Abstract

In spite of rapidly increasing demand and various applications of infrared (IR) detectors, their design process for the performance improvement has been mostly dependent on researchers’ intuition and knowledge. We present two-dimensional unit structure design of the absorbing layer in IR detectors. A systematic approach is introduced to enhance the absorbing efficiency of incident beam in the near-infrared wavelength range. We derived a layered structure composed of a silicon nitride (Si3N4) layer and an amorphous silicon (a-Si) one in turn by the so called topology optimization in association with the time variant finite element analysis (FEA). It is confirmed that thickness at each layer is in associated with the IR wavelength so that detail dimensions of each layer are inferred. A prototype of the layered structure was fabricated and its performance has been verified through experimental measurement.

© 2013 Optical Society of America

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References

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2013 (1)

2012 (3)

H. Soh, J. Yoo, D. Kim, “Optimal design of the light absorbing layer in thin film silicon solar cells,” Sol. Energy 86(7), 2095–2105 (2012).
[CrossRef]

H. Soh, J. Yoo, “Texturing design for a light trapping system using topology optimization,” IEEE Trans. Magn. 48(2), 227–230 (2012).
[CrossRef]

M. Yuan, X. Zhou, X. Yu, “Study on Infrared Absorption Characteristics of Ti and TiNx Nanofilms,” ECS Trans. 44, 1429–1435 (2012).
[CrossRef]

2011 (1)

2010 (4)

J. Li, H. Yu, S. M. Wong, G. Zhang, G. Lo, D. Kwong, “Si nanocone array optimization on crystalline Si thin films for solar energy harvesting,” J. Phys. D Appl. Phys. 43(25), 255101 (2010).
[CrossRef]

M. Laamanen, M. Blomberg, R. L. Puurunen, A. Miranto, H. Kattelus, “Thin film absorbers for visible, near-infrared, and short-wavelength infrared spectra,” Sensor Actuator A 162(2), 210–214 (2010).
[CrossRef]

J. Andkjær, S. Nishiwaki, T. Nomura, O. Sigmund, “Topology optimization of grating couplers for the efficient excitation of surface plasmons,” J. Opt. Soc. Am. B 27(9), 1828–1832 (2010).
[CrossRef]

R. Matzen, J. S. Jensen, O. Sigmund, “Topology optimization for transient response of photonic crystal structures,” J. Opt. Soc. Am. B 27(10), 2040–2050 (2010).
[CrossRef]

2009 (2)

T. Nomura, S. Nishiwaki, K. Sato, K. Hirayama, “Topology optimization for the design of periodic microstructures composed of electromagnetic materials,” Finite Elem. Anal. Des. 45(3), 210–226 (2009).
[CrossRef]

R. Dewan, D. Knipp, “Light trapping in thin-film silicon solar cells with integrated diffraction grating,” J. Appl. Phys. 106(7), 074901 (2009).
[CrossRef]

2008 (1)

A. Lin, J. Phillips, “Optimization of random diffraction gratings in thin-film solar cells using genetic algorithms,” Sol. Energy Mater. Sol. Cells 92(12), 1689–1696 (2008).
[CrossRef]

2007 (2)

C. Haase, H. Stiebig, “Thin-film silicon solar cells with efficient periodic light trapping texture,” Appl. Phys. Lett. 91(6), 061116 (2007).
[CrossRef]

D. Bergström, J. Powell, A. F. H. Kaplan, “The absorptance of steels to Nd:YLF and Nd:YAG laser light at room temperature,” Appl. Surf. Sci. 253(11), 5017–5028 (2007).
[CrossRef]

2005 (2)

2004 (1)

J. Springer, A. Poruba, L. Müllerova, M. Vanecek, “Absorption loss at nanorough silver back reflector of thin-film silicon solar cells,” J. Appl. Phys. 95(3), 1427–1429 (2004).
[CrossRef]

2003 (1)

A. Rogalski, “Infrared detector: status and trends,” Prog. Quantum Electron. 27(2-3), 59–210 (2003).
[CrossRef]

2002 (2)

A. Rogalski, “Infrared detectors: an overview,” Infrared Phys. Technol. 43(3-5), 187–210 (2002).
[CrossRef]

P. Norton, “HgCdTe infrared detectors,” Opto-Electron. Rev. 10, 159–174 (2002).

2001 (1)

