# Raman and Brillouin Devices

This is a continuation from the previous tutorial - bistable optical devices.

The nonparametric processes of stimulated Raman scattering and stimulated Brillouin scattering both cause a shift of the optical frequency, leading to a loss for the pump beam and a gain for a Stokes beam if the material is not originally excited or a gain for an anti-Stokes beam if it is excited. On the positive side, such processes can be utilized for optical frequency conversion and optical signal amplification. On the negative side, however, they also place some serious limitations on the performance of certain optical devices and systems.

Both stimulated Raman scattering and stimulated Brillouin scattering can be characterized by the imaginary part of a complex third-order nonlinear susceptibility of the form $$\boldsymbol{\chi}^{(3)}(\omega_\text{S}=\omega_\text{S}+\omega_\text{p}-\omega_\text{p})$$ for the Stokes interaction and one of the form $$\boldsymbol{\chi}^{(3)}(\omega_\text{AS}=\omega_\text{AS}+\omega_\text{p}-\omega_\text{p})$$ for the anti-Stokes interaction. These susceptibilities are in resonance with a frequency $$\Omega=\omega_\text{p}-\omega_\text{S}=\omega_\text{AS}-\omega_\text{p}$$ that characterizes a material excitation and, consequently, have the property given in (9-71) [refer to the coupled-wave analysis of nonlinear optical interactions tutorial]. In most device applications using a Raman or Brillouin process, a material is initially in its normal state without being excited. Therefore, we shall consider only the Stokes process in the following discussions.

Being nonparametric, the Raman and Brillouin processes are automatically phase matched. The corresponding material excitation in an interaction picks up any phase mismatch between the interacting optical waves. Indeed, a Raman or Brillouin Stokes process can be viewed as a parametric interaction among a pump wave, a Stokes wave, and a material excitation wave. The material excitation wave is characterized by a frequency $$\Omega$$ and a wavevector $$\mathbf{K}$$. From this viewpoint, it is easy to see that a Stokes interaction is governed by the following conditions:

$\tag{9-179}\omega_\text{S}=\omega_\text{p}-\Omega$

$\tag{9-180}\mathbf{k}_\text{S}=\mathbf{k}_\text{p}-\mathbf{K}$

Clearly, phase matching among the pump wave, the Stokes wave, and the material excitation wave is needed, but it is automatically achieved when the pump wave generates a material excitation that allows a Raman or Brillouin process to occur. Figure 9-30 below illustrates the relations among the three interacting waves in a Stokes process.

As mentioned in the nonlinear optical interactions tutorial, the fundamental difference between the Raman and the Brillouin processes lies in the different mode of material excitation associated with each process. This difference leads to very different considerations for these two processes.

Raman gain

Because an excitation that is responsible for Raman scattering is associated with a transition at the molecular or atomic level, the Raman frequency shift is determined by the resonance frequency, $$\Omega_\text{R}$$, of the Raman transition. This Raman frequency is an intrinsic property of a material and is independent of the frequency of the pump optical wave.

Such an excitation is also nondispersive. Because a nondispersive excitation can take any wavevector $$\mathbf{K}$$ independently of its frequency, the phase-matching condition in (9-180) is satisfied for any combination of $$\mathbf{k}_\text{p}$$ and $$\mathbf{k}_\text{S}$$ independently of the condition in (9-179). As a consequence, Raman scattering in all directions has the same frequency shift that is specific to a given material. Spontaneous Raman scattering has a nearly isotropic emission pattern in all directions, whereas stimulated Raman scattering occurs predominantly in the forward and backward directions due to the fact that a stimulated signal grows in strength as the interaction length increases.

The Raman frequency shift, which is usually quoted per centimeter, is typically in the range of $$300-3000\text{ cm}^{-1}$$ for $$f_\text{R}/c=\Omega_\text{R}/2\pi{c}$$, equivalent to $$10-100\text{ THz}$$ for $$f_\text{R}$$, for most materials ($$1\text{ cm}^{-1}$$ is equivalent to $$30\text{ GHz}$$).

When a material is initially in its ground state of a Raman transition, the effective Raman susceptibility defined in (9-75) [refer to the coupled-wave analysis of nonlinear optical interactions tutorial] has a negative imaginary part: $$\chi_\text{R}''\lt0$$. This situation leads to a gain for the Raman Stokes signal at the expense of the pump wave. From (9-78) [refer to the coupled-wave analysis of nonlinear optical interactions tutorial], we find that the Raman gain factor for the Stokes signal is given by

$\tag{9-181}\tilde{g}_\text{R}=-\frac{3\omega_\text{S}\mu_0}{n_{\text{S},z}n_{\text{p},z}}\chi_\text{R}''$

which has a positive value when $$\chi_\text{R}''\lt0$$. The unit of $$\tilde{g}_\text{R}$$ is meters per watt.

