhang, H. Li, J. Karunamuni, P. Gemperline, R. A. Synowicki, M. L. Myrick, “Design and testing of a multivariate optical element: the first demonstration of multivariate optical computing for predictive spectroscopy,” Anal. Chem. 73, 1069–1079 (2001).
[Crossref]

M. P. Nelson, J. F. Aust, J. A. Dobrowolski, P. G. Verly, M. L. Myrick, “Multivariate optical computation for predictive spectroscopy,” Anal. Chem. 70, 73–82 (1998).
[Crossref] [PubMed]

Nelson, M. P.

M. P. Nelson, J. F. Aust, J. A. Dobrowolski, P. G. Verly, M. L. Myrick, “Multivariate optical computation for predictive spectroscopy,” Anal. Chem. 70, 73–82 (1998).
[Crossref] [PubMed]

Prakash, A. M. C.

K. S. Booksh, A. M. C. Prakash, C. M. Stellman, “Optical regression: a method for improving quantitative precision of multivariate predictions with single channel spectrometers,” Chemom. Intell. Lab. Syst. 46, 265–274 (1999).
[Crossref]

Ryabenko, V.

A. Kasparov, V. Ryabenko, “Numerical study of a pattern recognition multispectral system with optimal spectral splitting,” Pattern Recogn. Image Anal. 1, 347–354 (1991).

Schnabel, R. B.

J. E. Dennis, R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, N.J., 1983).

Shiza, M. V.

O. Soyemi, F. G. Haibach, A. E. Greer, M. V. Shiza, P. Gemperline, M. L. Myrick, “Nonlinear optimization algorithm for multivariate optical element design,” Appl. Spectrosc. 56, 396–398 (2002).
[Crossref]

Soyemi, O.

O. Soyemi, F. G. Haibach, A. E. Greer, M. V. Shiza, P. Gemperline, M. L. Myrick, “Nonlinear optimization algorithm for multivariate optical element design,” Appl. Spectrosc. 56, 396–398 (2002).
[Crossref]

O. Soyemi, D. Eastwood, L. Zhang, H. Li, J. Karunamuni, P. Gemperline, R. A. Synowicki, M. L. Myrick, “Design and testing of a multivariate optical element: the first demonstration of multivariate optical computing for predictive spectroscopy,” Anal. Chem. 73, 1069–1079 (2001).
[Crossref]

Stellman, C. M.

K. S. Booksh, A. M. C. Prakash, C. M. Stellman, “Optical regression: a method for improving quantitative precision of multivariate predictions with single channel spectrometers,” Chemom. Intell. Lab. Syst. 46, 265–274 (1999).
[Crossref]

Synowicki, R. A.

O. Soyemi, D. Eastwood, L. Zhang, H. Li, J. Karunamuni, P. Gemperline, R. A. Synowicki, M. L. Myrick, “Design and testing of a multivariate optical element: the first demonstration of multivariate optical computing for predictive spectroscopy,” Anal. Chem. 73, 1069–1079 (2001).
[Crossref]

Verly, P. G.

M. P. Nelson, J. F. Aust, J. A. Dobrowolski, P. G. Verly, M. L. Myrick, “Multivariate optical computation for predictive spectroscopy,” Anal. Chem. 70, 73–82 (1998).
[Crossref] [PubMed]

Zhang, L.

