Abstract

We introduce a technique for measuring detection efficiency that is traceable to the primary standard, the cryogenic radiometer, through a reference silicon photodiode trap detector. The trap detector, used in conjunction with a switched integrator amplifier, can measure signals down to the 0.1 pW (3 x 105 photons second−1) level with 0.1% uncertainty in a total integration time of 300 seconds. This provides a convenient calibration standard for measurements at these levels across the optical spectrum (UV – near IR). A second technique is also described, based on correlated photons produced via parametric down-conversion. This can be used to directly measure detection efficiency in the photon counting regime, and provides a route for expanding the formulation of the candela in terms of photon flux to enable it to address the needs of emerging quantum optical technologies and applications. The two independent techniques were cross-validated by a comparison carried out at 702.2 nm, which showed agreement to within 0.2%.

©2011 Optical Society of America

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    [Crossref]
  26. J. Y. Cheung, M. P. Vaughan, J. R. M. Mountford, and C. J. Chunnilall, “Correlated photon metrology of detectors and sources,” Proc. SPIE 5161, 365–376 (2004).
    [Crossref]
  27. S. Castelletto, I. P. Degiovanni, and M. L. Rastello, “Evaluation of statistical noise in measurements based on correlated photons,” J. Opt. Soc. Am. B 19(6), 1247–1258 (2002).
    [Crossref]
  28. W. S. Hartree, E. Theocharous, and N. P. Fox, “A wavelength tunable, quasi-cw laser source for high accuracy spectrometric measurement in the 200 nm to 500 nm region,” Proc. SPIE 4826, 104–112 (2003).
    [Crossref]
  29. E. Usadi and L. Crane, “Laser-based spectrophotometry at NPL,” Proc. SPIE 5192, 36–45 (2003).
    [Crossref]
  30. J. Y. Cheung, J. L. Gardner, A. Migdall, S. Polyakov, and M. Ware, “High accuracy dual lens transmittance measurements,” Appl. Opt. 46(22), 5396–5403 (2007).
    [Crossref] [PubMed]
  31. G. Porrovechio, M. Smid, and J. Y. Cheung, “Monochromator-based technique for measuring the focal length of lenses at different wavelengths,” (In preparation).
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    [Crossref]
  35. S. V. Polyakov and A. Migdall, “Quantum radiometry,” J. Mod. Opt. 56(9), 1045–1052 (2009).
    [Crossref]

2010 (2)

J. C. Zwinkels, E. Ikonen, N. P. Fox, G. Ulm, and M. L. Rastello, “Photometry, radiometry and 'the candela': evolution in the classical and quantum world,” Metrologia 47(5), R15–R32 (2010).
[Crossref]

E. Theocharous, M. A. Itzler, J. Y. Cheung, and C. J. Chunnilall, “The characterisation of the linearity of response and spatial uniformity of response of two InGaAs/InP Geiger-mode avalanche photodiodes,” IEEE J. Quantum Electron. 46(11), 1561–1567 (2010).
[Crossref]

2009 (2)

A. R. Beaumont, J. Y. Cheung, C. J. Chunnilall, J. Ireland, and M. G. White, “Providing reference standards and metrology for the few photon - photon counting community,” Nucl. Instrum. Methods Phys. Res. A 610(1), 183–187 (2009).
[Crossref]

S. V. Polyakov and A. Migdall, “Quantum radiometry,” J. Mod. Opt. 56(9), 1045–1052 (2009).
[Crossref]

2008 (2)

2007 (3)

2004 (3)

M. Ware and A. L. Migdall, “Single photon detector characterization using correlated photons: the march from feasibility to metrology,” J. Mod. Opt. 51, 1549–1557 (2004).

J. Y. Cheung, C. J. Chunnilall, and J. Wang, “Radiometric applications of correlated photon metrology,” Proc. SPIE 5551, 220–230 (2004).
[Crossref]

J. Y. Cheung, M. P. Vaughan, J. R. M. Mountford, and C. J. Chunnilall, “Correlated photon metrology of detectors and sources,” Proc. SPIE 5161, 365–376 (2004).
[Crossref]

2003 (2)

W. S. Hartree, E. Theocharous, and N. P. Fox, “A wavelength tunable, quasi-cw laser source for high accuracy spectrometric measurement in the 200 nm to 500 nm region,” Proc. SPIE 4826, 104–112 (2003).
[Crossref]

E. Usadi and L. Crane, “Laser-based spectrophotometry at NPL,” Proc. SPIE 5192, 36–45 (2003).
[Crossref]

2002 (1)

2000 (3)

G. Brida, M. Genovese, and C. Novero, “On the measurement of photon flux in parametric down-conversion,” Eur. Phys. J. D 8(2), 273–275 (2000).
[Crossref]

G. P. Eppeldauer and D. C. Lynch, “Opto-mechanical and electronic design of a tunnel-trap Si radiometer,” J. Res. Natl. Inst. Stand. Technol. 105, 813–818 (2000).

