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Научни трудове на Съюза на учените в България-Пловдив. Серия В. Техника и технологии, естествен ии хуманитарни науки, том XVI., Съюз на учените сесия "Международна конференция на младите учени" 13-15 юни 2013. Scientific research of the Union of Scientists in Bulgaria-Plovdiv, series C. Natural Sciences and Humanities, Vol. XVI, ISSN 1311-9192, Union of Scientists, International Conference of Young Scientists, 13 - 15 June 2013, Plovdiv.
Phase modulated pulses into a nonlinear regime of propagation
Valeri Slavchev(1'2), Lubomir Kovachev(2) (1): Medical university of Plovdiv, bul. V. Aprilov 15-A, Plovdiv (2): Institute of electronics, BAS - Sofia, bul. Tzarigradsko chaussee 72
Abstract
This work examines the influence of a bi-convex short throw lens on the evolution of optical pulses in nonlinear dispersion medium. We investigated different regime of propagation by changing the value of the coefficients characterizing the dispersion and nonlinearity of the medium. We have made a numerical modulation of these processes. The optimal regime for a nonlinear focusing of a laser pulse, depending on the sign of dispersion and focal distance of convex lens, is found.
1. Introduction
With development of communication systems particularly important became the study of the evolution of optical pulses in different media. The linear regime of propagation without initial phase modulation is well known [1]. The behavior of optical pulses without initial phase modulation in nonlinear dispersive media has been studied analytically [1-3]. The initial spatially phase modulation and the width of the pulse have a significant influence on the simultaneous action of the dispersion and diffraction. There is different linear and nonlinear mechanisms for spatio - temporal modulation of the pulse. We examined different modes of propagation by changing of the focal length of the lens.
2. Basic equation
The scalar paraxial (3D+1) amplitude equation describing the propagation of an optical pulse in a nonlinear dispersive medium has the form:
.dA 1 d2A | ,i2
(1)
Where A(x,y,z,t) is the scalar amplitude function, characterizing the pulse envelope, k0 - is the wave number. p = Zd'ff/ is a dimensionless parameter giving the ration between the
/ Zd'sp
dispersion and the diffraction length of the pulse, zd'Sp = tQ / k" is the dispersion length, Zdiff = k0rl - the diffraction length. The nonlinearity of the medium is of Kerr type. It is
2 2 1 12 d2 d2 determined by the expression y = k0r_Ln2 A0 . a =__i__is the Laplace operator.
dx2 dx2
Equation (1) is more complex than commonly used nonlinear Schrodinger equation [4, 5]. However, it accounts effects of diffraction, higher orders of the nonlinear dispersion and
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nonlinearity of the medium. Therefore, we assume that the equation (1) describes more detail and correctly the evolution of three dimensional optical pulses. 3. Lens as a phase corrector
The influence of the double convex lens on the phase of the optical pulse is examined in [6]. The lens has a behavior of a quadratic phase corrector when laser beam passes through it. The additional phase, which it obtains, is:
f
n
0(x, y) =
(2)
[a2 - (x2 + y2)]
where X0 is the carrier wavelength, and a and f are respectively the radius of the aperture of the lens and the focal distance.
4. Numerical modeling of the evolution of optical pulses
We investigated the following regimes of propagation of optical pulses with carrier wavelength X0 = 800nm and z diff = 7.85:
1) Nonlinear regime ofpropagation of initial unmodulated pulse
2) Nonlinear regime ofpropagation of initial modulated pulse
4.1 Nonlinear regime of propagation of initial unmodulated pulse
We investigated the evolution of an optical pulse without initial phase modulation in dispersion medium with Kerr type nonlinearity. We assumed that coefficient of nonlinearity is slightly higher than the critical for self - focusing y = 1.8. We examined the behavior of a pulse in
media without dispersion P = 0.0005, with normal (P = 0.5) and anomalous
dispersion (P = -0.5) . The results are shown in Figure 1, Figure 2 and Figure 3.
i — a — r* IT ii 7. — ru * — fz
Figure 1: Propagation of initial unmodulated optical pulse in a medium where the ratio between the dispersive and diffraction length is negligible. A typical example of fi<<1 is the evolution of nano and picoseconds optical pulse in air. Coefficient of nonlinearity is y =1.8
Figure 1 shows the initial self - focusing and the formation of a broad pedestal. The evolution of optical pulse in absence of dispersion is similar to the evolution of an optical beam.
t™0 Z-iMliJ i-iiiir t-iUJlii
Figure 2: Propagation of initial unmodulated optical pulse in a medium with normal dispersion fi = 0.5 and nonlinearity coefficient y =1.8.
