Nonlinear Optical Effects

The response of any dielectric to light becomes nonlinear for intense electromagnetic fields, and optical fibers are no exception. Even though silica is intrinsically not a highly nonlinear material, the waveguide geometry that confines light to a small cross section over long fiber lengths makes nonlinear effects quite important in the design of modern lightwave systems. We discuss in this tutorial the nonlinear phenomena that are most relevant for fiber-optic communications.

1. Stimulated Light Scattering

Rayleigh scattering, discussed in the optical fiber loss tutorial, is an example of elastic scattering for which the frequency (or the photon energy) of scattered light remains unchanged. By contrast, the frequency of scattered light is shifted downward during inelastic scattering. Two examples of inelastic scattering are Raman scattering and Brillouin scattering. Both of them can be understood as scattering of a photon to a lower energy photon such that the energy difference appears in the form of a phonon. The main difference between the two is that optical phonons participate in Raman scattering, whereas acoustic phonons participate in Brillouin scattering. Both scattering processes result in a loss of power at the incident frequency. However, their scattering cross sections are sufficiently small that loss is negligible at low power levels.

At high power levels, the nonlinear phenomena of stimulated Raman scattering (SRS) and stimulated Brillouin scattering (SBS) become important. The intensity of the scattered light in both cases grows exponentially once the incident power exceeds a threshold value. SRS and SBS were first observed in optical fibers during the 1970s. Even though SRS and SBS are quite similar in their origin, different dispersion relations for acoustic and optical phonons lead to the following differences between the two in single-mode fibers: (i) SBS occurs only in the backward direction whereas SRS can occur in both directions; (ii) The scattered light is shifted in frequency by about 10 GHz for SBS but by 13 THz for SRS (this shift is called the Stokes shift); and (iii) the Brillouin gain spectrum is extremely narrow (bandwidth < 100 MHz) compared with the Raman-gain spectrum that extends over 20-30 THz. The origin of these differences lies in a relatively small value of the ratio v_A/c (~ 10^-5), where v_A is the acoustic velocity in silica and c is the velocity of light.

Stimulated Brillouin Scattering

The physical process behind Brillouin scattering is the tendency of materials to become compressed in the presence of an electric field - a phenomenon termed electrostriction. For an oscillating electric field at the pump frequency Ω_p, this process generates an acoustic wave at some frequency Ω. Spontaneous Brillouin scattering can be viewed as scattering of the pump wave from this acoustic wave, resulting in creation of a new wave at the pump frequency Ω_s. The scattering process must conserve both the energy and the momentum. The energy conservation requires that the Stokes shift Ω equals ω_p - ω_s. The momentum conservation requires that the wave vectors satisfy k_A = k_p - k_s. Using the dispersion relation |k_A| = Ω/v_A, where v_A is the acoustic velocity, this condition determines the acoustic frequency as

Ω =  |k_A|v_A = 2v_A|k_p| sin(θ/2)

where  |k_p| ≈ |k_s| was used and θ represents the angle between the pump and scattered waves. Note that Ω vanishes in the forward direction (θ = 0) and is maximum in the backward direction (θ = π). In single-mode fibers, light can travel only in the forward and backward directions. As a result, SBS occurs in the backward direction with a frequency shift Ω_B = 2v_A|k_p|. Using , where λ_p is the pump wavelength, the Brillouin shift is given by

where  is the mode index. Using v_A = 5.96 km/s and = 1.45 as typical values for silica fibers, ν_B = 11.1 GHz at λ_p = 1.55 μm. This equation shows that ν_B scales inversely with the pump wavelength.

Once the scattered wave is generated spontaneously, it beats with the pump and creates a frequency component at the beat frequency ω_p - ω_s, which is automatically equal to the acoustic frequency Ω. As a result, the beating term acts as source that increases the amplitude of the sound wave, which in turn increases the amplitude of the scattered wave, resulting in a positive feedback loop. SBS has its origin in this positive feedback, which ultimately can transfer all power from the pump to the scattered wave. The feedback process is governed by the following set of two coupled equations:

where I_p and I_s are the intensities of the pump and Stokes fields, g_B is the SBS gain, and α_p and α_s account for fiber losses.

The SBS gain g_B is frequency dependent because of a finite damping time T_B of acoustic waves (the lifetime of acoustic phonons). If the acoustic waves decay as exp(-t/T_B), the Brillouin gain has a Lorentzian spectral profile given by

The following figure shows the Brillouin gain spectra at λ_p = 1.525 μm for three different kinds of single-mode silica fibers.

