Stimulated Raman Scattering - Third Order Nonlinear Optical Effects

Chapter 1 Introduction

1.3 Third Order Nonlinear Optical Effects

1.3.5 Stimulated Raman Scattering

) is a photon-phonon interaction. It is described quan

Fig. 1.3 Raman-gain spectrum for fused silica at a pump wavelength λp = 1 μm.

Stimulated Raman scattering (SRS

tum-mechanically as scattering of a photon by one of the molecules to a lower frequency photon, while the molecule makes transition to a higher energy vibrational state. SRS can yield gain for a probe wave co-propagating with a pump wave and whose wavelength is located within the Raman gain bandwidth. The normalized Raman gain spectrum of silica is shown in Fig. 1.3 as a function of frequency difference between the pump and probe waves [26]. The Raman gain of the MFs is comparable to that of silica fibers [27]. The gain bandwidth is 40 THz with a peak located at 13.2 THz from the pump frequency. For an ultra-short pulse, the spectral width of the pulse is large enough that the Raman gain can amplify the low-frequency (red) spectral components of the pulse, with high-frequency (blue) components of the same pulse acting as a pump. This effect is called intrapulse Raman scattering [21]. As a result, the pulse spectrum shifts toward the low-frequency (red) side as the pulse propagates inside the fiber, a phenomenon referred to as the self-frequency shift.

1.3.6

results from the dispersion of the third-order susceptibility, i.e., the red

.3.7 Higher-Order Dispersion

ect becomes important in optical fibers when the carrier Self-Steepening

Self-steepening (SS)

frequency components experience a lower nonlinearity than the blue frequency components. In the time domain, SS can be thought as the intensity dependence of the group velocity: The peak of the pulse moves at a slower velocity than the wings which induces the trailing edge of the pulse to become steeper as the pulse propagates [21]. In combination with SPM, SS results in a more pronounced broadening of the blue frequency components compared to the red ones. The process of self-frequency shift is substantially reduced by SS since the nonlinearity decreases as the center wavelength of the soliton shifts towards the red.

Higher-order dispersion (HOD) eff

frequency is close to the zero dispersion point. Once the spectrum extends beyond the zero dispersion point to the normal dispersion region, the spectrum will be disturbed by the HOD to generate a new dispersive wave [see Fig 1.4]. This is because when accounting the higher-order dispersion, the wavenumber of the propagating pulse is the same as the dispersive wave so that the energy can transfer from the pulse to the dispersive wave.

Fig. 1.4 The dispersive wave generated at the normal dispersion due to the perturbation of HOD [24].

1.4 Motivation

There are so many applications of supercontinuum generated by microstructured fiber like optical switching [28], wavelength conversion [29], tunable filters [30], etc., and a broad spectrum from UV to NIR will be a key factor to suffice these applications. The spectrum on the IR part can be easily generated by SSFS effect because of the Raman gain of silica fiber, and the spectrum on the UV part can be generated by the dispersive waves from soliton fission and the XPM between the solitons and the dispersive waves. To obtain a dispersive wave at a certain wavelength, phase-matching condition has to be fulfilled. And in contrast, it has been shown theoretically and experimentally that the blue dispersive wave can be further shifted by cross-phase modulation (XPM) initiated by the infrared soliton [31].

Additionally, a theoretical suggestion has been given to increase the bandwidth in the visible by co-propagating the pump pulse with a pulse in the visible and thus, exploiting the XPM

between the two [32]. Although some experiments have been done using two spectrally distinct ns, ps and fs pulses to achieve additional broadening [33], the XPM induced frequency shifts in supercontinuum generation with dual pumped fundamental and second-harmonic femtosecond pulses has not yet been reported experimentally. In this thesis, we investigate the XPM induced shift in the case of supercontinuum generation numerically.

1.5 Organization of this Thesis

In Chapter 2, we will describe how to simulate the coupled nonlinear Schrödinger equation. Then, we will introduce our experiment in Chapter 3, including the pumping laser system, the specification of our MFs, the experimental setup single and dual wavelength pumping. Chapter 4 is the experimental results and discussion. We will compare the experimental results with the numerical results in this chapter. Finally, we will give a conclusion and the future works in Chapter 5.

Chapter 2 Simulation of Nonlinear Schrödinger Equation

2.1 Coupled Nonlinear Schrödinger Equations

An electromagnetic field propagating in a medium induces a polarization of the electric dipoles. The evolution of the electromagnetic field in the medium can be described by a propagation equation derived from the general wave equation [21]

where E is the electric field, P the induced polarization, μ0 the vacuum permeability and c the speed of light in vacuum. For intense radiation such as ps or fs laser pulses, the response of the medium becomes nonlinear and the induced polarization consists of a linear and a nonlinear parts. In the scalar approximation, the linear and nonlinear induced polarizations are related to the electromagnetic field as [21]

E inversion symmetry of silica glass at the molecular level results in negligible even-order susceptibilities. Moreover, susceptibilities of the order higher than three are not significant for

silica glass. Therefore, the relevant nonlinear effects in optical fibers are mainly induced by χ⁽³⁾ [21].