2000 (1)

J. Yoo, N. Kikuchi, J. L. Volakis, “Structural optimization in magnetic devices by the homogenization design method,” IEEE Trans. Magn. 36(3), 574–580 (2000).
[CrossRef]

1998 (1)

1988 (1)

M. P. Bendsøe, N. Kikuchi, “Generating optimal topologies in optimal design using a homogenization method,” Comput. Method Appl. M. 71(2), 197–224 (1988).
[CrossRef]

1987 (1)

P. Campbell, M. Green, “Light trapping properties of pyramidally textured surfaces,” J. Appl. Phys. 62(1), 243–249 (1987).
[CrossRef]

Andkjær, J.

Atwater, H. A.

Aydil, E. S.

J. B. Baxter, E. S. Aydil, “Nanowire-based dye-sensitized solar cells,” Appl. Phys. Lett. 86(5), 053114 (2005).
[CrossRef]

Baxter, J. B.

J. B. Baxter, E. S. Aydil, “Nanowire-based dye-sensitized solar cells,” Appl. Phys. Lett. 86(5), 053114 (2005).
[CrossRef]

Becker, C.

Bendsøe, M. P.

M. P. Bendsøe, N. Kikuchi, “Generating optimal topologies in optimal design using a homogenization method,” Comput. Method Appl. M. 71(2), 197–224 (1988).
[CrossRef]

Bergström, D.

D. Bergström, J. Powell, A. F. H. Kaplan, “The absorptance of steels to Nd:YLF and Nd:YAG laser light at room temperature,” Appl. Surf. Sci. 253(11), 5017–5028 (2007).
[CrossRef]

Blomberg, M.

M. Laamanen, M. Blomberg, R. L. Puurunen, A. Miranto, H. Kattelus, “Thin film absorbers for visible, near-infrared, and short-wavelength infrared spectra,” Sensor Actuator A 162(2), 210–214 (2010).
[CrossRef]

Campbell, P.

P. Campbell, M. Green, “Light trapping properties of pyramidally textured surfaces,” J. Appl. Phys. 62(1), 243–249 (1987).
[CrossRef]

Dewan, R.

R. Dewan, D. Knipp, “Light trapping in thin-film silicon solar cells with integrated diffraction grating,” J. Appl. Phys. 106(7), 074901 (2009).
[CrossRef]

Green, M.

P. Campbell, M. Green, “Light trapping properties of pyramidally textured surfaces,” J. Appl. Phys. 62(1), 243–249 (1987).
[CrossRef]

Haase, C.

C. Haase, H. Stiebig, “Thin-film silicon solar cells with efficient periodic light trapping texture,” Appl. Phys. Lett. 91(6), 061116 (2007).
[CrossRef]

Hanssen, L.

Hirayama, K.

T. Nomura, S. Nishiwaki, K. Sato, K. Hirayama, “Topology optimization for the design of periodic microstructures composed of electromagnetic materials,” Finite Elem. Anal. Des. 45(3), 210–226 (2009).
[CrossRef]

Jensen, J. S.

Kaplan, A. F. H.

D. Bergström, J. Powell, A. F. H. Kaplan, “The absorptance of steels to Nd:YLF and Nd:YAG laser light at room temperature,” Appl. Surf. Sci. 253(11), 5017–5028 (2007).
[CrossRef]

Kattelus, H.

M. Laamanen, M. Blomberg, R. L. Puurunen, A. Miranto, H. Kattelus, “Thin film absorbers for visible, near-infrared, and short-wavelength infrared spectra,” Sensor Actuator A 162(2), 210–214 (2010).
[CrossRef]

Kikuchi, N.

J. Yoo, N. Kikuchi, J. L. Volakis, “Structural optimization in magnetic devices by the homogenization design method,” IEEE Trans. Magn. 36(3), 574–580 (2000).
[CrossRef]

M. P. Bendsøe, N. Kikuchi, “Generating optimal topologies in optimal design using a homogenization method,” Comput. Method Appl. M. 71(2), 197–224 (1988).
[CrossRef]

Kim, D.

H. Soh, J. Yoo, D. Kim, “Optimal design of the light absorbing layer in thin film silicon solar cells,” Sol. Energy 86(7), 2095–2105 (2012).
[CrossRef]

Knipp, D.