G. Brida, S. Castelletto, I. P. Degiovanni, C. Novero, and M. L. Rastello, “Quantum efficiency and dead time of single-photon counting photodiodes: a comparison between two measurement techniques,” Metrologia 37(5), 625–628 (2000).
[Crossref]

1999 (1)

S. D. Biller, N. A. Jelley, N. M. D. Thorman, N. P. Fox, and T. H. Ward, “Measurements of photomultiplier single photon counting efficiency for the Sudbury Neutrino Observatory,” Nucl. Instrum. Methods Phys Res. A 432(2-3), 364–373 (1999).
[Crossref]

1994 (1)

1991 (2)

1985 (1)

J. E. Martin, N. P. Fox, and P. J. Key, “A cryogenic radiometer for absolute radiometric measurements,” Metrologia 21(3), 147–155 (1985).
[Crossref]

1983 (1)

1977 (1)

D. N. Klyshko, “Utilization of vacuum fluctuations as an optical brightness standard,” Sov. J. Quantum Electron. 7(5), 591–595 (1977).
[Crossref]

1970 (1)

D. C. Burnham and D. L. Weinberg, “Observation of simultaneity in parametric production of optical photon pairs,” Phys. Rev. Lett. 25(2), 84–87 (1970).
[Crossref]

1965 (1)

J. E. Midwinter and J. Warner, “The effects of phase matching method and of uniaxial crystal symmetry on the polar distribution of second-order non-linear optical polarization,” Br. J. Appl. Phys. 16(8), 1135–1142 (1965).
[Crossref]

Beaumont, A. R.

A. R. Beaumont, J. Y. Cheung, C. J. Chunnilall, J. Ireland, and M. G. White, “Providing reference standards and metrology for the few photon - photon counting community,” Nucl. Instrum. Methods Phys. Res. A 610(1), 183–187 (2009).
[Crossref]

Biller, S. D.

S. D. Biller, N. A. Jelley, N. M. D. Thorman, N. P. Fox, and T. H. Ward, “Measurements of photomultiplier single photon counting efficiency for the Sudbury Neutrino Observatory,” Nucl. Instrum. Methods Phys Res. A 432(2-3), 364–373 (1999).
[Crossref]

Brida, G.

G. Brida, S. Castelletto, I. P. Degiovanni, C. Novero, and M. L. Rastello, “Quantum efficiency and dead time of single-photon counting photodiodes: a comparison between two measurement techniques,” Metrologia 37(5), 625–628 (2000).
[Crossref]

G. Brida, M. Genovese, and C. Novero, “On the measurement of photon flux in parametric down-conversion,” Eur. Phys. J. D 8(2), 273–275 (2000).
[Crossref]

Burnham, D. C.

D. C. Burnham and D. L. Weinberg, “Observation of simultaneity in parametric production of optical photon pairs,” Phys. Rev. Lett. 25(2), 84–87 (1970).
[Crossref]

Castelletto, S.

S. Castelletto, I. P. Degiovanni, and M. L. Rastello, “Evaluation of statistical noise in measurements based on correlated photons,” J. Opt. Soc. Am. B 19(6), 1247–1258 (2002).
[Crossref]

G. Brida, S. Castelletto, I. P. Degiovanni, C. Novero, and M. L. Rastello, “Quantum efficiency and dead time of single-photon counting photodiodes: a comparison between two measurement techniques,” Metrologia 37(5), 625–628 (2000).
[Crossref]

Cheung, E. C.

Cheung, J. Y.

