Abstract

Si:As blocked impurity band detectors have been partially deprocessed and measured by Fourier transform spectroscopy to determine their transmittance and reflectance at cryogenic temperatures over the wavelength range 2μm to 40μm. A method is presented by which the propagation constants can be extracted from an inversion of the transmittance and reflectance data. The effective propagation constants for the active layer from 2μm to 20μm were calculated as well as the absorption cross section of arsenic in silicon, which agrees well with previous results from the literature. The infrared absorptance of the full detector was determined, and the analytical method also provides an estimate of absorption in the active layer alone. Infrared absorptance of the active layer is compared to the quantum yield measured by photoelectric means on similar detectors. The optical methods outlined here, in conjunction with standard electronic measurements, could be used to predict the performance of such detectors from measurements of the blanket films from which they are to be fabricated.

© 2011 Optical Society of America

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References

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  1. M. D. Petroff and M. G. Stapelbroek, “Blocked impurity band detectors,” U.S. patent 4,568,960 (February 4, 1986).
  2. R. D. Campbell, “Characterization of the Si:As blocked impurity band (BIB) detector in Keck’s long wavelength spectrometer (LWS),” Exp. Astron. 14, 57–60 (2002).
    [CrossRef]
  3. A. K. Mainzer, H. Hogue, M. Stapelbroek, D. Molyneux, J. Hong, M. Werner, M. Ressler, and E. Young, “Characterization of a megapixel mid-infrared array for high background applications,” Proc. SPIE 7021, 70210T-1–70210T-6 (2008).
  4. B. L. Cardozo, E. E. Haller, L. A. Reichertz, and J. W. Beeman, “Far infrared absorption in GaAs: Te liquid phase epitaxial films,” Appl. Phys. Lett. 83 (19), 3990–3992 (2003).
    [CrossRef]
  5. A. C. Carter, S. R. Lorentz, T. M. Jung, B. J. Klemme, and R. U. Datla, “NIST Facility for spectral calibration of detectors: calibration of arsenic doped silicon blocked impurity band detectors,” Proc. SPIE 4028, 420–425 (2000).
    [CrossRef]
  6. T. M. Jung, A. C. Carter, S. R. Lorentz, and R. U. Datla, “NIST-BMDO Transfer Radiometer (BXR),” Proc. SPIE 4028, 404–410 (2000).
    [CrossRef]
  7. A. C. Carter, S. I. Woods, S. M. Carr, T. M. Jung, and R. U. Datla, “Absolute cryogenic radiometer and solid state trap detectors for IR power scales down to 1 pW with 0.1% uncertainty,” Metrologia 46 (4), S146–S150 (2009).
    [CrossRef]
  8. E. J. Iglesias, A. W. Smith, and S. G. Kaplan, “A sensitive, spatially uniform photodetector for broadband infrared spectrophotometry,” Appl. Opt. 47 (13), 2430–2436 (2008).
    [CrossRef] [PubMed]
  9. M. D. Petroff and M. G. Stapelbroek, “Responsivity and noise models of blocked impurity band detectors. IRIA-IRIS,” in Proceedings of the Meeting of the Specialty Group on Infrared Detectors, Volume  2 (ERIM, 1984).
  10. F. Szmulowicz, F. L. Madarsz, and J. Diller, “Temperature dependence for the figures of merit for blocked impurity band detectors,” J. Appl. Phys. 63 (11), 5583–5588 (1988).
    [CrossRef]
  11. J. Geist, “Infrared absorption cross section of arsenic in silicon in the impurity band region of concentration,” Appl. Opt. 28, 1193–1199 (1989).
    [CrossRef] [PubMed]
  12. Reference is made to commercial laboratories and products to adequately specify the experimental procedures involved. Such identification does not imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it imply that these laboratories or products are the best for the purpose specified.
  13. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975), pp. 323–329.
  14. D. F. Edwards, “Silicon (Si),” in Handbook of Optical Constants of Solids, E.D.Palik, ed. (Academic, 1985) pp. 547–569.
  15. M. D. Petroff and M. G. Stapelbroek, “Spectral response, gain, and noise models for IBC detectors,” in IRIA-IRIS, Proceedings of the Meeting of the Specialty Group on Infrared Detectors (ERIM, 1985).
  16. Jon Geist (NIST), private communication.