R. Dewan, D. Knipp, “Light trapping in thin-film silicon solar cells with integrated diffraction grating,” J. Appl. Phys. 106(7), 074901 (2009).
[CrossRef]

Kosten, E. D.

Kwong, D.

J. Li, H. Yu, S. M. Wong, G. Zhang, G. Lo, D. Kwong, “Si nanocone array optimization on crystalline Si thin films for solar energy harvesting,” J. Phys. D Appl. Phys. 43(25), 255101 (2010).
[CrossRef]

Laamanen, M.

M. Laamanen, M. Blomberg, R. L. Puurunen, A. Miranto, H. Kattelus, “Thin film absorbers for visible, near-infrared, and short-wavelength infrared spectra,” Sensor Actuator A 162(2), 210–214 (2010).
[CrossRef]

Li, J.

J. Li, H. Yu, S. M. Wong, G. Zhang, G. Lo, D. Kwong, “Si nanocone array optimization on crystalline Si thin films for solar energy harvesting,” J. Phys. D Appl. Phys. 43(25), 255101 (2010).
[CrossRef]

Lin, A.

A. Lin, J. Phillips, “Optimization of random diffraction gratings in thin-film solar cells using genetic algorithms,” Sol. Energy Mater. Sol. Cells 92(12), 1689–1696 (2008).
[CrossRef]

Lo, G.

J. Li, H. Yu, S. M. Wong, G. Zhang, G. Lo, D. Kwong, “Si nanocone array optimization on crystalline Si thin films for solar energy harvesting,” J. Phys. D Appl. Phys. 43(25), 255101 (2010).
[CrossRef]

Lockau, D.

Matzen, R.

Miranto, A.

M. Laamanen, M. Blomberg, R. L. Puurunen, A. Miranto, H. Kattelus, “Thin film absorbers for visible, near-infrared, and short-wavelength infrared spectra,” Sensor Actuator A 162(2), 210–214 (2010).
[CrossRef]

Müllerova, L.

J. Springer, A. Poruba, L. Müllerova, M. Vanecek, “Absorption loss at nanorough silver back reflector of thin-film silicon solar cells,” J. Appl. Phys. 95(3), 1427–1429 (2004).
[CrossRef]

Nishiwaki, S.

J. Andkjær, S. Nishiwaki, T. Nomura, O. Sigmund, “Topology optimization of grating couplers for the efficient excitation of surface plasmons,” J. Opt. Soc. Am. B 27(9), 1828–1832 (2010).
[CrossRef]

T. Nomura, S. Nishiwaki, K. Sato, K. Hirayama, “Topology optimization for the design of periodic microstructures composed of electromagnetic materials,” Finite Elem. Anal. Des. 45(3), 210–226 (2009).
[CrossRef]

Nomura, T.

J. Andkjær, S. Nishiwaki, T. Nomura, O. Sigmund, “Topology optimization of grating couplers for the efficient excitation of surface plasmons,” J. Opt. Soc. Am. B 27(9), 1828–1832 (2010).
[CrossRef]

T. Nomura, S. Nishiwaki, K. Sato, K. Hirayama, “Topology optimization for the design of periodic microstructures composed of electromagnetic materials,” Finite Elem. Anal. Des. 45(3), 210–226 (2009).
[CrossRef]

Norton, P.

P. Norton, “HgCdTe infrared detectors,” Opto-Electron. Rev. 10, 159–174 (2002).

Phillips, J.

A. Lin, J. Phillips, “Optimization of random diffraction gratings in thin-film solar cells using genetic algorithms,” Sol. Energy Mater. Sol. Cells 92(12), 1689–1696 (2008).
[CrossRef]

Poruba, A.

J. Springer, A. Poruba, L. Müllerova, M. Vanecek, “Absorption loss at nanorough silver back reflector of thin-film silicon solar cells,” J. Appl. Phys. 95(3), 1427–1429 (2004).
[CrossRef]

Powell, J.

D. Bergström, J. Powell, A. F. H. Kaplan, “The absorptance of steels to Nd:YLF and Nd:YAG laser light at room temperature,” Appl. Surf. Sci. 253(11), 5017–5028 (2007).
[CrossRef]

Puurunen, R. L.