E. Theocharous, M. A. Itzler, J. Y. Cheung, and C. J. Chunnilall, “The characterisation of the linearity of response and spatial uniformity of response of two InGaAs/InP Geiger-mode avalanche photodiodes,” IEEE J. Quantum Electron. 46(11), 1561–1567 (2010).
[Crossref]

A. R. Beaumont, J. Y. Cheung, C. J. Chunnilall, J. Ireland, and M. G. White, “Providing reference standards and metrology for the few photon - photon counting community,” Nucl. Instrum. Methods Phys. Res. A 610(1), 183–187 (2009).
[Crossref]

J. Y. Cheung, C. J. Chunnilall, E. R. Woolliams, N. P. Fox, J. R. Mountford, J. Wang, and P. J. Thomas, “The quantum candela: a re-definition of the standard units for optical radiation,” J. Mod. Opt. 54(2-3), 373–396 (2007).
[Crossref]

J. Y. Cheung, J. L. Gardner, A. Migdall, S. Polyakov, and M. Ware, “High accuracy dual lens transmittance measurements,” Appl. Opt. 46(22), 5396–5403 (2007).
[Crossref] [PubMed]

J. Y. Cheung, C. J. Chunnilall, and J. Wang, “Radiometric applications of correlated photon metrology,” Proc. SPIE 5551, 220–230 (2004).
[Crossref]

J. Y. Cheung, M. P. Vaughan, J. R. M. Mountford, and C. J. Chunnilall, “Correlated photon metrology of detectors and sources,” Proc. SPIE 5161, 365–376 (2004).
[Crossref]

Chunnilall, C. J.

E. Theocharous, M. A. Itzler, J. Y. Cheung, and C. J. Chunnilall, “The characterisation of the linearity of response and spatial uniformity of response of two InGaAs/InP Geiger-mode avalanche photodiodes,” IEEE J. Quantum Electron. 46(11), 1561–1567 (2010).
[Crossref]

A. R. Beaumont, J. Y. Cheung, C. J. Chunnilall, J. Ireland, and M. G. White, “Providing reference standards and metrology for the few photon - photon counting community,” Nucl. Instrum. Methods Phys. Res. A 610(1), 183–187 (2009).
[Crossref]

J. Y. Cheung, C. J. Chunnilall, E. R. Woolliams, N. P. Fox, J. R. Mountford, J. Wang, and P. J. Thomas, “The quantum candela: a re-definition of the standard units for optical radiation,” J. Mod. Opt. 54(2-3), 373–396 (2007).
[Crossref]

J. Y. Cheung, C. J. Chunnilall, and J. Wang, “Radiometric applications of correlated photon metrology,” Proc. SPIE 5551, 220–230 (2004).
[Crossref]

J. Y. Cheung, M. P. Vaughan, J. R. M. Mountford, and C. J. Chunnilall, “Correlated photon metrology of detectors and sources,” Proc. SPIE 5161, 365–376 (2004).
[Crossref]

Crane, L.

E. Usadi and L. Crane, “Laser-based spectrophotometry at NPL,” Proc. SPIE 5192, 36–45 (2003).
[Crossref]

Degiovanni, I. P.

S. Castelletto, I. P. Degiovanni, and M. L. Rastello, “Evaluation of statistical noise in measurements based on correlated photons,” J. Opt. Soc. Am. B 19(6), 1247–1258 (2002).
[Crossref]

G. Brida, S. Castelletto, I. P. Degiovanni, C. Novero, and M. L. Rastello, “Quantum efficiency and dead time of single-photon counting photodiodes: a comparison between two measurement techniques,” Metrologia 37(5), 625–628 (2000).
[Crossref]

Duda, C. R.

Eppeldauer, G. P.

G. P. Eppeldauer and D. C. Lynch, “Opto-mechanical and electronic design of a tunnel-trap Si radiometer,” J. Res. Natl. Inst. Stand. Technol. 105, 813–818 (2000).

Fox, N. P.

J. C. Zwinkels, E. Ikonen, N. P. Fox, G. Ulm, and M. L. Rastello, “Photometry, radiometry and 'the candela': evolution in the classical and quantum world,” Metrologia 47(5), R15–R32 (2010).
[Crossref]

J. Y. Cheung, C. J. Chunnilall, E. R. Woolliams, N. P. Fox, J. R. Mountford, J. Wang, and P. J. Thomas, “The quantum candela: a re-definition of the standard units for optical radiation,” J. Mod. Opt. 54(2-3), 373–396 (2007).
[Crossref]

W. S. Hartree, E. Theocharous, and N. P. Fox, “A wavelength tunable, quasi-cw laser source for high accuracy spectrometric measurement in the 200 nm to 500 nm region,” Proc. SPIE 4826, 104–112 (2003).
[Crossref]

S. D. Biller, N. A. Jelley, N. M. D. Thorman, N. P. Fox, and T. H. Ward, “Measurements of photomultiplier single photon counting efficiency for the Sudbury Neutrino Observatory,” Nucl. Instrum. Methods Phys Res. A 432(2-3), 364–373 (1999).
[Crossref]

N. P. Fox, “Trap detectors and their properties,” Metrologia 28(3), 197–202 (1991).
[Crossref]

J. E. Martin, N. P. Fox, and P. J. Key, “A cryogenic radiometer for absolute radiometric measurements,” Metrologia 21(3), 147–155 (1985).
[Crossref]

Gardner, J. L.

