FIBER OPTICS Prof. R.K. Shevgaonkar Department of Electrical Engineering Indian Institute of Technology, Bombay Lecture: 35 Self-Phase-Modulation (SPM) Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of Electrical Engineering, IIT Bombay Page 1
In the present discussion, we shall study the nature of pulse evolution inside an optical fiber in the presence of non-linear effect only i.e. we assume the effect of dispersion to be negligibly small. This situation arises when the physical length of the optical fiber under consideration is much greater than the characteristic non-linearity length L NL, but is negligibly smaller than the characteristic dispersion length L D. This is the 3 rd case that arises in the discussion of the solution of the Non-Linear Schrodinger (NLS) equation by the Split-Fourier Step method. For a considered length of an optical fiber, the above case occurs if we use a high optical power pulse of large pulse width. The large width of the pulse causes L D to be very large and consequently the considered physical length becomes much smaller compared to L D. However, due to high optical power, the beam is intense and the non-linear effect cannot be ignored. Also, the non-linearity length becomes much smaller compared to the physical length of the fiber. In this case, the refractive index of the optical fiber which the optical pulse sees is different at different locations within the pulse. Thus, the refractive index becomes a function of the intensity of the light. Therefore, the peak of the pulse sees a large refractive index whereas the refractive index decreases on either side of the peak. Thus, some kind of a refractive index profiling of the material gets generated within the optical pulse itself. As the pulse moves forward along the optical fiber, this profile also moves along with the pulse along the optical fiber. But, due to this change in refractive index within the pulse there is also a change in the phase within the pulse. Thus, due to the presence of Kerr non-linearity inside the optical fiber, the intensity variation within a pulse, in turn, causes the phase to vary within the pulse. Since the pulse itself causes the modulation in phase, the regime is known as Self-Phase Modulation (SPM). The following discussion deals with the SPM due to Kerr non-linearity in more detail. One can show that (proof not intended here) continuous optical propagation inside an optical fiber in the presence of non-linearity is highly unstable and the beam inherently splits into pulses due to presence of non-linearity. The case we are dealing with, here, is the one in which the physical length L of the optical fiber under consideration is such that: The terms L D and L NL are the characteristic dispersion and the non-linearity lengths respectively. The non-linear Schrodinger equation, defined in terms of normalized amplitude function U(z,T) (as given by equation (34.9)) is given as: (35.1) The terms in the above expression retain their meanings as discussed in our earlier discussion. For the analysis in our present case, we assume the contribution of the dispersion effect to be negligibly small and so, the second term on the LHS of (35.1) can be dropped. The resulting equation becomes: (35.2) Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of Electrical Engineering, IIT Bombay Page 2
One way to solve the above equation is to pre-assume a solution for U and then calculate the assumed parameters in the solution from the given equation. Therefore, let us assume U to be a function of some envelope function V and a non-linear phase function as shown below: Differentiating U w.r.t. z, we obtain: (35.3) Comparing the RHS of equations (35.2) and (35.4) we obtain: (35.4) (35.5) Equating the real and the imaginary parts in the above equation, we obtain: (35.6) (35.7) Equation (35.5) suggests the fact that the amplitude envelope does not vary (remains constant) with length z. However, the phase varies with distance in accordance to equation (35.7). Hence, we can write: ( ) ( ) (35.8) ( ) { } (35.9) The quantity within the parentheses in equation (35.9) is the effective length of the optical fiber over which the optical power remains above acceptable level or the SNR is acceptably large. Hence, it is known as the effective length L eff of the optical fiber. In other words, L eff may also be referred to as the length of the optical fiber over which the nonlinear effects on pulse propagation may be witnessed. Hence, from (35.9): ( ) (35.10) One may calculate the maximum amount of phase change that can occur over L eff as shown below: (35.11) Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of Electrical Engineering, IIT Bombay Page 3
The important point to note is that, the phase function is not a linear function of time which leads to variation in frequency within the pulse. Thus, SPM, too, produces some sort of frequency modulation. This change in frequency is given as: ( ) (35.12) The above expression is, however, a generalized for any pulse shape U(0,T). To be specific, let us re-consider a Gaussian pulse which is shown below: Figure 35.1: Gaussian Pulse If we plot the corresponding we obtain: corresponding to the above Gaussian pulse shape, Figure 35.2: Plot of Frequency change Since there is a variation of frequency within a pulse, we obtain the frequency chirping phenomenon in this case too. However, the frequency chirp, for a pulse of the above type, is always from negative (lower frequencies) to positive (higher frequencies) within the pulse as suggested by equation 35.12. In other words, the lower frequencies lie towards the leading edge and gradually the frequencies increase towards the trailing edge of the pulse. At z=0, there is no change in phase and the carrier signal within the pulse has a uniform phase distribution throughout the pulse. This can be verified by substituting z=0 in equation (35.9). As the pulse moves along the fiber, the phase modulation starts to Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of Electrical Engineering, IIT Bombay Page 4
