Abstract

Spectral cameras with integrated thin-film Fabry–Perot filters enable many different applications. Some applications require the detection of spectral features that are only visible at specific wavelengths, and some need to quantify small spectral differences that are undetectable with RGB color cameras. One factor that influences the central wavelength of thin-film filters is the angle of incidence. Therefore, when light is focused from an imaging lens onto the filter array, undesirable shifts in the measured spectra are observed. These shifts limit the use of the sensor in applications that require fast lenses or lenses with large chief ray angles. To increase flexibility and enable new applications, we derive an analytical model that explains and can correct the observed shifts in measured spectra. The model includes the size of the aperture and physical position of each filter on the sensor. We experimentally validate the model with two spectral cameras: one in the visible and near-infrared region and one in the short wave infrared region.

© 2018 Optical Society of America

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References

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  2. G. Lu and B. Fei, “Medical hyperspectral imaging: a review,” J. Biomed. Opt. 19, 010901 (2014).
    [Crossref]
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    [Crossref]
  4. N. Tack, A. Lambrechts, P. Soussan, and L. Haspeslagh, “A compact, high-speed, and low-cost hyperspectral imager,” Proc. SPIE 8266, 82660Q (2012).
    [Crossref]
  5. B. Geelen, N. Tack, and A. Lambrechts, “A compact snapshot multispectral imager with a monolithically integrated per-pixel filter mosaic,” Proc. SPIE 8974, 89740L (2014).
    [Crossref]
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    [Crossref]
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    [Crossref]
  19. P. Gonzalez, J. Pichette, B. Vereecke, B. Masschelein, L. Krasovitski, L. Bikov, and A. Lambrechts, “An extremely compact and high-speed line-scan hyperspectral imager covering the SWIR range,” Proc. SPIE 10656, 106560L (2018).
    [Crossref]
  20. P. Agrawal, K. Tack, B. Geelen, B. Masschelein, P. Mateo, A. Moran, A. Lambrechts, and M. Jayapala, “Characterization of VNIR hyperspectral sensors with monolithically integrated optical filters,” in Image Sensors and Imaging Systems (Society for Imaging Science and Technology, 2016), pp. 1–7.
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    [Crossref]

2018 (1)

P. Gonzalez, J. Pichette, B. Vereecke, B. Masschelein, L. Krasovitski, L. Bikov, and A. Lambrechts, “An extremely compact and high-speed line-scan hyperspectral imager covering the SWIR range,” Proc. SPIE 10656, 106560L (2018).
[Crossref]

2017 (1)

J. Pichette, W. Charle, and A. Lambrechts, “Fast and compact internal scanning CMOS-based hyperspectral camera: the Snapscan,” Proc. SPIE 10110, 1011014 (2017).
[Crossref]

2016 (1)

I. G. E. Renhorn, D. Bergström, J. Hedborg, D. Letalick, and S. Möller, “High spatial resolution hyperspectral camera based on a linear variable filter,” Opt. Eng. 55, 114105 (2016).
[Crossref]

2015 (1)

B. Geelen, C. Blanch, P. Gonzalez, N. Tack, and A. Lambrechts, “A tiny VIS-NIR snapshot multispectral camera,” Proc. SPIE 9374, 937414 (2015).
[Crossref]

2014 (3)

B. Geelen, N. Tack, and A. Lambrechts, “A compact snapshot multispectral imager with a monolithically integrated per-pixel filter mosaic,” Proc. SPIE 8974, 89740L (2014).
[Crossref]

G. Lu and B. Fei, “Medical hyperspectral imaging: a review,” J. Biomed. Opt. 19, 010901 (2014).
[Crossref]

T. Skauli, H. E. Torkildsen, S. Nicolas, T. Opsahl, T. Haavardsholm, I. Kåsen, and A. Rognmo, “Compact camera for multispectral and conventional imaging based on patterned filters,” Appl. Opt. 53, C64–C74 (2014).
[Crossref]

2013 (1)

N. Hagen and M. W. Kudenov, “Review of snapshot spectral imaging technologies,” Opt. Eng. 52, 090901 (2013).
[Crossref]

2012 (1)

N. Tack, A. Lambrechts, P. Soussan, and L. Haspeslagh, “A compact, high-speed, and low-cost hyperspectral imager,” Proc. SPIE 8266, 82660Q (2012).
[Crossref]

2007 (1)

