Abstract

We report on the impact of realistic gain and loss models on the bistable operation of nonlinear parity–time (PT) Bragg gratings. In our model we include both dispersive and saturable gain and show that levels of gain/loss saturation can have a significant impact on the bistable operation of a nonlinear PT Bragg grating based on GaAs material. The hysteresis of the nonlinear PT Bragg grating is analyzed for different levels of gain and loss and different saturation levels. We show that high saturation levels can improve the nonlinear operation by reducing the intensity at which the bistability occurs. However, when the saturation intensity is low, saturation inhibits the PT characteristics of the grating.

© 2014 Optical Society of America

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    [CrossRef]

2013

S. Phang, A. Vukovic, H. Susanto, T. M. Benson, P. Sewell, J. Opt. Soc. Am. B 30, 2984 (2013).
[CrossRef]

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, A. Scherer, Nat. Mater. 12, 108 (2013).
[CrossRef]

M. Kulishov, B. Kress, R. Slavík, Opt. Express 21, 9473 (2013).
[CrossRef]

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, H. Renner, Nat. Photonics 7, 579 (2013).
[CrossRef]

2012

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, U. Peschel, Nature 488, 167 (2012).
[CrossRef]

2011

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, D. N. Christodoulides, Phys. Rev. Lett. 106, 213901 (2011).
[CrossRef]

F. Nazari, M. Nazari, M. K. Moravvej-Farshi, Opt. Lett. 36, 4368 (2011).
[CrossRef]

2010

H. Ramezani, T. Kottos, R. El-Ganainy, D. N. Christodoulides, Phys. Rev. A 82, 043803 (2010).
[CrossRef]

A. A. Sukhorukov, Z. Xu, Y. S. Kivshar, Phys. Rev. A 82, 043818 (2010).
[CrossRef]

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

J. Čtyroký, V. Kuzmiak, S. Eyderman, Opt. Express 18, 21585 (2010).
[CrossRef]

2008

Z. Musslimani, K. Makris, R. El-Ganainy, D. N. Christodoulides, Phys. Rev. Lett. 100, 030402 (2008).
[CrossRef]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, Phys. Rev. Lett. 100, 103904 (2008).
[CrossRef]

2007

2005

M. Kulishov, J. M. Laniel, N. Bélanger, J. Azaña, D. V. Plant, Opt. Express 13, 3068 (2005).
[CrossRef]

V. Janyani, A. Vukovic, J. D. Paul, P. Sewell, T. M. Benson, Opt. Quantum Electron. 37, 3 (2005).
[CrossRef]

2004

S. Lan, A. V. Gopal, K. Kanamoto, H. Ishikawa, Appl. Phys. Lett. 84, 5124 (2004).
[CrossRef]

M. Greenberg, M. Orenstein, Opt. Express 12, 4013 (2004).
[CrossRef]

2003

A. Suryanto, E. van Groesen, M. Hammer, H. J. W. M. Hoekstra, Opt. Quantum Electron. 35, 313 (2003).
[CrossRef]

1999

C. M. Bender, S. Boettcher, P. N. Meisinger, J. Math. Phys. 40, 2201 (1999).
[CrossRef]

1996

S. C. Hagness, R. M. Joseph, A. Taflove, Radio Sci. 31, 931 (1996).
[CrossRef]

1995

M. Krumpholz, C. Huber, P. Russer, IEEE Trans. Microwave Theor. Tech. 43, 1935 (1995).
[CrossRef]

1989

S. Adachi, J. Appl. Phys. 66, 6030 (1989).
[CrossRef]

Adachi, S.

S. Adachi, J. Appl. Phys. 66, 6030 (1989).
[CrossRef]

Almeida, V. R.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, A. Scherer, Nat. Mater. 12, 108 (2013).
[CrossRef]

Azaña, J.

Baets, R.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, H. Renner, Nat. Photonics 7, 579 (2013).
[CrossRef]

Bass, M.

M. Bass, G. Li, E. van Stryland, Handbook of Optics, 3rd ed. (McGraw-Hill, 2010), Vol. 4.

Bélanger, N.

Bender, C. M.

C. M. Bender, S. Boettcher, P. N. Meisinger, J. Math. Phys. 40, 2201 (1999).
[CrossRef]

Benson, T. M.

S. Phang, A. Vukovic, H. Susanto, T. M. Benson, P. Sewell, J. Opt. Soc. Am. B 30, 2984 (2013).
[CrossRef]

V. Janyani, A. Vukovic, J. D. Paul, P. Sewell, T. M. Benson, Opt. Quantum Electron. 37, 3 (2005).
[CrossRef]

Bersch, C.