Optical nonlinear processes can be divided into two categories. Elastic processes correspond to photon-photon interaction with no energy exchange occurring between the electric field and the medium. Such effects include SPM, DFWM, and generation of dispersive wave. Inelastic processes correspond to photon-phonon interaction, which leads to energy exchange between the electric field and the nonlinear medium. Raman scattering is one of the effects of inelastic process. Treating the nonlinear part of the induced polarization as a perturbation in Eq. 2.1.1 and assuming that the electric field is of the form

)

where A(z,T) is the slowly varying envelope of the electric field, β is the propagation constant and ω0 is the center frequency of the field, one can derive the well-known NLSE. The NLSE models accurately the propagation of light along optical fibers for pulses as short as 30 fs [35].

In a frame of reference moving at the group velocity of the pulse, the NSE can be written as [21]

where α is the fiber loss and βn are the coefficients of the Taylor-series expansion of the propagation constant β around ω, and R(T) is the response function describing the interaction

between the photon and medium. The response function should include both the instantaneous response (interaction between electron and photon) and the delayed Raman response (interaction between photon and phonon) and is given by

where fR represents the fractional contribution of the delayed Raman response function hR(T).

The value of fR is typically 0.18 and hR(T) can be presented as [21] delayed Raman response hR(T) can describe the phenomenon of intrapulse Raman scattering referred to the self-frequency shift. The right-hand side of Eq. 2.1.5 contains the nonlinear effects such as SPM, intrapulse Raman scattering and SS, the differential term which accounts for the dispersion of the nonlinear coefficient. On the left side of the Eq. 2.1.5, it presents not only the dispersion effect but also the fiber loss.

Fig. 2.1 Temporal variation of delayed Raman response function hR(T) for silica fibers [35].

In order to add the XPM effect to our simulation, we change the NLSE into a coupled NLSE as shown in Eq.2.1.8 and Eq. 2.1.9. [34] nonlinear coefficient, loss and dispersion of the second harmonic pulse, respectively. By simulating the NLSE, we can get the evolution of an optical pulse propagating in fibers and therefore realize the causes of the SC.

2.2 Fourier Transform

The NLSE is an important tool to analyze the evolution of a pulse propagating in fibers. By solving the NLSE, we can get the final spectrum of the pulse out of the fiber. In general, the NLSE is a nonlinear partial differential equation and doesn’t have an analytic solution. A numerical approach is therefore often necessary for understanding the nonlinear effects in optical fibers. A large number of numerical methods can be used for this purpose. These can be classified into two broad categories known as: (i) the finite-difference methods and (ii) the pseudospecral methods. Generally speaking, pseudospecral methods are faster by up to an order of magnitude to achieve the same accuracy [37]. It has been used extensively to solve the pulse-propagation problem in nonlinear dispersive media is the split-step Fourier method [38] [39].

In this thesis, we solve the NLSE using the split-step Fourier method. Other concepts such as the discrete Fourier transform (DFT), fast Fourier transform (FFT) and convolution theory should also be used in the simulation of NLSE. In this chapter, we will introduce the DFT, FFT and the convolution theory from 2.2.1 to 2.2.2. Then we will show how to use these tools to solve NLSE by the split-step Fourier method in detail in 2.3.

2.2.1 Discrete Fourier Transform

A physical process can be described in the time domain t and the frequency domain f as functions of h(t) and H(f) respectively. For many purposes it is useful to think h(t) and H(f) as being two different representations of a physical process. One goes back and forth between these two representations by means of the Fourier transform. We can get H(f) by the Fourier transform of h(t) or h(t) by the inverse Fourier transform of H(f) [40] .

∫

∞

In the most computational work, the function we deal is usually a train of sampled data at evenly spaced intervals in time. If we sample a continuous function h(t) to N consecutive values, h(t) can be presented as where ∆ is the time interval. According to the sampling theory, a continuous function h(t), sampled at an interval ∆, happens to be bandwidth limited to frequencies from - fc to fc where fc=1/2Δ [47]. Let us sample the frequency to N consecutive values inside the bandwidth, i.e.,

that there are N+1 values of n. It turns out that the two extreme values of n are not independent (in fact they are equal), but all the others are. This reduces the count to N.