2009 (1)

A. C. Carter, S. I. Woods, S. M. Carr, T. M. Jung, and R. U. Datla, “Absolute cryogenic radiometer and solid state trap detectors for IR power scales down to 1 pW with 0.1% uncertainty,” Metrologia 46 (4), S146–S150 (2009).
[CrossRef]

2008 (2)

E. J. Iglesias, A. W. Smith, and S. G. Kaplan, “A sensitive, spatially uniform photodetector for broadband infrared spectrophotometry,” Appl. Opt. 47 (13), 2430–2436 (2008).
[CrossRef] [PubMed]

A. K. Mainzer, H. Hogue, M. Stapelbroek, D. Molyneux, J. Hong, M. Werner, M. Ressler, and E. Young, “Characterization of a megapixel mid-infrared array for high background applications,” Proc. SPIE 7021, 70210T-1–70210T-6 (2008).

2003 (1)

B. L. Cardozo, E. E. Haller, L. A. Reichertz, and J. W. Beeman, “Far infrared absorption in GaAs: Te liquid phase epitaxial films,” Appl. Phys. Lett. 83 (19), 3990–3992 (2003).
[CrossRef]

2002 (1)

R. D. Campbell, “Characterization of the Si:As blocked impurity band (BIB) detector in Keck’s long wavelength spectrometer (LWS),” Exp. Astron. 14, 57–60 (2002).
[CrossRef]

2000 (2)

A. C. Carter, S. R. Lorentz, T. M. Jung, B. J. Klemme, and R. U. Datla, “NIST Facility for spectral calibration of detectors: calibration of arsenic doped silicon blocked impurity band detectors,” Proc. SPIE 4028, 420–425 (2000).
[CrossRef]

T. M. Jung, A. C. Carter, S. R. Lorentz, and R. U. Datla, “NIST-BMDO Transfer Radiometer (BXR),” Proc. SPIE 4028, 404–410 (2000).
[CrossRef]

1989 (1)

1988 (1)

F. Szmulowicz, F. L. Madarsz, and J. Diller, “Temperature dependence for the figures of merit for blocked impurity band detectors,” J. Appl. Phys. 63 (11), 5583–5588 (1988).
[CrossRef]

Beeman, J. W.

B. L. Cardozo, E. E. Haller, L. A. Reichertz, and J. W. Beeman, “Far infrared absorption in GaAs: Te liquid phase epitaxial films,” Appl. Phys. Lett. 83 (19), 3990–3992 (2003).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975), pp. 323–329.

Campbell, R. D.

R. D. Campbell, “Characterization of the Si:As blocked impurity band (BIB) detector in Keck’s long wavelength spectrometer (LWS),” Exp. Astron. 14, 57–60 (2002).
[CrossRef]

Cardozo, B. L.

B. L. Cardozo, E. E. Haller, L. A. Reichertz, and J. W. Beeman, “Far infrared absorption in GaAs: Te liquid phase epitaxial films,” Appl. Phys. Lett. 83 (19), 3990–3992 (2003).
[CrossRef]

Carr, S. M.

A. C. Carter, S. I. Woods, S. M. Carr, T. M. Jung, and R. U. Datla, “Absolute cryogenic radiometer and solid state trap detectors for IR power scales down to 1 pW with 0.1% uncertainty,” Metrologia 46 (4), S146–S150 (2009).
[CrossRef]

Carter, A. C.

A. C. Carter, S. I. Woods, S. M. Carr, T. M. Jung, and R. U. Datla, “Absolute cryogenic radiometer and solid state trap detectors for IR power scales down to 1 pW with 0.1% uncertainty,” Metrologia 46 (4), S146–S150 (2009).
[CrossRef]

T. M. Jung, A. C. Carter, S. R. Lorentz, and R. U. Datla, “NIST-BMDO Transfer Radiometer (BXR),” Proc. SPIE 4028, 404–410 (2000).
[CrossRef]

A. C. Carter, S. R. Lorentz, T. M. Jung, B. J. Klemme, and R. U. Datla, “NIST Facility for spectral calibration of detectors: calibration of arsenic doped silicon blocked impurity band detectors,” Proc. SPIE 4028, 420–425 (2000).
[CrossRef]

Datla, R. U.