Genovese, M.

G. Brida, M. Genovese, and C. Novero, “On the measurement of photon flux in parametric down-conversion,” Eur. Phys. J. D 8(2), 273–275 (2000).
[Crossref]

Hartree, W. S.

W. S. Hartree, E. Theocharous, and N. P. Fox, “A wavelength tunable, quasi-cw laser source for high accuracy spectrometric measurement in the 200 nm to 500 nm region,” Proc. SPIE 4826, 104–112 (2003).
[Crossref]

Ikonen, E.

J. C. Zwinkels, E. Ikonen, N. P. Fox, G. Ulm, and M. L. Rastello, “Photometry, radiometry and 'the candela': evolution in the classical and quantum world,” Metrologia 47(5), R15–R32 (2010).
[Crossref]

Ireland, J.

A. R. Beaumont, J. Y. Cheung, C. J. Chunnilall, J. Ireland, and M. G. White, “Providing reference standards and metrology for the few photon - photon counting community,” Nucl. Instrum. Methods Phys. Res. A 610(1), 183–187 (2009).
[Crossref]

Itzler, M. A.

E. Theocharous, M. A. Itzler, J. Y. Cheung, and C. J. Chunnilall, “The characterisation of the linearity of response and spatial uniformity of response of two InGaAs/InP Geiger-mode avalanche photodiodes,” IEEE J. Quantum Electron. 46(11), 1561–1567 (2010).
[Crossref]

Jelley, N. A.

S. D. Biller, N. A. Jelley, N. M. D. Thorman, N. P. Fox, and T. H. Ward, “Measurements of photomultiplier single photon counting efficiency for the Sudbury Neutrino Observatory,” Nucl. Instrum. Methods Phys Res. A 432(2-3), 364–373 (1999).
[Crossref]

Key, P. J.

J. E. Martin, N. P. Fox, and P. J. Key, “A cryogenic radiometer for absolute radiometric measurements,” Metrologia 21(3), 147–155 (1985).
[Crossref]

Klyshko, D. N.

D. N. Klyshko, “Utilization of vacuum fluctuations as an optical brightness standard,” Sov. J. Quantum Electron. 7(5), 591–595 (1977).
[Crossref]

Koch, K.

Liu, J. M.

Lynch, D. C.

G. P. Eppeldauer and D. C. Lynch, “Opto-mechanical and electronic design of a tunnel-trap Si radiometer,” J. Res. Natl. Inst. Stand. Technol. 105, 813–818 (2000).

Martin, J. E.

J. E. Martin, N. P. Fox, and P. J. Key, “A cryogenic radiometer for absolute radiometric measurements,” Metrologia 21(3), 147–155 (1985).
[Crossref]

Midwinter, J. E.

J. E. Midwinter and J. Warner, “The effects of phase matching method and of uniaxial crystal symmetry on the polar distribution of second-order non-linear optical polarization,” Br. J. Appl. Phys. 16(8), 1135–1142 (1965).
[Crossref]

Migdall, A.

Migdall, A. L.

S. V. Polyakov and A. L. Migdall, “High accuracy verification of a correlated-photon- based method for determining photoncounting detection efficiency,” Opt. Express 15(4), 1390–1407 (2007).
[Crossref] [PubMed]

M. Ware and A. L. Migdall, “Single photon detector characterization using correlated photons: the march from feasibility to metrology,” J. Mod. Opt. 51, 1549–1557 (2004).

Moore, G. T.

Mountford, J. R.

J. R. Mountford, G. Porrovecchio, M. Smid, and R. Smid, “Development of a switched integrator amplifier for high-accuracy optical measurements,” Appl. Opt. 47(31), 5821–5828 (2008).
[Crossref] [PubMed]

J. Y. Cheung, C. J. Chunnilall, E. R. Woolliams, N. P. Fox, J. R. Mountford, J. Wang, and P. J. Thomas, “The quantum candela: a re-definition of the standard units for optical radiation,” J. Mod. Opt. 54(2-3), 373–396 (2007).
[Crossref]

Mountford, J. R. M.

