Bragg solitons

Bragg Solitons

Research Scholar

Center for Nanoscience and Engineering

Indian Institute of Science, Bangalore

Course presentation: Nonlinear Photonics

Ajay Singh

Solitons Bragg Grating Bragg Solitons Mathematical Analysis Experimental Applications

Fiber Brag Grating

Optical solitons: From fibers to photonic crystals, “Yuri S. Kivshar and Govind P. Agrawal” cademic PressElsevier Science, USA (2003)


Fiber Brag Grating

How to write:

Irradiation from UV

Irradiate germanium doped

fiber with argon laser

Holographic technique

kg= 2π/Λ

Λ =~o.5 µm for λ=1.55 µm region



Fiber Brag Grating Coupled-Mode Equations

How to write:

Irradiation from UV

Irradiate germanium doped

fiber with argon laser


kg= 2π/Λ

Λ =~o.5 µm for λ=1.55 µm region


Coupled-Mode Equations and Nonlinear Propagation Equations



Coupled-Mode Equations and Nonlinear Propagation Equations

Anomalous

GVD

Normal

GVD


Optical solitons: From fibers to photonic crystals, “Yuri S. Kivshar and Govind P. Agrawal” cademic PressElsevier Science, USA (2003) Link


Introduction

Nonlinearity (SPM)

Dispersion (Material)

Soliton

Nonlinearity (SPM)

Dispersion (Grating)

Soliton Like Pulses

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Effective NLS Equation


Now introduce the speed reduction factor f:



Fig.. Intensity (solid line) and normalized spatial width (dot-dashed line) of fundamental soliton pulses, [6].





Fig.. Intensity (solid line) and normalized spatial width (dot-dashed line) of fundamental soliton pulses, [6].

Fig. Evoluation of (a) |Af|2 and (b) |Ab|2 [6].




Let us consider different cases of Bragg solitons

1-<v<1

Ψ=π/2: Center of the stop band: Combination of two counter-propogating waves VG = v.νg

If equal amplitude: νg= 0 Stationary gap soliton

|v|=1

Bragg soliton ceases to exist since the grating becomes ineffactive

General solution (coupled mode eqn) of shape preserving solitons: Solition exist in normal

Dispersion region also: Dark solitions

Video: Propagation of a single bragg soliton in a uniform grating [17]


B. J. Eggleton, R. E. Slusher (Bell Laboratories, Lucent Technologies, New Jersey 07974) and C.

Martijn de Sterke (University of Sydney): Vol. 16, No. 4/ April 1999/J. Opt. Soc. Am. B [7]

Fig. Schematic of our experimental setup.





Link





Link

818 m-1

3612 m-1

Link




Fig. 16. transmitted pulse for incoming pulse energies of 0.66 µJ Left experimental results; right follow numerical calculation.(solid curves) The values of the detuning are 876 m-

1 [traces (a) and (b)], 994 m-1 [traces (c) and (d)], 1318 m-1 [traces (e) and (f )], and 1847 m-1 [traces (g) and (h)].

Inte

nsi

ty (

a.u

.)

Link

Link




Link

Mathematical Analysis…..

• All-optical switching [5,8,9],

• Pulse compression [10,11,25],

• Pulse Limiting [12],

• Logic operations [13],

• Promising for the fiber-sensing technology [14],

• Optical communication systems [15,16],

• All optical buffers and storing devices can be based on such fibers [18],

• Pulse source for the soliton transmission system [19,20 ],

• All-optical modulation and demultiplexing systems [23],

• Tunable optical pulse source [24, 26],

• A possible way to trap a zero-velocity soliton is to use an attractive

finite-size or Local defect in BG [22] ,

• THz pulse generation !!!

• Suppercontinuum generation !!!


Applications of Bragg Solitons…..