We can approximate the integral in Eq. 2.2.1 by a discrete sum [40]:

∑

The relation between the DFT and continuous Fourier transform with a continuous function sampled at an interval ∆ can be written as

( )_n n

H f ≈H (2.2.7) From Eq. 2.2.4, we have seen that the index n varies from –N/2 to N/2. However, we find that Eq. 2.2.6 is periodic in n with period N. Therefore, H−n = HN−n, where n = 1, 2,…. With this conversion, we let Hn for n from 0 to N-1 form one complete period. When this convention is followed, we must remember that the zero frequency corresponds to n = 0, positive frequencies 0 < f < fc correspond to values 1 ≤ n ≤ N/2−1, while negative frequencies −fc < f <

0 correspond to N/2+1 ≤ n ≤ N−1. The value n = N/2 corresponds to both f = fc and f = −fc

[40].

The formula for the discrete inverse Fourier transform, which recovers the set of hk’s exactly

from the Hn’s is [40]: Notice that there are only two differences between Eq. 2.2.6 and Eq. 2.2.8. One is changing sign in the exponential and the other is further dividing by N. This means that a routine for calculating the DFT can also be used, with slight modification, to calculate the inverse transform.

2.2.2 Fast Fourier Transform

How much computation work is needed to compute the DFT? Let us introduce a new

complex number W: The vector hk is multiplied by a matrix W of the power n×k. This matrix multiplication finally requires N² complex multiplications, plus a smaller number of operations to generate the required powers of W. So, the DFT appears to be an order of N² processes. However, the computation work can be reduced to an order of N log2 N by means of FFT. The difference

weeks to finish N² computation whereas only 30 seconds for N log2 N for N=10⁶.

The DFT of length N can be rewritten as the sum of two DFTs, each of length N/2. One of the two is formed from the even-numbered points of original N, the other from the odd-numbered points [40].

denotes the k^th component of the Fourier transform of length N/2 formed from the even components, while is the corresponding transform of length N/2 formed from the odd

components. The dichotomy of the DFT can be used recursively. We can do the same reduction of to the two DFTs, each of length N/4. In other words, we can define and to be the DFTs of the points which are respectively even-even and even-odd on the successive subdivisions of the data. If we treat N as an integer power of 2, it is evident that we can continue applying the dichotomy until we have subdivided the data all the way down to transforms of length 1.

transform that is just one of the input numbers fn. The final step is to figure out which value of n corresponds to which pattern of e’s and o’s in Eq. 2.2.12. By using this information, we can

calculate Fk. This is what we call the FFT.

In our simulation, we use Compaq Visual Fortran 6.6. In the library of Compaq Visual Fortran 6.6, the instructions of the FFT and the inverse FFT are DFFTCF and DFFTCB.

The output of DFFTCF is Hn. That means we need to multiply the output by the interval Δ to get the Fourier transform H(fn). For DFFTCB, the output should be divided the sampled number N to get hk as Eq. 2.2.8. We should also notice that the sampled number N, the input in DFFTCF and DFFTCB, should be an integer of the power of 2.

2.2.3 Convolution Theory

A system is said to be linear if it satisfies the principle of superposition, i.e., if its response to the sum of any two inputs is the sum of its responses to each of the inputs separately. The output at time t is, in general, a weighted superposition of the input contributions at difference time τ [41],

τ τ τ f d t

h t

f₂( )

∫

^∞ (; ) ₁( )

∞

−

= , (2.2.13)

where f1(t), f2(t) and h(t;τ) are the input, the output of the linear system and a weighting

A linear system is said to be time-invariant or shift-invariant, if when the input of a linear system is shifted in time, its output shifts by an equal time, but otherwise remains the same.

Thenh(t;τ)can be written ash(t−τ)and Eq. 2.2.13 becomes [41] is known as the convolution theory presented as [41]

) part in NSE by the inverse Fourier transform of Eq. 3.1.14.

2.3 Split-step Fourier Method

One of the pseudospecral methods that have been used extensively to solve the pulse-propagation problem in nonlinear dispersive media is the split-step Fourier method. The main reason for the faster speed of the split-step method compared with the most finite-difference schemes is the use of the FFT.

To understand the philosophy behind the split-step Fourier method, it is useful to rewrite Eq. 2.1.5 in the form [21]

(

^Dˆ ^Nˆ

)

A= +

∂

∂ , (2.3.1)

where is a differential operator that accounts for the dispersion and absorption in a media and Nˆ is a nonlinear operator that presents the effect of fiber nonlinearities on pulse propagation.

These operators are given by [21]

Dˆ

In general, the dispersion and nonlinearity effects act together along the length of the fiber.