The Raman susceptibility is only very weakly dependent on the individual optical frequencies, $$\omega_\text{p}$$ and $$\omega_\text{S}$$, but it is a strong function of $$\Omega=\omega_\text{p}-\omega_\text{S}$$ with a resonance at $$\Omega=\Omega_\text{R}$$.

In the simple case when there is only one Raman resonance frequency in a material, $$\chi_\text{R}''$$ as a function of $$\Omega$$ has a Lorentzian lineshape as that of the linear susceptibility given in (176) [refer to the material dispersion tutorial]. Therefore, according to (9-181) above, the corresponding Raman gain factor has the form:

$\tag{9-182}\tilde{g}_\text{R}=\tilde{g}_\text{R0}\frac{\gamma_\text{R}^2}{(\Omega-\Omega_\text{R})^2+\gamma_\text{R}^2}=\tilde{g}_\text{R0}\frac{\gamma_\text{R}^2}{(\omega_\text{p}-\omega_\text{S}-\Omega_\text{R})^2+\gamma_\text{R}^2}$

where $$\tilde{g}_\text{R0}$$ is the peak Raman gain factor and $$\gamma_\text{R}$$ is the relaxation constant for the Raman excitation.

This Raman gain factor has a FWHM linewidth given by $$\Delta\Omega_\text{R}=2\gamma_\text{R}$$, or $$\Delta{f}_\text{R}=\gamma_\text{R}/\pi$$.

Note that both the Raman frequency $$f_\text{R}$$ and the Raman spectral linewidth $$\Delta{f}_\text{R}$$ are independent of the pump and the Stokes optical frequencies, but the peak Raman gain factor varies linearly with the Stokes optical frequency:

$\tilde{g}_\text{R0}\propto\omega_\text{S}\propto1/\lambda_\text{S}$

The response time of a Raman process is measured by $$\tau_\text{R}=\gamma_\text{R}^{-1}$$, which is the relaxation lifetime of the Raman excitation such as an optical phonon or a molecular vibration.

Typical Raman response times range from a few hundred picoseconds in molecules through a few picoseconds in crystalline solids to tens of femtoseconds in amorphous solids such as glasses.

Accordingly, the Raman linewidth $$\Delta{f}_\text{R}$$ ranges from a few gigahertz to the order of $$10\text{ THz}$$, depending on the properties of the materials.

A Raman process can efficiently respond only to an optical signal that has a bandwidth narrower than the Raman linewidth. For an optical pulse, this means that its pulsewidth has to be greater than the Raman response time. Therefore, depending on the specific material used, it is possible for a Raman device to function in steady state or in quasi-steady state with optical signals ranging from CW waves to picosecond or even sub-picosecond optical pulses.

When an optical signal varies faster than the Raman relaxation time, the interaction is characterized by transient stimulated Raman scattering with a reduced Raman gain among other features that are different from those of steady-state stimulated Raman scattering. In this tutorial, we consider only Raman devices operating in the steady-state regime.

Forward and backward Raman interactions have the same Raman gain factor. However, the value of $$\tilde{g}_\text{R0}$$ is a function of the polarization states of the pump and the Stokes waves because $$\boldsymbol{\chi}^{(3)}$$ is a tensor and the effective Raman susceptibility defined in (9-75) [refer to the coupled-wave analysis of nonlinear optical interactions tutorial] depends on $$\hat{e}_\text{p}$$ and $$\hat{e}_\text{S}$$.

In an isotropic medium, the maximum value of $$\tilde{g}_\text{R0}$$ is found when the pump and the Stokes waves are linearly polarized in the same direction. Therefore, special attention has to be paid to the polarization states of the optical waves throughout the course of interaction in evaluating the efficiency of a Raman process.

As an example, Figure 9-31 below shows the spectral dependence of the Raman gain factor of fused silica glass measured at a pump optical wavelength of $$1\text{ μm}$$ for pump and Stokes waves that are linearly polarized in the same direction.

This spectrum is very broad and does not have the ideal Lorentzian lineshape because there are many closely clustered Raman resonances in such an amorphous solid material.

This Raman spectral shape remains more or less the same for other pump wavelengths, but its peak value scales with the pump wavelength as $$\tilde{g}_\text{R0}\approx(1\times10^{-13}/\lambda_\text{S})\text{ m W}^{-1}$$, where $$\lambda_\text{S}$$ is in micrometers.