Solitons Bragg Grating Bragg Solitons Mathematical Analysis Applications

References:

[1] K. O. Hill, Y. Fujii, D. C. Johnson, and B. S. Kawasaki, Appl. Phys. Lett., Vol. 32, pp. 647, 1978. [2] E. Yablonovitch and T. J. Gmitter, Phys. Rev. Lett., Vol. 63, pp. 1950, 1989. [3] A. A. Sukhorukov, Y. S. Kivshar, H. S. Eisenberg and Y. Silberberg, IEEE J. Quantum Electron, Vol. 39, pp. 31, 2003. [4] W. Chen and D. L. Mills, Phys. Rev. Lett., Vol. 58, pp. 160, 1987. [5] S. Larochelle, V. Mizrahi, and G. Stegeman, Electron. Lett., Vol. 26, pp. 1459, 1990. [6] A. B. Aceves and S. Wabnitz, Phys. Lett. A, Vol. 141, pp. 37, 1989. [7] B. J. Eggleton, C. M. de, Sterke, and R. E. Slusher, J. Opt. Soc. Am. B, Vol. 16, pp. 587, 1999. [8] N. G. R. Broderick, D. Taverner, and D. J. Richardson, Opt. Express, Vol. 3, pp. 447, 1998. [9] Amer Kotb and Kyriakos, Optical Engineering (SPIE), Optical Engineering , Vol. 55(8), pp. 087109 (1-7), 2016. [10] N. G. R. Broderick, D. Taverner, D. J. Richardson, M. Ibsen and R. I. Laming, Opt. Lett. Vol. 22, pp. 1837, 1997. [11] N. G. R. Broderick, D. Taverner, D. J. Richardson, M. Ibsen and R. I. Laming, Phys. Rev. Lett.,Vol.79, pp. 4566, 1997. [12] D. E. Pelinovsky, L. Brzozowski, and E. H. Sargent. Phys. Rev. E, Vol. 62, pp. 4536, 2000. [13]. L. Brzozowski and E. H. Sargent, IEEE J. Quantum Electron, Vol. 36, pp. 550, 2000. [14] W. C. K. Mak, B. A. Malomed, and P. L. Chu, J. Opt. Soc. Am. B, Vol. 20, pp. 725, 2003. [15] R. H. Goodman, R. E. Slusher, and M. I. Weinstein, J. Opt. Soc. Am. B, Vol. 19, pp. 1635, 2002. [16] G. P. Agrawal, Nonlinear Fiber Optics. (New York: Academic), 1989. [17]. https://www.youtube.com/watch?v=GxmjYNuaoSE [18]. Xiaolu Li, Yuesong Jiang, and Lijun Xu, Communication and Network, Vol. 2, pp. 44, 2010 [19] N. Dogru and M. S. Ozyazici, LFNM. September 2004, pp. 115. [20]. N. Dogru, 2005, IEEE NUSOD’05, September 2005, pp.89. [21] C. M. de Sterke, B. J. Eggleton, and P. A. Krug, J. Lightwave Technol, Vol. 15, pp. 1494, 1997. [22] K. T. Mc-Donald, Am. J. Phys., Vol. 68, pp. 293, 2000. [23]. J. H. Lee, L. Katsuo, K. S. Berg, A. T. Clausen, D. J. Richardson, and P. Jeppesen, J. Lightwave Technol, Vol. 21, pp. 2518, 2003. [24]. J. H. Lee, Y. M. Chang, Y. G. Han, S. H. Kim, H. Chung,and S. B. Lee, IEEE Photon. Technol. Lett., Vol. 17, pp. 34, 2005. [25]. G. Lenz and B. J. Eggleton, J. Opt. Soc. Am. B, Vol. 15, pp. 2980, 1998. [26] Norihiko Nishizawa, Yoshimichi Andou, Emiko Omoda, Hiromichi Kataura, and Youichi Sakakibara, Opt. Exp. 23403, Vol. 24, 2016.

1553.2 nm Q-switching of the laser at a repetition rate of 500 Hz. train of pulses, each with a duration of approximately 80 ps. To avoid thermal effects, an electro-optic pulse selector was used to select one pulse in each train. fast photodiode with a response time of 9 ps The net time resolution, including trigger jitter and oscilloscope resolution, was approximately 20 ps. 75-mm-long unchirped, apodized fiber grating

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Historical prospective.…

Schematic illustration of a fiber grating. Dark and light shaded regions withi n the fiber core show periodic variations of the refractive index.