The split-step Fourier method assumes that the dispersive and nonlinear effects can be pretended to act independently inside a small distance h and therefore obtains an approximation solution. More specifically, propagation along the fiber from the position z to z + h is carried out in two steps. In the first step, the nonlinearity acts alone, and =0 in Eq. 2.3.1. In the second step, dispersion acts alone, and =0. Mathematically [21],

Dˆ The first step can be evaluated in the time domain while the second step should be calculated in

the frequency domain. The process is shown as the following prescription

(

^z ^h,^T

)

(

^exp^[^h^D^ˆ⁽^iω^)] ^F ^[exp(^h^N^ˆ⁾^A⁽^z,^T^)]

)

A + ≈ _T⁻¹ ∗ _T (2.3.5)

whereF_T ,F_T⁻¹,Dˆ(iω)are the FFT operation, the inverse FFT operation and the Fourier

byiω

Nˆ

, just a number in the frequency domain. That’s the reason why the dispersion effect should be calculated in the frequency space. After finishing the dispersion effect on , we should change the calculation from the frequency domain to the time domain by the inverse FFT. The use of the FFT makes numerical evaluation of Eq. 2.3.5 relatively fast. It is for this reason that the split-step Fourier method is faster up to two orders of magnitude compared with most of the finite-difference schemes.

) , ( Tz A

To estimate the accuracy of the split-step Fourier method, we note that a formally exact solution of Eq. 2.3.1 is given by

) if is assumed to be z independent. At this point, it is useful to recall the Baker -Hausdorff formula [35] for two noncommutating operatorsaˆ andbˆ ,

.... where . A comparison of Eq. 2.3.4 and Eq. 2.3.6 shows that the split-step Fourier method ignores the noncommutating feature of the operators and . By using Eq.

2.3.7 with and , the dominant error term is found resulting from the single commutator

The accuracy of the split-step method can be improved by adopting a different procedure to

propagate the optical pulse over one segment from z to z+h. In this procedure Eq. 2.3.4 is

The procedure divides into 3 parts [See Fig. 2.2]. At first, the dispersion effect acts alone in the first half of distance h. Then the effect of nonlinearity acts alone in the middle of segment.

Finally the dispersion effect acts again in the rest of length h/2. Similar to Eq. 2.3.5, the dispersion effects at the both sides of the segment is calculated in the frequency domain by the FFT whereas the nonlinear effect at the middle part is calculated in time domain.

Dispersion Only

Z=0 h

A(z,T) Nonlinear effect Only

Fig. 2.2 Schematic illustration of the symmetrized split-step Fourier method. Fiber length is divided into a large number of segments of width h. Within the segment, the effect of nonlinearity acts at the midplane shown by a dashed line and the effect of dispersion acts at the edge of the segment shown by a solid line.

Because of the symmetric form of Eq. 2.3.8, this scheme is known as the symmetrized split-step Fourier method [43]. The integral in the middle exponential considers the z

dependent of the nonlinear operator . If the step size h is small enough, the integral can be approximated by . The most important advantage of using the symmetrized form of Eq. 2.3.8 is that the leading error term comes from the double commutator in Eq. 2.3.7 is of the third order in the step size h. This can be proved by applying Eq. 2.3.8 twice in Eq. 2.3.7.

Nˆ ˆ)

exp( Nh

The accuracy of the symmetrized split-step Fourier method can be further improved by evaluating the integral in Eq. 2.3.8 more accurately than approximating it by . A simple approach is to employ the trapezoidal rule and approximate the integral by [44]

)

However, the implementation of Eq. 2.3.9 is not simple because is unknown at the mid-segment located at . It is necessary to use an iterative procedure that is initiated by replacing by . Equation is than used to estimate which in turn is used to calculate the new value of . Although the iteration procedure is time-consuming, it can still reduce the overall computing time if the step size h can be increased because of the improved accuracy of the numerical algorithm. Two iterations are generally enough in practice.

Let us see Eq. 2.3.3 again, the nonlinear operatorNˆ contains an integral part and the

differential part which corresponds to the Raman effect and the SS. It is more complicated to deal with them. We rewriteNˆ by using Eq. 2.1.6 and Eq. 2.3.3 as the following form:

⎪⎭

The integral part can be solved by the convolution theory by inversely FFT the product of )

nonlinear operator in use of the convolution theory and the FFT algorithm. In many papers, it is usually solved the SS term using the Runge-Kutta method with treating the differential term as a perturbation [40][45][46]. However, we use the FFT algorithm, which is simpler and more straightforward than the Runge-Kutta method. We therefore simulate the evolution of the pulse spectrum using the split-step Fourier method. We also combine the plug-in program

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