A. C. Carter, S. I. Woods, S. M. Carr, T. M. Jung, and R. U. Datla, “Absolute cryogenic radiometer and solid state trap detectors for IR power scales down to 1 pW with 0.1% uncertainty,” Metrologia 46 (4), S146–S150 (2009).
[CrossRef]

A. C. Carter, S. R. Lorentz, T. M. Jung, B. J. Klemme, and R. U. Datla, “NIST Facility for spectral calibration of detectors: calibration of arsenic doped silicon blocked impurity band detectors,” Proc. SPIE 4028, 420–425 (2000).
[CrossRef]

T. M. Jung, A. C. Carter, S. R. Lorentz, and R. U. Datla, “NIST-BMDO Transfer Radiometer (BXR),” Proc. SPIE 4028, 404–410 (2000).
[CrossRef]

Diller, J.

F. Szmulowicz, F. L. Madarsz, and J. Diller, “Temperature dependence for the figures of merit for blocked impurity band detectors,” J. Appl. Phys. 63 (11), 5583–5588 (1988).
[CrossRef]

Edwards, D. F.

D. F. Edwards, “Silicon (Si),” in Handbook of Optical Constants of Solids, E.D.Palik, ed. (Academic, 1985) pp. 547–569.

Geist, J.

Haller, E. E.

B. L. Cardozo, E. E. Haller, L. A. Reichertz, and J. W. Beeman, “Far infrared absorption in GaAs: Te liquid phase epitaxial films,” Appl. Phys. Lett. 83 (19), 3990–3992 (2003).
[CrossRef]

Hogue, H.

A. K. Mainzer, H. Hogue, M. Stapelbroek, D. Molyneux, J. Hong, M. Werner, M. Ressler, and E. Young, “Characterization of a megapixel mid-infrared array for high background applications,” Proc. SPIE 7021, 70210T-1–70210T-6 (2008).

Hong, J.

A. K. Mainzer, H. Hogue, M. Stapelbroek, D. Molyneux, J. Hong, M. Werner, M. Ressler, and E. Young, “Characterization of a megapixel mid-infrared array for high background applications,” Proc. SPIE 7021, 70210T-1–70210T-6 (2008).

Iglesias, E. J.

Jung, T. M.

A. C. Carter, S. I. Woods, S. M. Carr, T. M. Jung, and R. U. Datla, “Absolute cryogenic radiometer and solid state trap detectors for IR power scales down to 1 pW with 0.1% uncertainty,” Metrologia 46 (4), S146–S150 (2009).
[CrossRef]

T. M. Jung, A. C. Carter, S. R. Lorentz, and R. U. Datla, “NIST-BMDO Transfer Radiometer (BXR),” Proc. SPIE 4028, 404–410 (2000).
[CrossRef]

A. C. Carter, S. R. Lorentz, T. M. Jung, B. J. Klemme, and R. U. Datla, “NIST Facility for spectral calibration of detectors: calibration of arsenic doped silicon blocked impurity band detectors,” Proc. SPIE 4028, 420–425 (2000).
[CrossRef]

Kaplan, S. G.

Klemme, B. J.

A. C. Carter, S. R. Lorentz, T. M. Jung, B. J. Klemme, and R. U. Datla, “NIST Facility for spectral calibration of detectors: calibration of arsenic doped silicon blocked impurity band detectors,” Proc. SPIE 4028, 420–425 (2000).
[CrossRef]

Lorentz, S. R.

A. C. Carter, S. R. Lorentz, T. M. Jung, B. J. Klemme, and R. U. Datla, “NIST Facility for spectral calibration of detectors: calibration of arsenic doped silicon blocked impurity band detectors,” Proc. SPIE 4028, 420–425 (2000).
[CrossRef]

T. M. Jung, A. C. Carter, S. R. Lorentz, and R. U. Datla, “NIST-BMDO Transfer Radiometer (BXR),” Proc. SPIE 4028, 404–410 (2000).
[CrossRef]

Madarsz, F. L.

F. Szmulowicz, F. L. Madarsz, and J. Diller, “Temperature dependence for the figures of merit for blocked impurity band detectors,” J. Appl. Phys. 63 (11), 5583–5588 (1988).
[CrossRef]

Mainzer, A. K.