M. Laamanen, M. Blomberg, R. L. Puurunen, A. Miranto, H. Kattelus, “Thin film absorbers for visible, near-infrared, and short-wavelength infrared spectra,” Sensor Actuator A 162(2), 210–214 (2010).
[CrossRef]

Rech, B.

Ristau, D.

Rogalski, A.

A. Rogalski, “Infrared detector: status and trends,” Prog. Quantum Electron. 27(2-3), 59–210 (2003).
[CrossRef]

A. Rogalski, “Infrared detectors: an overview,” Infrared Phys. Technol. 43(3-5), 187–210 (2002).
[CrossRef]

Rudigier-Voigt, E.

Sato, K.

T. Nomura, S. Nishiwaki, K. Sato, K. Hirayama, “Topology optimization for the design of periodic microstructures composed of electromagnetic materials,” Finite Elem. Anal. Des. 45(3), 210–226 (2009).
[CrossRef]

Schmidt, F.

Sigmund, O.

Soh, H.

H. Soh, J. Yoo, D. Kim, “Optimal design of the light absorbing layer in thin film silicon solar cells,” Sol. Energy 86(7), 2095–2105 (2012).
[CrossRef]

H. Soh, J. Yoo, “Texturing design for a light trapping system using topology optimization,” IEEE Trans. Magn. 48(2), 227–230 (2012).
[CrossRef]

Sontheimer, T.

Springer, J.

J. Springer, A. Poruba, L. Müllerova, M. Vanecek, “Absorption loss at nanorough silver back reflector of thin-film silicon solar cells,” J. Appl. Phys. 95(3), 1427–1429 (2004).
[CrossRef]

Stiebig, H.

C. Haase, H. Stiebig, “Thin-film silicon solar cells with efficient periodic light trapping texture,” Appl. Phys. Lett. 91(6), 061116 (2007).
[CrossRef]

Vanecek, M.

J. Springer, A. Poruba, L. Müllerova, M. Vanecek, “Absorption loss at nanorough silver back reflector of thin-film silicon solar cells,” J. Appl. Phys. 95(3), 1427–1429 (2004).
[CrossRef]

Volakis, J. L.

J. Yoo, N. Kikuchi, J. L. Volakis, “Structural optimization in magnetic devices by the homogenization design method,” IEEE Trans. Magn. 36(3), 574–580 (2000).
[CrossRef]

Warren, E. L.

Welsch, E.

Willamowski, U.

Wong, S. M.

J. Li, H. Yu, S. M. Wong, G. Zhang, G. Lo, D. Kwong, “Si nanocone array optimization on crystalline Si thin films for solar energy harvesting,” J. Phys. D Appl. Phys. 43(25), 255101 (2010).
[CrossRef]

Yoo, J.

H. Soh, J. Yoo, “Texturing design for a light trapping system using topology optimization,” IEEE Trans. Magn. 48(2), 227–230 (2012).
[CrossRef]

H. Soh, J. Yoo, D. Kim, “Optimal design of the light absorbing layer in thin film silicon solar cells,” Sol. Energy 86(7), 2095–2105 (2012).
[CrossRef]

J. Yoo, N. Kikuchi, J. L. Volakis, “Structural optimization in magnetic devices by the homogenization design method,” IEEE Trans. Magn. 36(3), 574–580 (2000).
[CrossRef]

Yu, H.

J. Li, H. Yu, S. M. Wong, G. Zhang, G. Lo, D. Kwong, “Si nanocone array optimization on crystalline Si thin films for solar energy harvesting,” J. Phys. D Appl. Phys. 43(25), 255101 (2010).
[CrossRef]

Yu, X.

M. Yuan, X. Zhou, X. Yu, “Study on Infrared Absorption Characteristics of Ti and TiNx Nanofilms,” ECS Trans. 44, 1429–1435 (2012).
[CrossRef]

Yuan, M.

M. Yuan, X. Zhou, X. Yu, “Study on Infrared Absorption Characteristics of Ti and TiNx Nanofilms,” ECS Trans. 44, 1429–1435 (2012).
[CrossRef]

Zhang, G.

J. Li, H. Yu, S. M. Wong, G. Zhang, G. Lo, D. Kwong, “Si nanocone array optimization on crystalline Si thin films for solar energy harvesting,” J. Phys. D Appl. Phys. 43(25), 255101 (2010).
[CrossRef]

Zhou, X.