A. Gowen, C. O’Donnell, P. Cullen, G. Downey, and J. Frias, “Hyperspectral imaging—an emerging process analytical tool for food quality and safety control,” Trends Food Sci. Technol. 18, 590–598 (2007).
[Crossref]

2004 (1)

P. Shippert, “Why use hyperspectral imagery?” Earth Sci. 70, 377–379 (2004).

2000 (2)

N. Gat, “Imaging spectroscopy using tunable filters: a review,” Proc. SPIE 4056, 50–64 (2000).
[Crossref]

P. Thenkabail, R. Smith, and E. De Pauw, “Hyperspectral vegetation indices and their relationships with agricultural crop characteristics,” Remote Sens. Environ. 71, 158–182 (2000).
[Crossref]

1985 (1)

1979 (1)

1966 (1)

1964 (1)

Agrawal, P.

P. Agrawal, K. Tack, B. Geelen, B. Masschelein, P. Mateo, A. Moran, A. Lambrechts, and M. Jayapala, “Characterization of VNIR hyperspectral sensors with monolithically integrated optical filters,” in Image Sensors and Imaging Systems (Society for Imaging Science and Technology, 2016), pp. 1–7.

Bergström, D.

I. G. E. Renhorn, D. Bergström, J. Hedborg, D. Letalick, and S. Möller, “High spatial resolution hyperspectral camera based on a linear variable filter,” Opt. Eng. 55, 114105 (2016).
[Crossref]

Bikov, L.

P. Gonzalez, J. Pichette, B. Vereecke, B. Masschelein, L. Krasovitski, L. Bikov, and A. Lambrechts, “An extremely compact and high-speed line-scan hyperspectral imager covering the SWIR range,” Proc. SPIE 10656, 106560L (2018).
[Crossref]

Blanch, C.

B. Geelen, C. Blanch, P. Gonzalez, N. Tack, and A. Lambrechts, “A tiny VIS-NIR snapshot multispectral camera,” Proc. SPIE 9374, 937414 (2015).
[Crossref]

Charle, W.

J. Pichette, W. Charle, and A. Lambrechts, “Fast and compact internal scanning CMOS-based hyperspectral camera: the Snapscan,” Proc. SPIE 10110, 1011014 (2017).
[Crossref]

Cullen, P.

A. Gowen, C. O’Donnell, P. Cullen, G. Downey, and J. Frias, “Hyperspectral imaging—an emerging process analytical tool for food quality and safety control,” Trends Food Sci. Technol. 18, 590–598 (2007).
[Crossref]

De Pauw, E.

P. Thenkabail, R. Smith, and E. De Pauw, “Hyperspectral vegetation indices and their relationships with agricultural crop characteristics,” Remote Sens. Environ. 71, 158–182 (2000).
[Crossref]

Downey, G.

A. Gowen, C. O’Donnell, P. Cullen, G. Downey, and J. Frias, “Hyperspectral imaging—an emerging process analytical tool for food quality and safety control,” Trends Food Sci. Technol. 18, 590–598 (2007).
[Crossref]

Fei, B.

G. Lu and B. Fei, “Medical hyperspectral imaging: a review,” J. Biomed. Opt. 19, 010901 (2014).
[Crossref]

Frias, J.

A. Gowen, C. O’Donnell, P. Cullen, G. Downey, and J. Frias, “Hyperspectral imaging—an emerging process analytical tool for food quality and safety control,” Trends Food Sci. Technol. 18, 590–598 (2007).
[Crossref]

Gat, N.

N. Gat, “Imaging spectroscopy using tunable filters: a review,” Proc. SPIE 4056, 50–64 (2000).
[Crossref]

Geelen, B.

B. Geelen, C. Blanch, P. Gonzalez, N. Tack, and A. Lambrechts, “A tiny VIS-NIR snapshot multispectral camera,” Proc. SPIE 9374, 937414 (2015).
[Crossref]

B. Geelen, N. Tack, and A. Lambrechts, “A compact snapshot multispectral imager with a monolithically integrated per-pixel filter mosaic,” Proc. SPIE 8974, 89740L (2014).
[Crossref]

P. Agrawal, K. Tack, B. Geelen, B. Masschelein, P. Mateo, A. Moran, A. Lambrechts, and M. Jayapala, “Characterization of VNIR hyperspectral sensors with monolithically integrated optical filters,” in Image Sensors and Imaging Systems (Society for Imaging Science and Technology, 2016), pp. 1–7.