Both the Brillouin shift ν_B and the gain bandwidth Δν_B can vary from fiber to fiber because of the guided nature of light and the presence of dopants in the fiber core. The fiber labeled (a) has a core of nearly pure silica (germania concentration of about 0.3% per mole). The measured Brillouin shift ν_B = 11.25 GHz is in agreement with its equation. The Brillouin shift is reduced for fibers (b) and (c) of a higher germania concentration in the fiber core. The double-peak structure for fiber (b) results from inhomogeneous distribution of germania within the core. The gain bandwidth in the figure is larger than that expected for bulk silica (Δν_B ≈ 17 MHz at λ_p = 1.525 μm). A part of the increase is due to the guided nature of acoustic modes in optical fibers. However, most of the increase in bandwidth can be attributed to variations in the core diameter along the fiber length.  Because such variations are specific to each fiber, the SBS gain bandwidth is generally different for different fibers and can exceed 100 MHz; typical values are ~ 50 MHz for λ_p near 1.55 μm.

The peak value of the Brillouin gain in the last equation occurs for Ω = Ω_B and depends on various material parameters such as the density and the elasto-optic coefficient. For silica fibers g_B ≈ 5 x 10^-11 m/W. The threshold power level for SBS can be estimated by solving the equations and finding at what value of I_p and I_s grows from noise to a significant level. The threshold power P_th = I_pA_eff , where A_eff is the effective mode area, satisfies the condition

g_BP_thL_eff /A_eff ≈ 21

where L_eff is the effective interaction length defined as

L_eff  = [1 - exp(- αL)]/α

and α represents fiber losses. For optical communication systems  L_eff can be approximated by 1/α as αL >> 1 in practice. Using A_eff = πw², where w is the spot size, P_th can be as low as 1mW depending on the values of w and α. Once the power launched into an optical fiber exceeds the threshold level, most of the light is reflected backward through SBS. Clearly, SBS limits the launched power to a few milliwatts because of its low threshold.

The preceding estimate of P_th applies to a narrowband CW beam as it neglects the temporal and spectral characteristics of the incident light. In a lightwave system, the signal is in the form of a bit stream. For a single short pulse whose width is much smaller than the phonon lifetime, no SBS is expected to occur. However, for a high-speed bit stream, pulses come at such as fast rate that successive pulses build up the acoustic wave, similar to the case of a CW beam, although SBS threshold increases. The exact value of the average threshold power depends on the modulation format (RZ versus NRZ) and is typically ~5 mW. It can be increased to 10 mW or more by increasing the bandwidth of the optical carrier to >200 MHz through phase modulation. SBS does not produce interchannel crosstalk in WDM systems because the 10-GHz frequency shift is much smaller than typical channel spacing.

Stimulated Raman Scattering

Spontaneous Raman scattering occurs in optical fibers when a pump wave is scattered by the silica molecules. It can be understood using the energy-level diagram shown in (b) of the following figure.

Some pump photons give up their energy to create other photons of reduced energy at a lower frequency; the remaining energy is absorbed by silica molecules, which end up in an excited vibrational state. An important difference from Brillouin scattering is that the vibrational energy levels of silica dictate the value of the Raman shift Ω_R = ω_p - ω_s. As an acoustic wave is not involved, spontaneous Raman scattering is an isotropic process and occurs in all directions.

Similar to the SBS case, the Raman scattering process becomes stimulated if the pump power exceeds a threshold value. SRS can occur in both the forward and backward directions in optical fibers. Physically speaking, the beating of the pump and with the scattered light in these two directions creates a frequency component at the beat frequency ω_p - ω_s, which acts as a source that derives molecular oscillations. Since the amplitude of the scattered wave increases in response to these oscillations, a positive feedback loop sets in. In the case of forward SRS, the feedback process is governed by the following set of two coupled equations

where g_R is the SRS gain. In the case of backward SRS, a minus sign is added in front of the derivative in the 2nd equation above, and this set of equations becomes identical to the SBS case.

The spectrum of the Raman gain depends on the decay time associated with the excited vibrational state. In the case of a molecular gas or liquid, the decay time is relatively long (~ 1 ns), resulting in a Raman-gain bandwidth of ~1 GHz. In the case for optical fibers, the bandwidth exceeds 10 THz. The figure above shows the Raman-gain spectrum of silica fibers. The broadband and multipeak nature of the spectrum is due to the amorphous nature of glass. More specifically, vibrational energy levels of silica molecules merge together to form a band. As a result, the Stokes frequency ω_s can differ from the pump frequency ω_p over a wide range. The maximum gain occurs when the Raman shift Ω_R ≡ ω_p - ω_s is about 13 THz. Another major peak occurs near 15 THz while minor peaks persist for values of Ω_R as large as 35 THz. The peak value of the Raman gain g_R is about 1 x 10^-13 m/W at a wavelength of 1 μm. This value scales linearly with ω_p (or inversely with the pump wavelength λ_p), resulting in g_R ≈ 6 x 10^-13 m/W at 1.55 μm.