M. Yuan, X. Zhou, X. Yu, “Study on Infrared Absorption Characteristics of Ti and TiNx Nanofilms,” ECS Trans. 44, 1429–1435 (2012).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (2)

C. Haase, H. Stiebig, “Thin-film silicon solar cells with efficient periodic light trapping texture,” Appl. Phys. Lett. 91(6), 061116 (2007).
[CrossRef]

J. B. Baxter, E. S. Aydil, “Nanowire-based dye-sensitized solar cells,” Appl. Phys. Lett. 86(5), 053114 (2005).
[CrossRef]

Appl. Surf. Sci. (1)

D. Bergström, J. Powell, A. F. H. Kaplan, “The absorptance of steels to Nd:YLF and Nd:YAG laser light at room temperature,” Appl. Surf. Sci. 253(11), 5017–5028 (2007).
[CrossRef]

Comput. Method Appl. M. (1)

M. P. Bendsøe, N. Kikuchi, “Generating optimal topologies in optimal design using a homogenization method,” Comput. Method Appl. M. 71(2), 197–224 (1988).
[CrossRef]

ECS Trans. (1)

M. Yuan, X. Zhou, X. Yu, “Study on Infrared Absorption Characteristics of Ti and TiNx Nanofilms,” ECS Trans. 44, 1429–1435 (2012).
[CrossRef]

Finite Elem. Anal. Des. (1)

T. Nomura, S. Nishiwaki, K. Sato, K. Hirayama, “Topology optimization for the design of periodic microstructures composed of electromagnetic materials,” Finite Elem. Anal. Des. 45(3), 210–226 (2009).
[CrossRef]

IEEE Trans. Magn. (2)

H. Soh, J. Yoo, “Texturing design for a light trapping system using topology optimization,” IEEE Trans. Magn. 48(2), 227–230 (2012).
[CrossRef]

J. Yoo, N. Kikuchi, J. L. Volakis, “Structural optimization in magnetic devices by the homogenization design method,” IEEE Trans. Magn. 36(3), 574–580 (2000).
[CrossRef]

Infrared Phys. Technol. (1)

A. Rogalski, “Infrared detectors: an overview,” Infrared Phys. Technol. 43(3-5), 187–210 (2002).
[CrossRef]

J. Appl. Phys. (3)

R. Dewan, D. Knipp, “Light trapping in thin-film silicon solar cells with integrated diffraction grating,” J. Appl. Phys. 106(7), 074901 (2009).
[CrossRef]

P. Campbell, M. Green, “Light trapping properties of pyramidally textured surfaces,” J. Appl. Phys. 62(1), 243–249 (1987).
[CrossRef]

J. Springer, A. Poruba, L. Müllerova, M. Vanecek, “Absorption loss at nanorough silver back reflector of thin-film silicon solar cells,” J. Appl. Phys. 95(3), 1427–1429 (2004).
[CrossRef]

J. Opt. Soc. Am. B (3)

J. Phys. D Appl. Phys. (1)

J. Li, H. Yu, S. M. Wong, G. Zhang, G. Lo, D. Kwong, “Si nanocone array optimization on crystalline Si thin films for solar energy harvesting,” J. Phys. D Appl. Phys. 43(25), 255101 (2010).
[CrossRef]

Opt. Express (2)

Opto-Electron. Rev. (1)

P. Norton, “HgCdTe infrared detectors,” Opto-Electron. Rev. 10, 159–174 (2002).

Prog. Quantum Electron. (1)

A. Rogalski, “Infrared detector: status and trends,” Prog. Quantum Electron. 27(2-3), 59–210 (2003).
[CrossRef]

Sensor Actuator A (1)

M. Laamanen, M. Blomberg, R. L. Puurunen, A. Miranto, H. Kattelus, “Thin film absorbers for visible, near-infrared, and short-wavelength infrared spectra,” Sensor Actuator A 162(2), 210–214 (2010).
[CrossRef]

Sol. Energy (1)

H. Soh, J. Yoo, D. Kim, “Optimal design of the light absorbing layer in thin film silicon solar cells,” Sol. Energy 86(7), 2095–2105 (2012).
[CrossRef]

Sol. Energy Mater. Sol. Cells (1)