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, U. Peschel, Nature 488, 167 (2012).
[CrossRef]

Boettcher, S.

C. M. Bender, S. Boettcher, P. N. Meisinger, J. Math. Phys. 40, 2201 (1999).
[CrossRef]

Cao, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, D. N. Christodoulides, Phys. Rev. Lett. 106, 213901 (2011).
[CrossRef]

Chen, Y.-F.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, A. Scherer, Nat. Mater. 12, 108 (2013).
[CrossRef]

Christodoulides, D. N.

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, U. Peschel, Nature 488, 167 (2012).
[CrossRef]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, D. N. Christodoulides, Phys. Rev. Lett. 106, 213901 (2011).
[CrossRef]

H. Ramezani, T. Kottos, R. El-Ganainy, D. N. Christodoulides, Phys. Rev. A 82, 043803 (2010).
[CrossRef]

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, Phys. Rev. Lett. 100, 103904 (2008).
[CrossRef]

Z. Musslimani, K. Makris, R. El-Ganainy, D. N. Christodoulides, Phys. Rev. Lett. 100, 030402 (2008).
[CrossRef]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, Z. H. Musslimani, Opt. Lett. 32, 2632 (2007).
[CrossRef]

Christopoulos, C.

C. Christopoulos, The Transmission-Line Modeling Method: TLM (IEEE, 1995).

Collin, R. E.

R. E. Collin, Field Theory of Guided Waves, 2nd ed. (IEEE, 1991).

Ctyroký, J.

Doerr, C. R.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, H. Renner, Nat. Photonics 7, 579 (2013).
[CrossRef]

Eich, M.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, H. Renner, Nat. Photonics 7, 579 (2013).
[CrossRef]

Eichelkraut, T.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, D. N. Christodoulides, Phys. Rev. Lett. 106, 213901 (2011).
[CrossRef]

El-Ganainy, R.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

H. Ramezani, T. Kottos, R. El-Ganainy, D. N. Christodoulides, Phys. Rev. A 82, 043803 (2010).
[CrossRef]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, Phys. Rev. Lett. 100, 103904 (2008).
[CrossRef]

Z. Musslimani, K. Makris, R. El-Ganainy, D. N. Christodoulides, Phys. Rev. Lett. 100, 030402 (2008).
[CrossRef]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, Z. H. Musslimani, Opt. Lett. 32, 2632 (2007).
[CrossRef]

Eyderman, S.

Fan, S.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, H. Renner, Nat. Photonics 7, 579 (2013).
[CrossRef]

Fegadolli, W. S.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, A. Scherer, Nat. Mater. 12, 108 (2013).
[CrossRef]

Feng, L.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, A. Scherer, Nat. Mater. 12, 108 (2013).
[CrossRef]

Freude, W.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, H. Renner, Nat. Photonics 7, 579 (2013).
[CrossRef]

Gopal, A. V.

S. Lan, A. V. Gopal, K. Kanamoto, H. Ishikawa, Appl. Phys. Lett. 84, 5124 (2004).
[CrossRef]

Greenberg, M.

Hagness, S. C.

S. C. Hagness, R. M. Joseph, A. Taflove, Radio Sci. 31, 931 (1996).
[CrossRef]

Hammer, M.

A. Suryanto, E. van Groesen, M. Hammer, H. J. W. M. Hoekstra, Opt. Quantum Electron. 35, 313 (2003).
[CrossRef]

Hoekstra, H. J. W. M.

A. Suryanto, E. van Groesen, M. Hammer, H. J. W. M. Hoekstra, Opt. Quantum Electron. 35, 313 (2003).
[CrossRef]

Huber, C.

M. Krumpholz, C. Huber, P. Russer, IEEE Trans. Microwave Theor. Tech. 43, 1935 (1995).
[CrossRef]

Ishikawa, H.

S. Lan, A. V. Gopal, K. Kanamoto, H. Ishikawa, Appl. Phys. Lett. 84, 5124 (2004).
[CrossRef]

Jalas, D.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, H. Renner, Nat. Photonics 7, 579 (2013).
[CrossRef]

Janyani, V.

V. Janyani, A. Vukovic, J. D. Paul, P. Sewell, T. M. Benson, Opt. Quantum Electron. 37, 3 (2005).
[CrossRef]

Joannopoulos, J. D.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, H. Renner, Nat. Photonics 7, 579 (2013).
[CrossRef]

Joseph, R. M.