A. K. Mainzer, H. Hogue, M. Stapelbroek, D. Molyneux, J. Hong, M. Werner, M. Ressler, and E. Young, “Characterization of a megapixel mid-infrared array for high background applications,” Proc. SPIE 7021, 70210T-1–70210T-6 (2008).

Molyneux, D.

A. K. Mainzer, H. Hogue, M. Stapelbroek, D. Molyneux, J. Hong, M. Werner, M. Ressler, and E. Young, “Characterization of a megapixel mid-infrared array for high background applications,” Proc. SPIE 7021, 70210T-1–70210T-6 (2008).

Petroff, M. D.

M. D. Petroff and M. G. Stapelbroek, “Responsivity and noise models of blocked impurity band detectors. IRIA-IRIS,” in Proceedings of the Meeting of the Specialty Group on Infrared Detectors, Volume  2 (ERIM, 1984).

M. D. Petroff and M. G. Stapelbroek, “Blocked impurity band detectors,” U.S. patent 4,568,960 (February 4, 1986).

M. D. Petroff and M. G. Stapelbroek, “Spectral response, gain, and noise models for IBC detectors,” in IRIA-IRIS, Proceedings of the Meeting of the Specialty Group on Infrared Detectors (ERIM, 1985).

Reichertz, L. A.

B. L. Cardozo, E. E. Haller, L. A. Reichertz, and J. W. Beeman, “Far infrared absorption in GaAs: Te liquid phase epitaxial films,” Appl. Phys. Lett. 83 (19), 3990–3992 (2003).
[CrossRef]

Ressler, M.

A. K. Mainzer, H. Hogue, M. Stapelbroek, D. Molyneux, J. Hong, M. Werner, M. Ressler, and E. Young, “Characterization of a megapixel mid-infrared array for high background applications,” Proc. SPIE 7021, 70210T-1–70210T-6 (2008).

Smith, A. W.

Stapelbroek, M.

A. K. Mainzer, H. Hogue, M. Stapelbroek, D. Molyneux, J. Hong, M. Werner, M. Ressler, and E. Young, “Characterization of a megapixel mid-infrared array for high background applications,” Proc. SPIE 7021, 70210T-1–70210T-6 (2008).

Stapelbroek, M. G.

M. D. Petroff and M. G. Stapelbroek, “Blocked impurity band detectors,” U.S. patent 4,568,960 (February 4, 1986).

M. D. Petroff and M. G. Stapelbroek, “Responsivity and noise models of blocked impurity band detectors. IRIA-IRIS,” in Proceedings of the Meeting of the Specialty Group on Infrared Detectors, Volume  2 (ERIM, 1984).

M. D. Petroff and M. G. Stapelbroek, “Spectral response, gain, and noise models for IBC detectors,” in IRIA-IRIS, Proceedings of the Meeting of the Specialty Group on Infrared Detectors (ERIM, 1985).

Szmulowicz, F.

F. Szmulowicz, F. L. Madarsz, and J. Diller, “Temperature dependence for the figures of merit for blocked impurity band detectors,” J. Appl. Phys. 63 (11), 5583–5588 (1988).
[CrossRef]

Werner, M.

A. K. Mainzer, H. Hogue, M. Stapelbroek, D. Molyneux, J. Hong, M. Werner, M. Ressler, and E. Young, “Characterization of a megapixel mid-infrared array for high background applications,” Proc. SPIE 7021, 70210T-1–70210T-6 (2008).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975), pp. 323–329.

Woods, S. I.

A. C. Carter, S. I. Woods, S. M. Carr, T. M. Jung, and R. U. Datla, “Absolute cryogenic radiometer and solid state trap detectors for IR power scales down to 1 pW with 0.1% uncertainty,” Metrologia 46 (4), S146–S150 (2009).
[CrossRef]

Young, E.

A. K. Mainzer, H. Hogue, M. Stapelbroek, D. Molyneux, J. Hong, M. Werner, M. Ressler, and E. Young, “Characterization of a megapixel mid-infrared array for high background applications,” Proc. SPIE 7021, 70210T-1–70210T-6 (2008).