A. Lin, J. Phillips, “Optimization of random diffraction gratings in thin-film solar cells using genetic algorithms,” Sol. Energy Mater. Sol. Cells 92(12), 1689–1696 (2008).
[CrossRef]

Other (7)

A. Rogalski, Infrared Detectors (CRC Press, 2011).

J. M. Palmer, Handbook of optics, (McGraw-Hill, 1995).

P. Ye, Optical waves in layered media (Wiley, 1998).

D. W. Driscoll and W. Vaughan, Handbook of optics (McGraw-Hill, New York, 1978)

D. N. Wang, J. M. White, K. S. Law, and C. Leung, “Thermal CVD/PECVD reactor and use for thermal chemical vapor deposition of silicon dioxide and in-situ multi-step planarized process,” US Patent, 5000113 (1991).

G. S. Sandhu and T. W. Buley, “Low-pressure chemical vapor deposition process for depositing high-density, highly-conformal titanium nitride films of low bulk resistivity,” US Patent, 5246881 (1993).

M. P. Bendsøe and O. Sigmund, Topology optimization: theory, methods, and applications (Springer-Verlag, 2003).

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Figures (8)

Fig. 1
Fig. 1

Model for analysis and design. (a) simplified schematic of photovoltaic IR detector and (b) its initial model for analysis.

Fig. 2
Fig. 2

Hz contour plot of the initial model and its time history at the measuring domain with the definition of the time integration period. It is measured for 1064nm incident beam wavelength.

Fig. 3
Fig. 3

Hz Topology optimization concept of a traditional structural design in a 2D problem. Design domain is subjected to boudary and load conditions and its optimal material distribution is determined according to the density value of each element

Fig. 4
Fig. 4

Hz Convergence histories of the objective function and the Si3N4/a-Si layer shape changes during the topology optimization process in the 1064nm incident beam wavelength for different initial shapes (a) wedge shaped initial case and (b) full Si3N4 initial case.

Fig. 5
Fig. 5

Absorbing layer structures for (a) wedge initial case, (b) Si3N4 mono layer case and (c) optimal case suggested. Efficiencies are computed as 0.0988, 0.3686 and 0.4480, respectively.

Fig. 6
Fig. 6

Wave propagation plots in cases of (a) wedge initial model, (b) Si3N4 mono layer model and (c) optimal model suggested.

Fig. 7
Fig. 7

SEM images of prototypes for (a) Si3N4 mono layer model and (b) Si3N4 and a-Si multi-layer model.

Fig. 8
Fig. 8

Experiment set-ups for (a) reflectance measurement using an integrating sphere and (b) transmittance measurement.

Tables (1)

Tables Icon

Table 1 Measured values of transmittance, reflectance and resultant absorptance of Si3N4 mono layer model and Si3N4/a-Si multi-layered model

Equations (15)

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1 ε r H z = 1 c 0 2 2 t 2 H z
1 ε r H z x + 1 ε r H z y = i ω 2 1 c 2 ε r ( H z H 0 )
E = Re [ 1 j ω ε 0 ( 1 ε r H z y H z * ) ]
P o y o b j = Ω o b j E d Ω / A m e a s u r e
P o y i n c = Γ i n c E d Ω / L i n c
ψ o b j = t t + Δ t P o y o b j d τ
ψ i n c = t t + Δ t P o y i n c d τ
e f f t r = ψ o b j ψ i n c
ε r = ( γ p ε S i 3 N 4 + ( 1 γ p ) ε a S i ) + j ( γ p ε S i 3 N 4 + ( 1 γ p ) ε a S i )
M a x i m i z e e f f t r = ψ o b j ψ i n c S u b j e c t t o (Equilibrium of Eq . (2)) and (No volume constraint)
1 ( Δ t ) 2 1 c 2 ( H z ) n + 1 ( 1 ε r 2 ) n ( H z ) n + 2 ( Δ t ) 2 1 c 2 ( H z ) n 1 ( Δ t ) 2 1 c 2 ( H z ) n 1
K φ = f
d F ( t i , Δ t ) d γ = F ( t i , Δ t ) γ λ T ( t i , Δ t ) ( K γ H z ( t ) f γ )
F ( t i , Δ t ) = t i t i + Δ t F d τ
K λ ( t i , Δ t ) T = F ( t i , Δ t ) H z ( t )

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