J. Y. Cheung, M. P. Vaughan, J. R. M. Mountford, and C. J. Chunnilall, “Correlated photon metrology of detectors and sources,” Proc. SPIE 5161, 365–376 (2004).
[Crossref]

Novero, C.

G. Brida, S. Castelletto, I. P. Degiovanni, C. Novero, and M. L. Rastello, “Quantum efficiency and dead time of single-photon counting photodiodes: a comparison between two measurement techniques,” Metrologia 37(5), 625–628 (2000).
[Crossref]

G. Brida, M. Genovese, and C. Novero, “On the measurement of photon flux in parametric down-conversion,” Eur. Phys. J. D 8(2), 273–275 (2000).
[Crossref]

Penin, A. N.

Polyakov, S.

Polyakov, S. V.

Porrovecchio, G.

Rastello, M. L.

J. C. Zwinkels, E. Ikonen, N. P. Fox, G. Ulm, and M. L. Rastello, “Photometry, radiometry and 'the candela': evolution in the classical and quantum world,” Metrologia 47(5), R15–R32 (2010).
[Crossref]

S. Castelletto, I. P. Degiovanni, and M. L. Rastello, “Evaluation of statistical noise in measurements based on correlated photons,” J. Opt. Soc. Am. B 19(6), 1247–1258 (2002).
[Crossref]

G. Brida, S. Castelletto, I. P. Degiovanni, C. Novero, and M. L. Rastello, “Quantum efficiency and dead time of single-photon counting photodiodes: a comparison between two measurement techniques,” Metrologia 37(5), 625–628 (2000).
[Crossref]

Sergienko, A. V.

Smid, M.

Smid, R.

Theocharous, E.

E. Theocharous, M. A. Itzler, J. Y. Cheung, and C. J. Chunnilall, “The characterisation of the linearity of response and spatial uniformity of response of two InGaAs/InP Geiger-mode avalanche photodiodes,” IEEE J. Quantum Electron. 46(11), 1561–1567 (2010).
[Crossref]

E. Theocharous, “Absolute linearity characterization of lock-in amplifiers,” Appl. Opt. 47(8), 1090–1096 (2008).
[Crossref] [PubMed]

W. S. Hartree, E. Theocharous, and N. P. Fox, “A wavelength tunable, quasi-cw laser source for high accuracy spectrometric measurement in the 200 nm to 500 nm region,” Proc. SPIE 4826, 104–112 (2003).
[Crossref]

Thomas, P. J.

J. Y. Cheung, C. J. Chunnilall, E. R. Woolliams, N. P. Fox, J. R. Mountford, J. Wang, and P. J. Thomas, “The quantum candela: a re-definition of the standard units for optical radiation,” J. Mod. Opt. 54(2-3), 373–396 (2007).
[Crossref]

Thorman, N. M. D.

S. D. Biller, N. A. Jelley, N. M. D. Thorman, N. P. Fox, and T. H. Ward, “Measurements of photomultiplier single photon counting efficiency for the Sudbury Neutrino Observatory,” Nucl. Instrum. Methods Phys Res. A 432(2-3), 364–373 (1999).
[Crossref]

Ulm, G.

J. C. Zwinkels, E. Ikonen, N. P. Fox, G. Ulm, and M. L. Rastello, “Photometry, radiometry and 'the candela': evolution in the classical and quantum world,” Metrologia 47(5), R15–R32 (2010).
[Crossref]

Usadi, E.

E. Usadi and L. Crane, “Laser-based spectrophotometry at NPL,” Proc. SPIE 5192, 36–45 (2003).
[Crossref]

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[Crossref]

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A. R. Beaumont, J. Y. Cheung, C. J. Chunnilall, J. Ireland, and M. G. White, “Providing reference standards and metrology for the few photon - photon counting community,” Nucl. Instrum. Methods Phys. Res. A 610(1), 183–187 (2009).
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Figures (14)

Fig. 1
Fig. 1 Experimental set-up for measurement of detection efficiency, top view, ND = neutral density filter, IS = integrating sphere, L = lens, P = polarizer, SIA = switch integration amplifier.

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Fig. 2
Fig. 2 Experimental set-up for measurement of detection efficiency, side view.