Gonzalez, P.

P. Gonzalez, J. Pichette, B. Vereecke, B. Masschelein, L. Krasovitski, L. Bikov, and A. Lambrechts, “An extremely compact and high-speed line-scan hyperspectral imager covering the SWIR range,” Proc. SPIE 10656, 106560L (2018).
[Crossref]

B. Geelen, C. Blanch, P. Gonzalez, N. Tack, and A. Lambrechts, “A tiny VIS-NIR snapshot multispectral camera,” Proc. SPIE 9374, 937414 (2015).
[Crossref]

Gowen, A.

A. Gowen, C. O’Donnell, P. Cullen, G. Downey, and J. Frias, “Hyperspectral imaging—an emerging process analytical tool for food quality and safety control,” Trends Food Sci. Technol. 18, 590–598 (2007).
[Crossref]

Haavardsholm, T.

Hagen, N.

N. Hagen and M. W. Kudenov, “Review of snapshot spectral imaging technologies,” Opt. Eng. 52, 090901 (2013).
[Crossref]

Haspeslagh, L.

N. Tack, A. Lambrechts, P. Soussan, and L. Haspeslagh, “A compact, high-speed, and low-cost hyperspectral imager,” Proc. SPIE 8266, 82660Q (2012).
[Crossref]

Hedborg, J.

I. G. E. Renhorn, D. Bergström, J. Hedborg, D. Letalick, and S. Möller, “High spatial resolution hyperspectral camera based on a linear variable filter,” Opt. Eng. 55, 114105 (2016).
[Crossref]

Hernandez, G.

Janesick, J. R.

J. R. Janesick, Photon Transfer (SPIE, 2007).

Jayapala, M.

P. Agrawal, K. Tack, B. Geelen, B. Masschelein, P. Mateo, A. Moran, A. Lambrechts, and M. Jayapala, “Characterization of VNIR hyperspectral sensors with monolithically integrated optical filters,” in Image Sensors and Imaging Systems (Society for Imaging Science and Technology, 2016), pp. 1–7.

Kåsen, I.

Kingslake, R.

R. Kingslake, Optics in Photography, Vol. 6 of SPIE Press Monograph (SPIE, 1992).

Krasovitski, L.

P. Gonzalez, J. Pichette, B. Vereecke, B. Masschelein, L. Krasovitski, L. Bikov, and A. Lambrechts, “An extremely compact and high-speed line-scan hyperspectral imager covering the SWIR range,” Proc. SPIE 10656, 106560L (2018).
[Crossref]

Kudenov, M. W.

N. Hagen and M. W. Kudenov, “Review of snapshot spectral imaging technologies,” Opt. Eng. 52, 090901 (2013).
[Crossref]

Lambrechts, A.

P. Gonzalez, J. Pichette, B. Vereecke, B. Masschelein, L. Krasovitski, L. Bikov, and A. Lambrechts, “An extremely compact and high-speed line-scan hyperspectral imager covering the SWIR range,” Proc. SPIE 10656, 106560L (2018).
[Crossref]

J. Pichette, W. Charle, and A. Lambrechts, “Fast and compact internal scanning CMOS-based hyperspectral camera: the Snapscan,” Proc. SPIE 10110, 1011014 (2017).
[Crossref]

B. Geelen, C. Blanch, P. Gonzalez, N. Tack, and A. Lambrechts, “A tiny VIS-NIR snapshot multispectral camera,” Proc. SPIE 9374, 937414 (2015).
[Crossref]

B. Geelen, N. Tack, and A. Lambrechts, “A compact snapshot multispectral imager with a monolithically integrated per-pixel filter mosaic,” Proc. SPIE 8974, 89740L (2014).
[Crossref]

N. Tack, A. Lambrechts, P. Soussan, and L. Haspeslagh, “A compact, high-speed, and low-cost hyperspectral imager,” Proc. SPIE 8266, 82660Q (2012).
[Crossref]

P. Agrawal, K. Tack, B. Geelen, B. Masschelein, P. Mateo, A. Moran, A. Lambrechts, and M. Jayapala, “Characterization of VNIR hyperspectral sensors with monolithically integrated optical filters,” in Image Sensors and Imaging Systems (Society for Imaging Science and Technology, 2016), pp. 1–7.

Letalick, D.