During the evolution of the initial unmodulated pulse in a medium with normal dispersion it is observed a suppression of the nonlinearity and an increasing of the threshold for self -focusing. This effect is similar to the nonlinear regime of propagation for an optical pulse in a positive region of dispersion of the optical fiber.
j—If—1 1 £
x-Q t - i-iM ; - iiMttl
Figure 3: Propagation of an unmodulated optical pulse in a medium with anomalous dispersion p = - 0.5 and nonlinearity coefficient y=1.8.
Figure 3 shows the propagation of a pulse in a medium with anomalous dispersion. The spot of the pulse is self - compressed and it is observed an amplification, self - focusing and significant increasing of the amplitude.
4.2 Nonlinear regime of propagation of initial modulated pulse
The initial phase modulation of the pulse is achieved, when it passes through the short throw double convex lens. It is determined by the formula (2). We have assumed that the focus of the lens is f = 7.85cm . We have investigated the evolution of an initial modulated pulse in a medium with normal (ft = 0.5) dispersion, anomalous dispersion(ft = -0.5) and a coefficient of nonlinearity y = 1.8 . The numerical results are shown in Figure 4 and Figure 5.
L ~ 0 ' ~ ?i.\\ - Z — Akfl L ~ yi 2
Figure 4: Propagation of initial modulated pulse in a medium with normal dispersion p=0.5
and y =1.8.
t — a t. — j+n ra 7 — K — .HiiflVj
Figure 5: Propagation of initial modulated pulse in a medium with anomalous dispersion
fi= - 0.5 and y =1.8.
The phase modulation obtained from the lens significantly influences the behavior of the pulses. In the case of propagation of the modulated pulse in a medium with normal dispersion (Fig. 4) initially (z = zdiff) it is observed a focusing of pulse, after which the pulse is
expanded and the amplitude is reduced. In a case of propagation of the modulated pulse in a medium with anomalous dispersion it is observed very strong self - focusing which leads to a compression of the pulse and a significant increasing of the amplitude in comparison with the results obtained without a lens (Fig. 4).
5. Conclusion
The initial phase modulation caused by the short throw double convex lens also affects the propagation of optical pulses in an isotropic nonlinear dispersive medium. During the evolution of a pulse in a medium with a positive linear dispersion, we initially observe a self - focusing. After the focus of the lens, the paraxiality of the diffraction is preserved and the pulse diffracted in several diffraction lengths without wavefront distortion. The propagation in a medium with negative dispersion the initial phase modulation increases the compression of the pulse. Numerical modeling of the optical pulses in a nonlinear regime of propagation with initial phase modulation caused by different optical elements has important application in the modern optical engineering.
6. References
1. Lubomir M. Kovachev and Kamen Kovachev, "Linear and Nonlinear Femtosecond Optics in Isotropic Media. Ionization-free Filamentation", Laser Pulses / Book 1, chapter, ISBN 978-953-307-429-0, InTech, (2011).
2. Kovachev L. M., //J.of Modern Optics, val.56, 16, (2009).
3. Agrawal, G. P., Nonlinear fiber optics, Academic Press, INC, New York (2007).
4. Dakova, A., Dakova, D., "Nonlinear regime of propagation of femtosecond optical pulses in single-mode fiber", Proc. SPIE 8770, 17th International School on Quantum Electronics: Laser Physics and Applications (2013).
5. Tarasov, L. V., Physics of Processes in Generators of Coherent Optical Radiation, Radio i Svyaz', Moscow (1981).
6. Slavchev, V., Kovachev, L., „The lens as a phase corrector", University of Plovdiv „Paisii Hilendarski", Scientific studies, Physics, Vol. 36, Fasc. 4 (2011)h.