S. C. Hagness, R. M. Joseph, A. Taflove, Radio Sci. 31, 931 (1996).
[CrossRef]

Kanamoto, K.

S. Lan, A. V. Gopal, K. Kanamoto, H. Ishikawa, Appl. Phys. Lett. 84, 5124 (2004).
[CrossRef]

Kip, D.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

Kivshar, Y. S.

A. A. Sukhorukov, Z. Xu, Y. S. Kivshar, Phys. Rev. A 82, 043818 (2010).
[CrossRef]

Kottos, T.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, D. N. Christodoulides, Phys. Rev. Lett. 106, 213901 (2011).
[CrossRef]

H. Ramezani, T. Kottos, R. El-Ganainy, D. N. Christodoulides, Phys. Rev. A 82, 043803 (2010).
[CrossRef]

Kress, B.

Krumpholz, M.

M. Krumpholz, C. Huber, P. Russer, IEEE Trans. Microwave Theor. Tech. 43, 1935 (1995).
[CrossRef]

Kulishov, M.

Kuzmiak, V.

Lan, S.

S. Lan, A. V. Gopal, K. Kanamoto, H. Ishikawa, Appl. Phys. Lett. 84, 5124 (2004).
[CrossRef]

Laniel, J. M.

Li, G.

M. Bass, G. Li, E. van Stryland, Handbook of Optics, 3rd ed. (McGraw-Hill, 2010), Vol. 4.

Lin, Z.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, D. N. Christodoulides, Phys. Rev. Lett. 106, 213901 (2011).
[CrossRef]

Lu, M.-H.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, A. Scherer, Nat. Mater. 12, 108 (2013).
[CrossRef]

Makris, K.

Z. Musslimani, K. Makris, R. El-Ganainy, D. N. Christodoulides, Phys. Rev. Lett. 100, 030402 (2008).
[CrossRef]

Makris, K. G.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, Phys. Rev. Lett. 100, 103904 (2008).
[CrossRef]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, Z. H. Musslimani, Opt. Lett. 32, 2632 (2007).
[CrossRef]

Meisinger, P. N.

C. M. Bender, S. Boettcher, P. N. Meisinger, J. Math. Phys. 40, 2201 (1999).
[CrossRef]

Melloni, A.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, H. Renner, Nat. Photonics 7, 579 (2013).
[CrossRef]

Miri, M.-A.

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, U. Peschel, Nature 488, 167 (2012).
[CrossRef]

Moravvej-Farshi, M. K.

Musslimani, Z.

Z. Musslimani, K. Makris, R. El-Ganainy, D. N. Christodoulides, Phys. Rev. Lett. 100, 030402 (2008).
[CrossRef]

Musslimani, Z. H.

Nazari, F.

Nazari, M.

Oliveira, J. E. B.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, A. Scherer, Nat. Mater. 12, 108 (2013).
[CrossRef]

Onishchukov, G.

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, U. Peschel, Nature 488, 167 (2012).
[CrossRef]

Orenstein, M.

Paul, J. D.

V. Janyani, A. Vukovic, J. D. Paul, P. Sewell, T. M. Benson, Opt. Quantum Electron. 37, 3 (2005).
[CrossRef]

Peschel, U.

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, U. Peschel, Nature 488, 167 (2012).
[CrossRef]

Petrov, A.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, H. Renner, Nat. Photonics 7, 579 (2013).
[CrossRef]

Phang, S.

Plant, D. V.

Popovic, M.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, H. Renner, Nat. Photonics 7, 579 (2013).
[CrossRef]

Ramezani, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, D. N. Christodoulides, Phys. Rev. Lett. 106, 213901 (2011).
[CrossRef]

H. Ramezani, T. Kottos, R. El-Ganainy, D. N. Christodoulides, Phys. Rev. A 82, 043803 (2010).
[CrossRef]

Regensburger, A.

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, U. Peschel, Nature 488, 167 (2012).
[CrossRef]

Renner, H.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, H. Renner, Nat. Photonics 7, 579 (2013).
[CrossRef]

Russer, P.

M. Krumpholz, C. Huber, P. Russer, IEEE Trans. Microwave Theor. Tech. 43, 1935 (1995).
[CrossRef]

Rüter, C. E.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

Scherer, A.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, A. Scherer, Nat. Mater. 12, 108 (2013).
[CrossRef]

Segev, M.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

Sewell, P.