I. G. E. Renhorn, D. Bergström, J. Hedborg, D. Letalick, and S. Möller, “High spatial resolution hyperspectral camera based on a linear variable filter,” Opt. Eng. 55, 114105 (2016).
[Crossref]

Lu, G.

G. Lu and B. Fei, “Medical hyperspectral imaging: a review,” J. Biomed. Opt. 19, 010901 (2014).
[Crossref]

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters (CRC Press, 2001).

Masschelein, B.

P. Gonzalez, J. Pichette, B. Vereecke, B. Masschelein, L. Krasovitski, L. Bikov, and A. Lambrechts, “An extremely compact and high-speed line-scan hyperspectral imager covering the SWIR range,” Proc. SPIE 10656, 106560L (2018).
[Crossref]

P. Agrawal, K. Tack, B. Geelen, B. Masschelein, P. Mateo, A. Moran, A. Lambrechts, and M. Jayapala, “Characterization of VNIR hyperspectral sensors with monolithically integrated optical filters,” in Image Sensors and Imaging Systems (Society for Imaging Science and Technology, 2016), pp. 1–7.

Mateo, P.

P. Agrawal, K. Tack, B. Geelen, B. Masschelein, P. Mateo, A. Moran, A. Lambrechts, and M. Jayapala, “Characterization of VNIR hyperspectral sensors with monolithically integrated optical filters,” in Image Sensors and Imaging Systems (Society for Imaging Science and Technology, 2016), pp. 1–7.

Möller, S.

I. G. E. Renhorn, D. Bergström, J. Hedborg, D. Letalick, and S. Möller, “High spatial resolution hyperspectral camera based on a linear variable filter,” Opt. Eng. 55, 114105 (2016).
[Crossref]

Moran, A.

P. Agrawal, K. Tack, B. Geelen, B. Masschelein, P. Mateo, A. Moran, A. Lambrechts, and M. Jayapala, “Characterization of VNIR hyperspectral sensors with monolithically integrated optical filters,” in Image Sensors and Imaging Systems (Society for Imaging Science and Technology, 2016), pp. 1–7.

Nicolas, S.

O’Donnell, C.

A. Gowen, C. O’Donnell, P. Cullen, G. Downey, and J. Frias, “Hyperspectral imaging—an emerging process analytical tool for food quality and safety control,” Trends Food Sci. Technol. 18, 590–598 (2007).
[Crossref]

Opsahl, T.

Pichette, J.

P. Gonzalez, J. Pichette, B. Vereecke, B. Masschelein, L. Krasovitski, L. Bikov, and A. Lambrechts, “An extremely compact and high-speed line-scan hyperspectral imager covering the SWIR range,” Proc. SPIE 10656, 106560L (2018).
[Crossref]

J. Pichette, W. Charle, and A. Lambrechts, “Fast and compact internal scanning CMOS-based hyperspectral camera: the Snapscan,” Proc. SPIE 10110, 1011014 (2017).
[Crossref]

Pidgeon, C. R.

Renhorn, I. G. E.

I. G. E. Renhorn, D. Bergström, J. Hedborg, D. Letalick, and S. Möller, “High spatial resolution hyperspectral camera based on a linear variable filter,” Opt. Eng. 55, 114105 (2016).
[Crossref]

Rognmo, A.

Shippert, P.

P. Shippert, “Why use hyperspectral imagery?” Earth Sci. 70, 377–379 (2004).

Skauli, T.

Smith, R.

P. Thenkabail, R. Smith, and E. De Pauw, “Hyperspectral vegetation indices and their relationships with agricultural crop characteristics,” Remote Sens. Environ. 71, 158–182 (2000).
[Crossref]

Smith, S. D.

Soussan, P.

N. Tack, A. Lambrechts, P. Soussan, and L. Haspeslagh, “A compact, high-speed, and low-cost hyperspectral imager,” Proc. SPIE 8266, 82660Q (2012).
[Crossref]

Tack, K.

P. Agrawal, K. Tack, B. Geelen, B. Masschelein, P. Mateo, A. Moran, A. Lambrechts, and M. Jayapala, “Characterization of VNIR hyperspectral sensors with monolithically integrated optical filters,” in Image Sensors and Imaging Systems (Society for Imaging Science and Technology, 2016), pp. 1–7.

Tack, N.