Appl. Opt. (2)

Appl. Phys. Lett. (1)

B. L. Cardozo, E. E. Haller, L. A. Reichertz, and J. W. Beeman, “Far infrared absorption in GaAs: Te liquid phase epitaxial films,” Appl. Phys. Lett. 83 (19), 3990–3992 (2003).
[CrossRef]

Exp. Astron. (1)

R. D. Campbell, “Characterization of the Si:As blocked impurity band (BIB) detector in Keck’s long wavelength spectrometer (LWS),” Exp. Astron. 14, 57–60 (2002).
[CrossRef]

J. Appl. Phys. (1)

F. Szmulowicz, F. L. Madarsz, and J. Diller, “Temperature dependence for the figures of merit for blocked impurity band detectors,” J. Appl. Phys. 63 (11), 5583–5588 (1988).
[CrossRef]

Metrologia (1)

A. C. Carter, S. I. Woods, S. M. Carr, T. M. Jung, and R. U. Datla, “Absolute cryogenic radiometer and solid state trap detectors for IR power scales down to 1 pW with 0.1% uncertainty,” Metrologia 46 (4), S146–S150 (2009).
[CrossRef]

Proc. SPIE (3)

A. K. Mainzer, H. Hogue, M. Stapelbroek, D. Molyneux, J. Hong, M. Werner, M. Ressler, and E. Young, “Characterization of a megapixel mid-infrared array for high background applications,” Proc. SPIE 7021, 70210T-1–70210T-6 (2008).

A. C. Carter, S. R. Lorentz, T. M. Jung, B. J. Klemme, and R. U. Datla, “NIST Facility for spectral calibration of detectors: calibration of arsenic doped silicon blocked impurity band detectors,” Proc. SPIE 4028, 420–425 (2000).
[CrossRef]

T. M. Jung, A. C. Carter, S. R. Lorentz, and R. U. Datla, “NIST-BMDO Transfer Radiometer (BXR),” Proc. SPIE 4028, 404–410 (2000).
[CrossRef]

Other (7)

M. D. Petroff and M. G. Stapelbroek, “Blocked impurity band detectors,” U.S. patent 4,568,960 (February 4, 1986).

M. D. Petroff and M. G. Stapelbroek, “Responsivity and noise models of blocked impurity band detectors. IRIA-IRIS,” in Proceedings of the Meeting of the Specialty Group on Infrared Detectors, Volume  2 (ERIM, 1984).

Reference is made to commercial laboratories and products to adequately specify the experimental procedures involved. Such identification does not imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it imply that these laboratories or products are the best for the purpose specified.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975), pp. 323–329.

D. F. Edwards, “Silicon (Si),” in Handbook of Optical Constants of Solids, E.D.Palik, ed. (Academic, 1985) pp. 547–569.

M. D. Petroff and M. G. Stapelbroek, “Spectral response, gain, and noise models for IBC detectors,” in IRIA-IRIS, Proceedings of the Meeting of the Specialty Group on Infrared Detectors (ERIM, 1985).

Jon Geist (NIST), private communication.

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Figures (11)

Fig. 1
Fig. 1

Schematic of the backside-illuminated BIB detectors measured in this study. The silicon substrate is about 350 μm thick and the Si:As IR-AL is about 15 μm thick.

Fig. 2
Fig. 2

SIMS data for a partially deprocessed BIB detector used for FTS measurements. Depth is measured from the frontside of the detector, which is the surface of the AL exposed by deprocessing. The AL arsenic doping is very close to 1.0 × 10 18 cm 3 .

Fig. 3
Fig. 3

Quantum yield data from calibrated photocurrent measurements on a BIB detector nominally identical to the device characterized optically in this study. Data supplied by Rockwell International, manufacturer of the BIB detectors [12].

Fig. 4
Fig. 4

Transmittance and reflectance data for a deprocessed detector at room temperature (black) and low temperature (red). The transmittance data for short and long wavelength ranges are shown at the top, and the reflectance data in the bottom two graphs. Estimates of the incoherent limits of the transmittance and reflectance from the data are represented by the blue datapoints.

Fig. 5
Fig. 5

Cold transmittance and reflectance data over a limited wavenumber range, showing two clear oscillation periods, one of about 4 cm 1 and the other about 95 cm 1 .