S. Phang, A. Vukovic, H. Susanto, T. M. Benson, P. Sewell, J. Opt. Soc. Am. B 30, 2984 (2013).
[CrossRef]

V. Janyani, A. Vukovic, J. D. Paul, P. Sewell, T. M. Benson, Opt. Quantum Electron. 37, 3 (2005).
[CrossRef]

Slavík, R.

Sukhorukov, A. A.

A. A. Sukhorukov, Z. Xu, Y. S. Kivshar, Phys. Rev. A 82, 043818 (2010).
[CrossRef]

Suryanto, A.

A. Suryanto, E. van Groesen, M. Hammer, H. J. W. M. Hoekstra, Opt. Quantum Electron. 35, 313 (2003).
[CrossRef]

Susanto, H.

Taflove, A.

S. C. Hagness, R. M. Joseph, A. Taflove, Radio Sci. 31, 931 (1996).
[CrossRef]

van Groesen, E.

A. Suryanto, E. van Groesen, M. Hammer, H. J. W. M. Hoekstra, Opt. Quantum Electron. 35, 313 (2003).
[CrossRef]

van Stryland, E.

M. Bass, G. Li, E. van Stryland, Handbook of Optics, 3rd ed. (McGraw-Hill, 2010), Vol. 4.

Vanwolleghem, M.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, H. Renner, Nat. Photonics 7, 579 (2013).
[CrossRef]

Vukovic, A.

S. Phang, A. Vukovic, H. Susanto, T. M. Benson, P. Sewell, J. Opt. Soc. Am. B 30, 2984 (2013).
[CrossRef]

V. Janyani, A. Vukovic, J. D. Paul, P. Sewell, T. M. Benson, Opt. Quantum Electron. 37, 3 (2005).
[CrossRef]

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Figures (5)

Fig. 1.
Fig. 1.

(a) Schematic of an N -period PTBG in a background material n B , (b) single period of a grating with n H and n L as the high and low real refractive indices at f B in even symmetry and gain (green) and loss (orange) in odd symmetry.

Fig. 2.
Fig. 2.

(a) Transmittance and (b) reflectance for left and (c) right incidence of PTBG as a function of frequency considering a linear medium ( n 2 = 0 ) at U condition ( | α 0 | = 1460.24 cm 1 ) with no gain/loss saturation ( S = 1 ). The results obtained using the T-matrix method for the idealized gain/loss model are included for reference.

Fig. 3.
Fig. 3.

Hysteresis of an NPTBG with high saturation intensity and for different gain/loss parameters | α 0 | = 800 cm 1 and 1460.24 cm 1 (U operation): (a) transmittance T L , (c) reflectance Γ L for the signal incident from the left; (b) transmittance T R , (d) reflectance Γ R or the signal incident from the right of the grating. Saturation intensity is I S = 5 × 10 13 Wm 2 . The dashed line represents the response of the NBG for reference.

Fig. 4.
Fig. 4.

Hysteresis of NPTBG operating at f = 337.7 THz as a function of input signal intensity for two different values of gain/loss parameters, | α 0 | = 800 cm 1 and 1460.24 cm 1 (U operation). Gain and loss saturation is turned off ( S = 1 ). (a) The transmittance T L and (c) reflectance Γ L for left incidence; (b) the transmittance T R and (d) reflectance Γ R for the right incidence. Hysteresis of the NBG is included for reference.

Fig. 5.
Fig. 5.

(a) Time and (b) frequency response of the transmitted electric field of the NPTBG at U operation ( | α 0 | = 1460.24 cm 1 ) with input intensity I = 2.6 × 10 14 Wm 2 and the incidence from the left side of the grating.

Equations (4)

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n ^ G ( z , ω , I , t ) = { n ^ H ( ω ) + n 2 I ( z , t ) + j c ω α ( ω , I ) , z < Λ 4 n ^ L ( ω ) + n 2 I ( z , t ) + j c ω α ( ω , I ) , Λ 4 < z < Λ 2 n ^ L ( ω ) + n 2 I ( z , t ) j c ω α ( ω , I ) , Λ 2 < z < 3 Λ 4 n ^ H ( ω ) + n 2 I ( z , t ) j c ω α ( ω , I ) , 3 Λ 4 < z < Λ ,
n ^ 2 = ( 1 + χ e ) + χ e 0 ω 0 D 2 2 j ω δ + ( ω 0 D 2 ω 2 ) .
| α | ( ω , I ) = S ( I ) ( | α 0 | 1 + j ( ω ω 0 σ ) τ + | α 0 | 1 + j ( ω + ω 0 σ ) τ ) .
S ( I ) = 1 1 + I / I s ,

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