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Fig. 3
Fig. 3 Fit to typical data of the relative detection efficiency of the channel photomultiplier supplied by Perkin Elmer (Fig. 3a), the full spectral range is in Fig. 3b.

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Fig. 4
Fig. 4 Linearity of channel photomultiplier measured at 600 nm.

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Fig. 5
Fig. 5 Direct linearity measurements of the photon counter.

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Fig. 6
Fig. 6 Type I down-conversion in BBO; matching shapes represent different correlated photon pairs, with the crosses representing degenerately down-converted photons.

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Fig. 7
Fig. 7 Schematic of set-up for detection efficiency measurements using correlated photons, F = filter, PBS = polarizing beam splitter, HWP = half wave plate, NLC = non linear crystal,, L = lens, DUT = Device Under Test, TRIG = trigger detector, D = delay unit. The whole apparatus sits in a light tight enclosure

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Fig. 8
Fig. 8 Graph of coincidence events per 78 ps time bin.

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Fig. 9
Fig. 9 Transmittance of the crystal, measured at 700 nm, at 6° to the incident beam. The values to the right are the values when the crystal is out of the beam. The colour bar represents transmittance.

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Fig. 10
Fig. 10 Schematic of set-up for the measurement of lens transmittance, SF = spatial filter,BS = beam splitter, MS = motorized shutter.

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Fig. 11
Fig. 11 a. Geometric alignment (measurements Spring 2008), Fig. 11b is a close up of the data points over which the relative variation is taken as the uncertainty in the geometric alignment.

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Fig. 12
Fig. 12 Down-conversion geometry viewed along pump axis. The dashed line represents the down-conversion wavelength of interest, λs, the shaded area represents the spread of the down-conversion wavelength, δλs.

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Fig. 13
Fig. 13 Intersection of viewing cone with plane perpendicular to pump direction.

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Fig. 14
Fig. 14 Transmittance profile of the DUT and trigger band pass filters measured at normal incidence using the NPL calibrated Cary 5 spectrometer.

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Tables (4)

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Table 1 Assessment of the Uncertainties of the Conventional Technique

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Table 2 Uncertainties for Final Set-up (Winter 2009) of Correlated Photon Technique

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Table 3 Final Value from Winter 2009 Measurements

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Table 4 Final results of Comparison (†Uncertainty Dominated by Reproducibility of Those Measurements, *Corresponds to 2009)

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Equations (18)

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η CPM = N CPM N trap
V out =P R   G SIA
N trap = P hc/λ
η CPM = N CPM  R   G SIA V out hc λ
Θ trap = R eff R cal ( 702.2 ) = ∫ λ L( λ )  F A ( λ )  R th ( λ )dλ R th ( 702.2 )  ∫ λ L( λ )  F A ( λ )dλ
R th ( λ )= ελe hc = λ( nm ) 1239.48
Θ CPM = η eff η( 702.2 ) = ∫ λ N ph ( λ )  F A ( λ )  η rel ( λ )dλ η rel ( 702.2 )  ∫ λ N ph ( λ )  F A ( λ )dλ = N CPMeff N CPM_702.2
η CPMeff = N CPMeff   R eff    G SIA V out hc λ
η CPMconventional ( 702.2 )= N CPMeff Θ CPM Θ trap R cal ( 702.2 ) G SIA V out hc λ
η DUT = N c /N TRIG
η DUT = N c −Ac T DUT ( N TRIGGER − N FALSE )
T DUT = t d . t f . t l . t b . t g
P s ( λ )∝ P p   λ p λ s 5 λ i 2 δ λ i  ;   1 λ p = 1 λ s + 1 λ i
N ph_s ( λ )∝ P p λ s 4 λ i 2 δ λ s
N sap ( λ s )= t cs ∑ j t pj λ i 4 λ s 2   ϕ s,j   t sj
S trig = ∫ λ N sap ( λ )  F IEN ( λ )   η dt ( λ ) dλ
T Aeff = ∫ λ N sap ( λ ¯ )  F IEN ( λ ¯ )  F A ( λ ) dλ ∫ λ N sap ( λ ¯ )  F IEN ( λ ¯ ) dλ
θ cc = η eff_cc η cc ( 702.2 ) = ∫ λ N sap ( λ ¯ )  F IEN ( λ ¯ )  F A ( λ )  η rel ( λ ) dλ η rel ( 702.2 ) ∫ λ N sap ( λ ¯ )  F IEN ( λ ¯ )  F A ( λ ) dλ =1.0015

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