Fig. 6
Fig. 6

(a) Schematic used in zeroth order calculations on the FTS data, where reflectance between Si:As and Si is considered negligible. It is assumed there is no absorption at the interfaces; (b) Schematic used in first order calculations on the FTS data, allowing for nonzero reflectance R AL Si .

Fig. 7
Fig. 7

Schematic used in the calculations of the cross section and estimated absorptance for the full BIB detector. The shaded layer on the right side of the figure is the aluminum high reflectance layer used to boost detector absorption.

Fig. 8
Fig. 8

Calculated values for transmissivity ξ AL (top) and reflectance R AL air (bottom). Estimates of the incoherent limits of ξ AL and R AL air from the data are represented by the blue datapoints.

Fig. 9
Fig. 9

Calculated values for the propagation constants k AL (top) and n AL (bottom). The data has not been smoothed, so the significant spectral oscillations associated with etalon seen in the raw data are also evident in these calculated values. Estimates of the incoherent limits of k AL and n AL from the data are represented by the blue datapoints.

Fig. 10
Fig. 10

Comparison of absorption cross section for As in Si published by Geist (blue) and determined in this study from the FTS measurements (black).

Fig. 11
Fig. 11

Comparison of AL absorptance determined from FTS optical measurements (red) and quantum yield determined from photocurrent (blue) measurements. The ratio of quantum yield to AL absorptance, which is approximately the product of collection efficiency and detector gain, is presented in the inset.

Equations (8)

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T tot ( 0 ) = ξ tot ( 1 R AL air ) ( 1 R Si air ) 1 1 ξ tot 2 R AL air R Si air R tot ( 0 ) = R AL air + ξ tot 2 ( 1 R AL air ) 2 R Si air 1 1 ξ tot 2 R AL air R Si air D tot ( 0 ) = ( 1 R AL air ) ( 1 ξ tot ) ( 1 + R Si air ξ tot ) 1 1 ξ tot 2 R AL air R Si air .
ξ tot = ( R tot ( 0 ) 1 ) ( 1 R Si air ) + [ ( R tot ( 0 ) 1 ) 2 ( 1 R Si air ) 2 + 4 ( T tot ( 0 ) ) 2 R Si air ] 1 / 2 2 T tot ( 0 ) R Si air R AL air = T tot ( 0 ) ξ tot ( 1 R Si air ) ξ tot [ T tot ( 0 ) ξ tot R Si air ( 1 R Si air ) ] .
ξ AL = ξ tot ξ Si α AL = ln ( ξ AL ) W AL D AL ( 0 ) = ( 1 R AL air ) ( 1 ξ AL ) ( 1 + R Si air ξ AL ) 1 1 ξ AL 2 R AL air R Si air σ AL = α AL N = ln ( ξ AL ) N W AL = ln ( ξ AL ) 0 W AL N d W ,
k AL = λ 4 π W AL ln ( ξ AL ) n AL = ( R AL air + 1 ) [ ( R AL air + 1 ) 2 ( 1 + k AL 2 ) ( R AL air 1 ) 2 ] 1 / 2 ( R AL air 1 ) .
D tot = ( 1 R Si air ) 1 1 ξ Si 2 R sub R Si air [ ( 1 ξ Si ) ( 1 + R sub ξ Si ) + ξ Si D sub ] where     R sub = R AL Si + ξ AL 2 ( 1 R AL Si ) 2 R metal 1 1 ξ AL 2 R AL Si R metal and D sub = ( 1 R AL Si ) ( 1 ξ AL ) ( 1 + R metal ξ AL ) 1 1 ξ AL 2 R AL Si R metal ,
D AL ξ Si ( 1 R Si air ) ( 1 R AL Si ) ( 1 ξ AL ) ( 1 + R metal ξ AL ) 1 1 ξ Si 2 ξ AL 2 R Si air R metal .
T tot ( δ ) = ξ tot ( 1 R AL air ) ( 1 R Si air ) 1 1 + ξ tot 2 R AL air R Si air ( 2 ξ tot R AL air R Si air ) cos δ ,
R tot ( δ ) = R + ξ tot ( 1 R ) R ξ tot ( 1 + R ) 2 cos δ 1 ξ tot 2 R 2 2 ξ tot R cos δ .

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