O. Soyemi, D. Eastwood, L. Zhang, H. Li, J. Karunamuni, P. Gemperline, R. A. Synowicki, M. L. Myrick, “Design and testing of a multivariate optical element: the first demonstration of multivariate optical computing for predictive spectroscopy,” Anal. Chem. 73, 1069–1079 (2001).
[Crossref]

Anal. Chem. (3)

S. E. Bialkowski, “A scheme for species discrimination and quantitative estimation using incoherent lidar optical signal processing,” Anal. Chem. 58, 2561–2563 (1986).
[Crossref]

M. P. Nelson, J. F. Aust, J. A. Dobrowolski, P. G. Verly, M. L. Myrick, “Multivariate optical computation for predictive spectroscopy,” Anal. Chem. 70, 73–82 (1998).
[Crossref] [PubMed]

O. Soyemi, D. Eastwood, L. Zhang, H. Li, J. Karunamuni, P. Gemperline, R. A. Synowicki, M. L. Myrick, “Design and testing of a multivariate optical element: the first demonstration of multivariate optical computing for predictive spectroscopy,” Anal. Chem. 73, 1069–1079 (2001).
[Crossref]

Appl. Spectrosc. (1)

O. Soyemi, F. G. Haibach, A. E. Greer, M. V. Shiza, P. Gemperline, M. L. Myrick, “Nonlinear optimization algorithm for multivariate optical element design,” Appl. Spectrosc. 56, 396–398 (2002).
[Crossref]

Chemom. Intell. Lab. Syst. (1)

K. S. Booksh, A. M. C. Prakash, C. M. Stellman, “Optical regression: a method for improving quantitative precision of multivariate predictions with single channel spectrometers,” Chemom. Intell. Lab. Syst. 46, 265–274 (1999).
[Crossref]

Pattern Recogn. Image Anal. (1)

A. Kasparov, V. Ryabenko, “Numerical study of a pattern recognition multispectral system with optimal spectral splitting,” Pattern Recogn. Image Anal. 1, 347–354 (1991).

Other (2)

J. E. Dennis, R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, N.J., 1983).

P. Geladi, H. Grahn, Multivariate Image Analysis (Wiley, New York, 1996).

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Figures (5)

Fig. 1
Fig. 1

Concept for use of MOEs. D T and D R are detectors for transmitted and reflected light, respectively.

Fig. 2
Fig. 2

Schematic of a T-format multivariate optical computation device. Radiation from the sample is split by a MOE as in Fig. 1. L 1 and L 2 are the focusing lenses. CCD1 and CCD2 are charge-coupled detectors for transmitted and reflected light, respectively. Light paths marked a, b, and c represent the maximum, mean, and minimum angles of incidence of the light passing through the MOE. Pixel-by-pixel color composites are created when the detected light in the transmitted channel is combined with that in the reflected channel. Pixels in both channels are matched together so that the portion of the source scene viewed by each is the same.

Fig. 3
Fig. 3

Transmittance (Tr.) spectra of a 13-layer MOE designed from the modified SVR algorithm at angles of incidence ranging from 42° to 48°.

Fig. 4
Fig. 4

Comparison of the SEP for a 15-layer MOE designed by the fixed-angle SVR algorithm (circles) with the MOE in Fig. 3 (squares) at angles of incidence ranging from 42° to 48°.

Fig. 5
Fig. 5

Vectorial illustration of an angle-insensitive IMOE spectrum in terms of three neighboring wavelengths, λ1, λ2, and λ3. 1, vector of ones. a, vector of an analyte; b, vector of an interfering species; m, vector representing an angle-insensitive MOE. The ellipse is in a plane perpendicular to b. The gray conic section is given by rotation of m around 1.

Tables (1)

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Table 1 Thirteen-Layer Polarization-Insensitive IMOE Design for the Determination of BB in a Binary Mixture with CV

Equations (9)

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yi=j=1N lλjxiλj,
Tλj=0.5+klλj,
Rλj=0.5-klλj.
Rt,i=j=1N0.5-klλjxiλj=0.5 j=1N xiλj-k j=1N lλjxiλj.
Tt,i=0.5 j=1N xiλj+k j=1N lλjxiλj.
Tt,i-Rt,i=2k j=1N lλjxiλj=2kyi.
fx=RMSEC=i=1syi-yˆi2s1/2,
yˆi,δ=xiG/m2Tλδ-1+off,
SEP=i=1pyi, val-yˆi, val2p1/2,

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