B. Geelen, C. Blanch, P. Gonzalez, N. Tack, and A. Lambrechts, “A tiny VIS-NIR snapshot multispectral camera,” Proc. SPIE 9374, 937414 (2015).
[Crossref]

B. Geelen, N. Tack, and A. Lambrechts, “A compact snapshot multispectral imager with a monolithically integrated per-pixel filter mosaic,” Proc. SPIE 8974, 89740L (2014).
[Crossref]

N. Tack, A. Lambrechts, P. Soussan, and L. Haspeslagh, “A compact, high-speed, and low-cost hyperspectral imager,” Proc. SPIE 8266, 82660Q (2012).
[Crossref]

Thenkabail, P.

P. Thenkabail, R. Smith, and E. De Pauw, “Hyperspectral vegetation indices and their relationships with agricultural crop characteristics,” Remote Sens. Environ. 71, 158–182 (2000).
[Crossref]

Torkildsen, H. E.

Vereecke, B.

P. Gonzalez, J. Pichette, B. Vereecke, B. Masschelein, L. Krasovitski, L. Bikov, and A. Lambrechts, “An extremely compact and high-speed line-scan hyperspectral imager covering the SWIR range,” Proc. SPIE 10656, 106560L (2018).
[Crossref]

Wilksch, P. A.

Appl. Opt. (4)

Earth Sci. (1)

P. Shippert, “Why use hyperspectral imagery?” Earth Sci. 70, 377–379 (2004).

J. Biomed. Opt. (1)

G. Lu and B. Fei, “Medical hyperspectral imaging: a review,” J. Biomed. Opt. 19, 010901 (2014).
[Crossref]

J. Opt. Soc. Am. (1)

Opt. Eng. (2)

I. G. E. Renhorn, D. Bergström, J. Hedborg, D. Letalick, and S. Möller, “High spatial resolution hyperspectral camera based on a linear variable filter,” Opt. Eng. 55, 114105 (2016).
[Crossref]

N. Hagen and M. W. Kudenov, “Review of snapshot spectral imaging technologies,” Opt. Eng. 52, 090901 (2013).
[Crossref]

Proc. SPIE (6)

N. Gat, “Imaging spectroscopy using tunable filters: a review,” Proc. SPIE 4056, 50–64 (2000).
[Crossref]

N. Tack, A. Lambrechts, P. Soussan, and L. Haspeslagh, “A compact, high-speed, and low-cost hyperspectral imager,” Proc. SPIE 8266, 82660Q (2012).
[Crossref]

B. Geelen, N. Tack, and A. Lambrechts, “A compact snapshot multispectral imager with a monolithically integrated per-pixel filter mosaic,” Proc. SPIE 8974, 89740L (2014).
[Crossref]

B. Geelen, C. Blanch, P. Gonzalez, N. Tack, and A. Lambrechts, “A tiny VIS-NIR snapshot multispectral camera,” Proc. SPIE 9374, 937414 (2015).
[Crossref]

J. Pichette, W. Charle, and A. Lambrechts, “Fast and compact internal scanning CMOS-based hyperspectral camera: the Snapscan,” Proc. SPIE 10110, 1011014 (2017).
[Crossref]

P. Gonzalez, J. Pichette, B. Vereecke, B. Masschelein, L. Krasovitski, L. Bikov, and A. Lambrechts, “An extremely compact and high-speed line-scan hyperspectral imager covering the SWIR range,” Proc. SPIE 10656, 106560L (2018).
[Crossref]

Remote Sens. Environ. (1)

P. Thenkabail, R. Smith, and E. De Pauw, “Hyperspectral vegetation indices and their relationships with agricultural crop characteristics,” Remote Sens. Environ. 71, 158–182 (2000).
[Crossref]

Trends Food Sci. Technol. (1)

A. Gowen, C. O’Donnell, P. Cullen, G. Downey, and J. Frias, “Hyperspectral imaging—an emerging process analytical tool for food quality and safety control,” Trends Food Sci. Technol. 18, 590–598 (2007).
[Crossref]

Other (5)

P. Agrawal, K. Tack, B. Geelen, B. Masschelein, P. Mateo, A. Moran, A. Lambrechts, and M. Jayapala, “Characterization of VNIR hyperspectral sensors with monolithically integrated optical filters,” in Image Sensors and Imaging Systems (Society for Imaging Science and Technology, 2016), pp. 1–7.

