Chapter 1 Introduction
1.2 Supercontinuum Generation
Supercontinuum (SC) generation is the formation of broad continuous spectra by propagation of high power pulses through nonlinear media [14]. Provided enough power, SC generation can be observed in a drop of water [15]. However, the nonlinear effects involved in the spectral broadening are highly depend on the dispersion of the media; and a clever dispersion design can significantly reduce the power required. The widest spectra are obtained when the pump pulses are launched close to the zero-dispersion wavelength of the nonlinear media. Due to the technology which can fabricate the shiftable dispersion profile and small core MFs, MFs become powerful tools to generate the SC and was first demonstrated in 1999 [16]. The zero-dispersion wavelength of MFs can be shifted close to the pumping wavelength on the visible region and the small core of MFs enhances the
nonlinear effects, mechanisms leading to the SC.
To generate the SC with MFs, femtosecond [17][18] and picosecond mode-locked laser systems were generally used as the pumping sources [19]. For femtosecond pumping, it’s easily to get higher peak power of the pumping pulse due to its short pulse duration and therefore to induce strong nonlinear effects. These nonlinear effects include high-order soliton breakup [20-21], soliton self-frequency shift (SSFS) [22] and four-wave mixing (FWM) [23]. Usually about mini-watts of average pumping power are needed to generate the supercontinuum for a femtosecond mode-locked laser system [18].
1.3 Third order optical nonlinear Effects
Nonlinear optical effects are the major mechanisms leading to the supercontinuum (SC).
With enough peak power, a pulse propagating in the fiber will induce several nonlinear effects.
Under the pulse pumping in the anomalous dispersion region, a pulse will experience the self-phase modulation (SPM) [21], one of the nonlinear effects leading to the spectral broadening of the pulse. The nonlinear phase induced by SPM will interact with the anomalous dispersion and generate pairs of new frequencies at each side of pumping. This phenomenon is what we call modulation instability (MI) [17],[21], which can be regarded as degenerate four-wave mixing (DFWM). Once the new frequencies of the solitons located in
scattering (SRS) and self-steepening (SS) which called the SSFS (soliton-self frequency shift) effect. And that will shift the spectrum further into longer wavelength and distort the shape of spectrum. Higher-order dispersion (HOD) (usually β3 and β4) should also be considered if the spectrum extends from the anomalous dispersion region to the normal dispersion region [24]. The new dispersive wave will be generated at the normal dispersion region. When two optical fields with different wavelengths co-propagate in a nonlinear medium, the optical field experiences a nonlinear phase-shift induced by the co-propagating optical filed. This nonlinear phase-shift is commonly referred to as cross-phase modulation (XPM). Those nonlinear effects mentioned above will be described more detail in the following.
1.3.1 Self-phase Modulation
SPM is a phenomenon that leads to spectral broadening of optical pulses. It originates from the intensity-dependence of the refractive index [25]:
2 2 A n n
n= L +
,
(1.3.1)where nL is the linear part of the refractive index, n2 is the nonlinear index coefficient and A2is the optical intensity. A typical value of n2 for silica material is 3.2×10-20 m2/W. For
an optical pulse, SPM refers to the self-induced nonlinear phase shift as it propagates along the fiber
2
2 ( )
) 2
( Ln AT
SPM T
NL λ
φ = π , (1.3.2)
where L is the length of the fiber. This nonlinear phase shift can induce a frequency chirp which leads to the spectral broadening of the pulse. A useful quantity γPp interprets the maximum nonlinear phase shift for a pulse propagating in fibers, where Pp is the peak power of the optical pulse and γ is the nonlinear coefficient [21]
cAeff
n ω
γ = 2
.
(1.3.3)Here is the effective area of the propagating mode inside the fiber and ω is the center
frequency of the optical field. The nonlinear coefficient γ represents the strength of nonlinear effects and is inversely proportional to the area of fiber core.
Aeff
1.3.2 Cross-phase Modulation
When two optical fields with different wavelengths co-propagate in a nonlinear medium, the refractive index seen by one of the fields not only depends on its own intensity but also on the intensity of the other field. Consequently, the optical field with a center wavelength λi
experiences a nonlinear phase-shift induced by the co-propagating optical filed at wavelength λj such that[21]
, (1.3.4)
modulation (XPM) and requires the optical fields to overlap temporally. Equation (1.3.4) shows that XPM is twice as effective as SPM.
.3.3 Degenerate Four-Wave Mixing
) is a process where two pump photons generate a St
1
Degenerate four wave mixing (DFWM okes photon and an anti-Stokes photon:
as respectively. Being a coherent process, four-wave mixing is efficient only if the phase-matching condition is fulfilled [18], i.e.,
0
Note that only the even terms of the series expansion of β contribute to the phase-matching condition and the odd terms will cancel one another. The nonlinear phase shift 2γPp due to SPM should be also included in the phase-matching condition. For a pump wavelength located in the anomalous dispersion region, the phase-matching condition is mainly governed by the induced nonlinear phase shift. Usually the process of DFWM in the anomalous region can be regarded as MI which we will discuss in next section.
1.3.4 Modulation Instability
ted in the anomalous dispersion region, it is possible to co
For a pump wavelength loca
mpensate the induced nonlinear phase shift 2γPp by the negative value of β2 and generate the corresponding Stokes and anti-Stokes components. The frequency difference between the pump and the Stokes (anti-Stokes) component calculated by Eq. 1.3.6 is [21]
2 1
2 ⎞
⎛ γP
2 ⎟⎟
⎜⎜ ⎠
±⎝
= β
Ωs p , (1.3.7)
if considering only the term of β2. This frequency shift can also be calculated by solving the standard nonlinear Schrödinger equation (NLSE) of CW light [21]. By using the perturbation theory, the CW light solution of NLSE will become unstable in the anomalous dispersion region and generate two new frequencies on either side of the pump frequency.
These two new frequencies calculated by perturbation of CW solution are as the same as the frequencies calculated by the DFWM. The new frequencies will break up the CW or quasi-CW radiation into a train of ultra short pulses. We call this phenomenon modulation instability (MI) which results from an interplay between the nonlinear and the dispersive effects. In fact, MI can be interpreted in terms of DFWM in the frequency domain, whereas in the time domain it results from an unstable growth of weak perturbation from the CW steady state.
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.
1
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]
2
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
13
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
15
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.
17
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.,
2
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 as19
( )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 kth 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
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