Similar to the case of SBS, the threshold power P_th is defined as the incident power at which half of the pump power is transferred to the Stokes field at the output end of a fiber of length L. It is estimated from

g_RP_thL_eff /A_eff ≈ 16

where g_R is the peak value of the Raman gain. As before, L_eff can be approximated by 1/α. If we replace A_eff by πw², where w is the spot size, P_th for SRS is given by

 P_th ≈ 16α(πw²)/g_R

If we use πw² = 50 μm² and α = 0.2 dB/km as the representative values, P_th is about 570 mW near 1.55 μm. It is important to emphasize that this equation provides an order-of-magnitude estimate only as many approximations are made in its derivation.

As channel powers in optical communication systems are typically below 10 mW, SRS is not a limiting factor for single-channel lightwave systems. However, it affects the performance of WDM systems considerably; this will be covered in another tutorial.

Both SRS and SBS can be used to advantage while designing optical communication systems because they can amplify an optical signal by transferring energy to it from a pump beam whose wavelength is suitably chosen. SRS is especially useful because of its extremely large bandwidth. Indeed, the Raman gain is used routinely for compensating fiber losses in modern lightwave systems.

2. Nonlinear Phase Modulation

The refractive index of silica was assumed to be power independent in the discussion of fiber modes. In reality, all materials behave nonlinearly at high intensities and their refractive index increases with intensity. The physical origin of this effect lies in the anharmonic response of electrons to optical fields, resulting in a nonlinear susceptibility. To include nonlinear refraction, we modify the core and cladding indices of a silica fiber as

where  is the nonlinear-index coefficient, P is the optical power, and A_eff is the effective mode area introduced earlier. The numerical value of  is about 2.6 x 10^-20 m²/W for silica fibers and varies somewhat with dopants used inside the core. Because of this relatively small value, the nonlinear part of the refractive index is quite small (< 10^-12  at a power level of 1 mW). Nevertheless, it affects modern lightwave systems considerably because of long fiber lengths. In particular, it leads to the phenomena of self- and cross-phase modulations.

Self-Phase Modulation

It we use first-order perturbation theory to see how fiber modes are affected by the nonlinear term in the equation above, we find that the mode shape does not change but the propagation constant becomes power dependent. It can be written as

where is an important nonlinear parameter with values ranging from 1 to 5 W^-1/km depending on the values of A_eff and the wavelength. Noting that the optical phase increases linearly with z, the γ term produces a nonlinear phase shift given by

where P(z) = P_inexp(- αz) accounts for fiber losses and L_eff is defined earlier.

In deriving this phase shift equation P_in was assumed to be constant. In practice, time dependence of P_in makes Φ_NL to vary with time. In fact, the optical phase changes with time in exactly the same fashion as the optical signal. Since this nonlinear phase modulation is self-induced, the nonlinear phenomenon responsible for it is called self-phase modulation (SPM). SPM leads to frequency chirping of optical pulses. In contrast with the linear chirp, the frequency chirp is proportional to the derivative dP_in/dt and depends on the pulse shape.

The figure shows variation of (a) the nonlinear phase shift Φ_NL and (b) the frequency chirp across the pulse for γP_inL_eff = 1 in the cases of a Gaussian pulse (m = 1) and a super-Gaussian pulse (m = 3). The SPM-induced chirp affects the pulse shape through GVD and often leads to additional pulse broadening. In general, spectral broadening of the pulse induced by SPM increases the signal bandwidth considerably and limits the performance of lightwave systems.

If fiber losses are compensated periodically using optical amplifiers, Φ_NL should be multiplied by the number of amplifiers N_A because the SPM-induced phase accumulates over multiple amplifiers. To reduce the impact of SPM in lightwave systems, it is necessary that Φ_NL << 1. If we use Φ_NL = 0.1 as the maximum tolerable value and replace L_eff by 1/α for long fibers, this condition can be written as a limit on the input peak power as

P_in < 0.1α/(γN_A)

For example, if γ = 2 W^-1/km, N_A = 10, and α = 0.2 dB/km, the input peak power is limited to below 2.2 mW. Clearly, SPM can be a major limiting factor for long-haul lightwave systems.