R. Kingslake, Optics in Photography, Vol. 6 of SPIE Press Monograph (SPIE, 1992).

J. R. Janesick, Photon Transfer (SPIE, 2007).

H. A. Macleod, Thin-Film Optical Filters (CRC Press, 2001).

Software Spectra, “Thin film design software for Windows, Version 3.5.15,” Software Spectra, 2009, http://sspectra.com/files/win_demo/manual.pdf .

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Figures (15)

Fig. 1.
Fig. 1. Schematic representation of how the aperture focuses light on the spectral imaging sensor with integrated thin-film optical filters.
Fig. 2.
Fig. 2. Basic Fabry–Perot etalon. Two parallel near-perfect mirrors are separated by a material of refractive index ns and thickness t.
Fig. 3.
Fig. 3. Decomposition in contributions with equal angles of incidence. The weight of a contribution is measured by the length of the blue arc within the aperture.
Fig. 4.
Fig. 4. Decomposition of the light cone from the aperture in contributions of equal angle of incidence ϕ. The weight of each contribution is the infinitesimal area dA. Here d is the distance of the pixel from the optical axis. (a) Top view; (b) cut section view.
Fig. 5.
Fig. 5. Shape of the kernel [Eq. (15)] for different ratios of θcone and θCRA. The mean value of each kernel according to Eq. (17) is marked with a circle.
Fig. 6.
Fig. 6. Transmittance of a thin-film optical filter simulated with TFCalc for different apertures compared to applying the convolution kernel to the transmittance at collimated conditions. In this example neff=1.75.
Fig. 7.
Fig. 7. Experimental setups. (a) VNIR Snapscan with color filter; (b) SWIR Snapscan with reflectance target.
Fig. 8.
Fig. 8. Experiment 1 (SWIR). The same sample is measured at different f-numbers. The shift becomes larger for smaller f-numbers, causing the spread of shifts in the graph. The shifts are corrected using Eq. (23). The working f-number f#,W [Eq. (21)] is used for correction; the legend shows the f-number f# as marked on the lens. (a) Uncorrected; (b) central wavelength, corrected.
Fig. 9.
Fig. 9. Experiment 2 (VNIR). The transmittance of the color filter is measured at different off-axis distances at f/8, which is a high f-number. The shifts are corrected using Eq. (24). (a) Uncorrected; (b) central wavelength, corrected.
Fig. 10.
Fig. 10. Image of the scene from Fig. 7(a) at the 721 nm band before and after central wavelength correction (Fig. 9). The uniformity of the scene is significantly improved. The corners are outside the image circle of the lens. (a) Before correction; (b) after correction.
Fig. 11.
Fig. 11. Experiment 3 (SWIR). The reflectance of the sample is measured at different f-numbers at an off-axis position with θCRA10.2°. The shifts are corrected using Eq. (25). As a reference, the measurement at f/5.6 at zero chief ray angle is used. (a) Uncorrected; (b) central wavelength, corrected.
Fig. 12.
Fig. 12. Experiment 3 simulated (SWIR). The offset and loss of detail are very similar to those observed in real measurements (Fig. 10). (a) Uncorrected; (b) central wavelength corrected.
Fig. 13.
Fig. 13. Approximation of the mean value compared to the numerical result (neff=1.7). For large chief ray angle, the shift is slightly overestimated.
Fig. 14.
Fig. 14. Approximation of the mean value compared to the numerical result (neff=1.7). For large chief ray angle, the estimated shift becomes too large.
Fig. 15.
Fig. 15. For very large values of θcone and θCRA, the higher-order approximation [Eq. (C1)] fits the numerical solution very well, while the more simple approximation deteriorates (neff=1.7).

Tables (2)

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Table 1. List of the Main Symbols and Their Meaning

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Table 2. Properties of the Experimental Setupa

Equations (49)