occur and at z=z 1 the frequency chirp pattern in the pulse is shown in figure 35.2. At point z=z 2 (z 2 >z 1 ), the frequency chirp pattern changes to the one as shown in figure 35.2. Therefore, as the pulse moves forward along the fiber, over the length L eff, the Kerr nonlinearity causes the slope of the frequency chirp pattern to increase with distance. However, one must note that, unlike the dispersion effect, the frequency chirp in this case is always from positive to negative. That is, at the leading edge, there is a red-shift in the frequencies and towards the trailing edge there is a blue-shift in the frequencies from the centre frequency of the optical carrier signal in the pulse. The envelope function of the pulse remains unchanged and so the time domain function is unaltered. However, due to the phase modulation, new frequencies get generated in the spectral domain which, thereby, changes the spectrum of the signal. This situation is the exact counter-situation of the dispersive situation discussed earlier in which the time domain function gets modified yet the spectrum of the signal remains constant. The spectral modulation due to nonlinearity in SPM scheme is schematically represented in the following figure: Figure 35.3: Spectral Broadening due to non-linearity If we compare this with the dispersion regime, the following figure illustrates the key differences in the signal and spectrum of the two regimes: Figure 35.4: Comparison of Dispersion and SPM regimes Let us now imagine a situation in which we have an optical fiber and an optical pulse such that the pulse propagation undergoes an anomalous dispersion and self-phase modulation inside the optical fiber. This situation can be achieved in practice by proper adjustment of parameters. In such a case, if we can cause the positive going frequency Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of Electrical Engineering, IIT Bombay Page 5
chirp in the pulse (due to SPM) to compensate for the negative going frequency chirp (due to anomalous dispersion), we would then achieve a pulse which is neither time domain nor frequency domain modified. Thus, there would be no pulse-broadening and we can realise high data rates. In other words, the non-linear effect can be used to kill the effect of dispersion. This turns out to be a very attractive proposition because, allowing one distorting effect to be present alone leads to pulse distortion, but allowing both the distorting phenomena to co-exist may, indeed, lead to distortion-less pulse generation. Hence, by increasing the power level in a pulse so that the frequency chirps created by dispersion and non-linearity cancel each other out, the resulting pulse is capable of travelling large lengths of optical fibers without undergoing distortion provided both the effects are maintained to balance each other on the optical fiber. Such a pulse is known as soliton. Optical Solitons An optical soliton can be generated when the group velocity dispersion (GVD) is balanced by self-phase-modulation (SPM) as discussed above. In this regime, the physical length L of the optical fiber is such that: (35.13) Where, L D and L NL are the characteristic dispersion and non-linearity lengths respectively of the optical fiber given by equations (33.25) and (33.26) as shown below (all the terms retain their original meanings): (35.14) (35.15) The above situation is, in fact, the fourth case out of the four cases generated during the numerical solution of the Non-linear Schrodinger (NLS) equation by the split- Fourier step method. Let us define a few normalized parameters as shown below: (35.16) In the above definitions U is the normalized amplitude, is the normalized distance and is the normalized time normalized to pulse-width T 0. Let us consider the loss in the optical fiber to be negligibly small (i.e. ). We may write the NLS using the above quantities as shown below: ( ) (35.17) The quantity N is defined as the ration of the characteristic dispersion length to the characteristic non-linearity length and is known as the order of a soliton. It is defined as: (35.18) Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of Electrical Engineering, IIT Bombay Page 6
For N=1, the resulting soliton is called the fundamental soliton. Let us define a parameter u given as: (35.19) Since we are dealing with solitons, we are implicitly in the anomalous dispersion regime where the sign of is negative. Hence, the NLS can be re-written as: (35.20) The above differential equation can be solved by the inverse scattering method for N>1. However, since the above equation assumes N=1, we shall solve it in a rather simpler way. As we argued that the shape of a soliton remains intact throughout the optical fiber, we may assume u which is a function of ( ) to be equal to the product of some time-varying function V and a phase function ( ). That is: ( ) ( ) ( ) (35.21) Substituting the assume value of u in equation (35.20), we obtain two distinct equation by equating the real and imaginary parts. One can show that the solution for the phase function is of the form: ( ) (35.22) In the above expression, K is analogous to the phase constant and the quantity is analogous to the frequency shift. If we assume the frequency shift to be zero as the pulse propagates along the fiber and substitute the resulting expression of into the equation for V, we obtain: ( ) (35.23) The above equation can be solved using a very intuitive way- by multiplying both sides of the equation by and then integrating both sides w.r.t.. The solution of the above equation and the corresponding expression for U shall be discussed in the subsequent sections. Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of Electrical Engineering, IIT Bombay Page 7