First photo-induced optical fiber Bragg gratings

by Hill and coworkers in 1978 [1]

Photonic band structure” concept: Yablonovitch

in the late 1980’s [2]

Light interaction with nonlinear periodic media

yields a diversity of fascinating phenomenan:

discrete (or lattice) solitons and gap (or Bragg)

solitons [11-12]

Larochelle, Hihino, Mizrahi and Stegeman (in

1990): experimental investigation of the optical

response of nonlinear periodic structures.

investigation of nonlinear pulse propagation in

uniform fiber gratings: the Bragg solitons easily

generated in the laboratory travel at 60–80% of

veocity of light in fiber absence of grating [5]

Fiber Brag Grating

How to write:

Irradiation from UV

Irradiate germanium

doped fiber with argon

laser


kg= 2π/Λ

Λ =~o.5 µm for λ=1.55 µm

region



Coupled-Mode Equations

And here dispersion comes into play…..

Figure : Dispersion curves showing variation of 6 with q and the existence of the photonic bandgap for a fiber grating.

Anomalous

GVD

Normal

GVD

o β2 changes its sign on the

two sides of the stop band

centered at the Bragg

wavelength, whose

location is easily

controlled and can be in

any region of the optical

spectrum.

o β2 is anomalous on the

shorter-wavelength side

of the stop band, unlike

for wavelengths longer

than the zero dispersion

wave-length in the fibers

o The magnitude of β2 >

100ps2/cm


o the negative sign appearing in front of the ∂Ab/∂Z term: because of backward propagation of Ab and

o the presence of linear coupling between the counter propagating waves governed by the parameter k

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Fig. 13. Comparison between the estimated pulse energy in the fiber and the deduced value of the peak intensity. The result, which is consistent with the expected proportionality between the two quantities, leads to a proportionality constant of 47 GW/cm2 µJ-1.

Fig. 8. Intensity versus time after propagation through the grating at a pulse energy of approximately 0.66 µJ, for a detuning close to the edge of the gap (solid curve) and far from the edge of the gap (dashed curve). The pulse tuned close to the edge of the gap is delayed by approximately 310 ps, corresponding to an average velocity of approximately 0.50V.

Fig. 7. Intensity versus time after propagation through the grating at a peak input intensity of 11 GW/cm2, for seven different values of the detuning (solid curve, 729 m-1; dotted curve, 788 m-1; short-dashed curve, 847 m-1; long-dashed curve, 935 m-1; short-dashed–dotted curve, 1053 m-1; long-dashed–dotted curve, 1406 m-1; long–short-dashed curve, 3612 m-1)

Fig. 6. Delay of the transmitted pulses versus detuning at low intensity. Experimental results are indicated by dots, numerical results by the solid curve.



Martijn de Sterke (University of Sydney): Vol. 16, No. 4/ April 1999/J. Opt. Soc. Am. B

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Fig. 16. transmitted pulse for incoming pulse energies of 0.66 µJ at 4 different detunings. Left figures are experimental results; right follow from solving Eq. (1) numerically and convolving the result with the estimated response of the detection system (solid curves) and from the NLSE followed by a convolution (dashed curves). The values of the detuning are 876 m-1 [traces (a) and (b)], 994 m-1 [traces (c) and (d)], 1318 m-1 [traces (e) and (f )], and 1847 m-1 [traces (g) and (h)].

Fig. 9. FWHM of the transmitted pulses versus detuning at an estimated pulse energy of 0.06 µJ. Experimental results are indicated by dots, numerical results obtained by solving Eq. (1) are indicated by the solid curve, and numerical obtained by solving q. (10) are indicated by the dashed curve; in both sets of numerical results the peak intensity of the incoming pulse is taken to be 3 GW/cm2.

Fig. 15. Solid curve: Estimated peak power of the fundamental soliton versus detuning with the parameters from the text. Dots: Similar results, but deduced from the experiments, with the criterion that the width does not change upon propagation.



Martijn de Sterke (University of Sydney): Vol. 16, No. 4/ April 1999/J. Opt. Soc. Am. B

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Fig. 5. Full-width at half-maximum (FWHM) of the transmitted pulses versus detuning at low intensity. Experimental results are indicated by dots, numerical results by the solid curve.





Fig. Intensity versus time after propagation through the grating at low intensity, for seven different values of the detuning (solid curve, 818 m-1; dotted curve, 847 m-1; short-dashed curve, 906 m-1; long-dashed curve, 965 m-1; short-dashed– dotted curve, 1023 m-1; long-dashed–dotted curve, 1171 m-1; long–short-dashed curve, 3612 m-

1).

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Bragg solitons

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