Cross-Phase Modulation

The intensity dependence of the refractive index can also lead to another nonlinear phenomenon known as cross-phase modulation (XPM). It occurs when two or more optical channels are transmitted simultaneously inside an optical fiber using the WDM technique. In such systems, the nonlinear phase shift for a specific channel depends not only on the power of that channel but also on the power of other channels. The phase shift for the jth channel becomes

where the sum extends over the number of channels. The factor of 2 in this equation has its origin in the form of the nonlinear susceptibility and indicates that XPM is twice as effective as SPM for the same amount of power. The total phase shift depends on the powers in all channels and would vary from bit to bit depending on the bit pattern of the neighboring channels. It we assume equal channel powers, the phase shift in the worst case in which all channels simultaneously carry 1 bits and all pulses overlap is given by

It is difficult to estimate the impact of XPM on the performance of multichannel lightwave systems. The reason is that the preceding discussion has implicitly assumed that XPM acts in isolation without dispersive effects and is valid only for CW optical beams. In practice, pulses in different channels travel at different speeds. The XPM-induced phase shift can occur only when two pulses overlap in time. For widely separated channels they overlap for such a short time that XPM effects are virtually negligible. On the other hand, pulses in neighboring channels will overlap long enough for XPM effects to accumulate. These arguments show that the above equation cannot be used to estimate the limiting input power.

A common method for studying the impact of SPM and XPM uses a numerical approach. It is generalized to include the SPM and XPM effects by adding a nonlinear term. The resulting equation is known as the nonlinear Schrödinger equation and has the form

where neglected the third-order dispersion and added the term containing α to account for fiber losses. This equation is quite useful for designing lightwave systems and will be used in later tutorials.

Since the nonlinear parameter γ depends inversely on the effective mode area, the impact of fiber nonlinearities can be reduced considerably by enlarging A_eff. A_eff is about 80 μm² for standard fibers but reduces to 50 μm² for dispersion-shifted fibers. A new kind of fiber known as large effective-area fiber (LEAF) has been developed for reducing the impact of fiber nonlinearities. The nonlinear effects are not always detrimental for lightwave systems. Numerical solutions of the equation above show that dispersion-induced broadening of optical pulses is considerably reduced in the case of anomalous dispersion. In fact, an optical pulse can propagate without distortion if the peak power of the pulse is chosen to correspond to a fundamental soliton. 

3. Four-Wave Mixing

The power dependence of the refractive index has its origin in the third-order nonlinear susceptibility denoted by Χ⁽³⁾. The nonlinear phenomenon, known as four-wave mixing (FWM), also originates from Χ⁽³⁾. If three optical fields with carrier frequencies ω₁, ω₂, and ω₃ copropagate inside the fiber simultaneously, Χ⁽³⁾ generates a fourth field whose frequency  ω₄ is related to other frequencies by a relation ω₄ = ω₁ ± ω₂ ± ω₃. Several frequencies  corresponding to different plus and minus sign combinations are possible in principle. In practice, most of these combinations do not build up because of a phase-matching requirement. Frequency combinations of the form ω₄ = ω₁ + ω₂ - ω₃ are often troublesome for multichannel communication systems since they can become nearly phase-matched when channel wavelengths lie close to the zero-dispersion wavelength. In fact, the degenerate FWM process for which ω₁ = ω₂ is often the dominant process and impacts the system performance most.

On a fundamental level, a FWM process can be viewed as a scattering proces in which two photons of energies ћω₁ and ћω₂ are destroyed, and their energy appears in the form of two new photons of energies ћω₃ and ћω₄. The phase-matching condition then stems from the requirement of momentum conservation. Since all four waves propagate in the same direction, the phase mismatch can be written as

Δ = β(ω₃) + β(ω₄) - β(ω₁) - β(ω₂)

where β(ω) is the propagation constant for an optical field with frequency ω. In the degenerate case, ω₂ = ω₁, ω₃ = ω₁ + Ω, and ω₄ = ω₁ - Ω, where Ω represents the channel spacing. Using the Taylor expansion, we find that the β₀ and β₁ terms cancel, and the phase mismatch is simply Δ = β₂Ω². The FWM process is completely phase matched when β₂ = 0. When β₂ is small (< 1 ps²/km) and channel spacing is also small (Ω < 100 GHz), this process can still occur and transfer power from each channel to its nearest neighbors. Such as power transfer not only results in the power loss for the channel but also induces interchannel crosstalk that degrades the system performance severely. Modern WDM systems avoid FWM by using the technique of dispersion management in which GVD is kept locally high in each fiber section even though it is low on average. Commercial dispersion shifted fibers are designed with a dispersion of about 4 ps/(km-nm), a value found large enough to suppress FWM.

FWM can also be useful for lightwave systems. It is often used for demultiplexing individual channels when time-division multiplexing is used in the optical domain. It can also be used for wavelength conversion. FWM in optical fibers is sometimes used for generating a spectrally inverted signal through the process of optical phase conjugation. This technique is useful for dispersion compensation.