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DN=λminλmaxT(λ)·L(λ)dλ.
DN=λminλmax(Kθcone,θCRA*T)(λ)·L(λ)dλ,
Δ(ϕ)=λcwl(1cosϕs)λcwl(1cosϕns),
Δ(ϕ)λcwl(1cosϕneff).
Tϕ(λ)T(λ)*δ(λλcwl(1cosϕneff)).
T^θcone,θCRA(λ)=ApertureTϕ(λ)dAAperturedA,
T^θcone,θCRA(λ)=T(λ)*Apertureδ(λΔ(ϕ))dAAperturedA
=T(λ)*Kθcone,θCRA(λ).
dA=2γ(r)rdr,
γ(r)=Re(arccosd2R2+r22dr).
dA=2x2γ(ϕ)tanϕcos2ϕdϕ,
γ(ϕ)=Re(arccostan2θCRAtan2θcone+tan2ϕ2tanθCRAtanϕ).
Kθcone,θCRA(λ)=ϕminϕmax2x2γ(ϕ)tanϕcos2ϕδ(λΔ(ϕ))dϕπR2,
ϕmin={0if  θCRA<θconearctan(tanθCRAtanθcone)if  θCRAθcone,ϕmax=arctan(tanθCRA+tanθcone),
Kθcone,θCRA(λ)2neff2λcwlγ(neff2λλcwl)πtan2θcone,
λ¯θcone,θCRA=λminλmaxλKθcone,θCRA(λ)dλλminλmaxKθcone,θCRA(λ)dλ=(normalized)λminλmaxλKθcone,θCRA(λ)dλ.
λ¯θcone,θCRAλcwlneff2(θcone24+θCRA22)for  θcone,θCRA0.
λcwlnew=λcwlλ¯θcone,θCRAλcwl(1θcone24neff2θCRA22neff2).
λcwlnewλcwl(1116f#,W2neff2d22x2neff2).
reflectance=DNsampleDNdarkDNwhiteDNdark.
f#,W(1+mP)f#,
neff=nL1nLnH+(nLnH)2,
λcwlnewλcwl(1116f#,W2neff2).
λcwlnewλcwl(1θCRA22neff2)=λcwl(1d22x2neff2),
λcwlnewλcwl(1116f#,W2neff2d22x2neff2).
Kθcone,θCRA(λ)=ϕminϕmax2x2γ(ϕ)tanϕcos2ϕδ(λΔ(ϕ))dϕπR2
=ϕminϕmax2x2γ(ϕ)tanϕcos2ϕδ(λΔ(ϕ))dϕπx2tan2θcone
=ϕminϕmax2γ(ϕ)tanϕcos2ϕδ(λΔ(ϕ))dϕπtan2θcone.
Kθcone,θCRA(λ)=2γ(ϕ)tanϕcos2ϕδ(λΔ(ϕ))Π(ϕ)dϕπtan2θcone,
Δ(ϕ)=λcwl(1cosϕs)
=λcwl(1cos(arcsinsinϕneff))
=λcwl(11sin2ϕneff2),
Δ1(λ)=arcsin(neffλλcwl(2λλcwl))
=neff2λλcwl+O(λ3/2λcwl3/2),λλcwl0.
du=λcwlcos(ϕ)sin(ϕ)neff21sin(ϕ)2neff2dϕ.
Kθcone,θCRA(λ)=2γ(Δ1(λ))tan(Δ1(λ))cos2(Δ1(λ))πtan2θcone·neff21sin2(Δ1(λ))neff2λcwlsin(Δ1(λ))cos(Δ1(λ))Π(Δ1(λ)).
Kθcone,θCRA(λ)=g(λ)γ(Δ1(λ))πtan2θconeΠ(Δ1(λ)),
g(λ)=2neff2λcwl·(1λλcwl12neff2λλcwl+neff2λ2λcwl2)
=2neff2λcwl+O(λλcwl),for  λλcwl0,
Kθcone,θCRA(λ)2neff2λcwl·γ(neff2λλcwl)πtan2θcone.
E(θcone,θCRA)=λ¯θcone,θCRA=λminλmaxλKθcone,θCRA(λ)dλ.
E(θcone,θCRA)E(0,0)+EθCRA|(0,0)θCRA+Eθcone|(0,0)θcone+122EθCRA2|(0,0)θCRA2+122Eθcone2|(0,0)θcone2+2EθconeθCRA|(0,0)θconeθCRA.
λminλmaxKθcone,θCRA(λ)dλ=1,θcone,θCRA>0.
0λmaxλg(λ)tan2θconedλ.
E(θcone,0)=01vλmax2g(vλmax)tan2θconedv.
E(θcone,0)=λcwl(θcone24neff2+(14neff2)24neff4θcone4+O(θcone6)),for  θcone0.
E(0,θCRA)=λcwl(θCRA22neff2+(34neff2)24neff4θCRA4+O(θCRA6)),for  θCRA0.
E(θcone,θCRA)λcwl(θcone24neff2+θCRA22neff2).
E(θcone,θCRA)λcwl(θcone24neff2+θCRA22neff2θCRA2θcone2neff2).

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