The Raman gain factor of fused silica is relatively small compared to those of many molecular substances such as benzene and $$\text{CS}_2$$. Many amorphous glass materials, such as $$\text{GeO}_2$$, $$\text{B}_2\text{O}_3$$, and $$\text{P}_2\text{O}_5$$, which are commonly used to dope silica fibers also have peak Raman gain factors that are five to ten times that of pure silica with corresponding frequency shifts ranging from $$400$$ to $$1400\text{ cm}^{-1}$$. In particular, the peak Raman gain factor of $$\text{GeO}_2$$ is $$9.2$$ times that of pure silica at a frequency shift of $$420\text{ cm}^{-1}$$.

Therefore, the peak value, the frequency shift corresponding to the peak, and the spectral shape of the Raman gain factor of a particular fiber all depend on the type and concentration of the dopants in the fiber.

Example 9-17

An optical wave at $$1.55\text{ μm}$$ wavelength propagates in a silica fiber that has a peak Raman gain factor as described above at a Raman frequency shift of $$460\text{ cm}^{-1}$$. If the optical intensity is sufficiently high to generate a Raman Stokes signal, what is the wavelength of the Stokes signal? What is the Raman frequency shift in hertz? What is the Raman gain factor for this signal?

Because $$\omega_\text{S}=\omega_\text{p}-\Omega_\text{R}$$, the Stokes wavelength can be found by using the relation

$\frac{1}{\lambda_\text{S}}=\frac{1}{\lambda_\text{p}}-\frac{f_\text{R}}{c}$

The Raman frequency shift quoted per centimeter is actually $$f_\text{R}/c$$ in the above relation. For the fiber considered here, we have $$f_\text{R}/c=460\text{ cm}^{-1}=4.6\times10^4\text{ m}^{-1}$$. Therefore, the wavelength of the Stokes signal is

$\lambda_\text{S}=\left(\frac{1}{1.55\times10^{-6}}-4.6\times10^4\right)^{-1}\text{ m}=1.669\text{ μm}$

Because $$1\text{ cm}^{-1}\equiv30\text{ GHz}$$, the Raman frequency shift is $$f_\text{R}/c=460\text{ cm}^{-1}=13.8\text{ THz}$$.

The Raman gain factor at this wavelength is

$\tilde{g}_\text{R0}=\frac{1\times10^{-13}}{\lambda_\text{S}}\text{ m W}^{-1}=\frac{1\times10^{-13}}{1.669}\text{ m W}^{-1}=5.99\times10^{-14}\text{ m W}^{-1}$

Brillouin gain

For Brillouin scattering, the relevant excitation is a long-range acoustic wave, which has a linear dispersion relation between the magnitude of its wavevector and its frequency as that given in (8-3): $$K=\Omega/v_\text{a}$$ [refer to the elastic waves tutorial].

The conditions in (9-179) and (9-180) for Brillouin Stokes scattering are the same as those for the first-order down-shifted Bragg diffraction discussed in the acousto-optic diffraction tutorial, except that in Brillouin scattering the acoustic wave is generated by the pump optical wave whereas in acousto-optic Bragg diffraction the acoustic wave is externally applied to the medium. Therefore, the amount of frequency shift in Brillouin scattering is a function of the pump optical frequency and the scattering angle, $$\theta$$, between $$\mathbf{k}_\text{S}$$ and $$\mathbf{k}_\text{p}$$.

In general, the Brillouin frequency shift is a few orders of magnitude smaller than the pump and the Stokes optical frequencies. For Brillouin scattering in an isotropic medium, the approximation $$k_\text{S}\approx{k}_\text{p}$$ is valid. Then, by using (9-179) and (9-180) together with the dispersion relation of the acoustic wave, we find the following angle-dependent frequency shift:

$\tag{9-183}\Omega=2v_\text{a}k_\text{p}\sin\frac{\theta_\text{def}}{2}=\frac{2nv_\text{a}}{c}\omega_\text{p}\sin\frac{\theta_\text{def}}{2}$

where $$n$$ is the index of refraction at the optical frequency $$\omega_\text{p}$$, $$v_\text{a}$$ is the acoustic velocity in the medium, and $$\theta_\text{def}=\theta_\text{d}-\theta_\text{i}$$ is the deflection angle of the acousto-optic Bragg diffraction as defined in the acousto-optic diffraction tutorial.

We see that, very differently from Raman scattering, Brillouin scattering does not have a constant frequency shift in all directions. In particular, there is no Brillouin Stokes scattering in the forward direction because $$\Omega$$ given in (9-183) vanishes for $$\theta_\text{def}=0$$.

Spontaneous Brillouin scattering appears in all other directions with a frequency shift that varies with the scattering angle. Stimulated Brillouin scattering occurs predominantly in the backward direction with a maximum frequency shift, known as the Brillouin frequency, which is determined by the phase-matching condition given in (9-180) to be

$\tag{9-184}\Omega_\text{B}=\frac{nv_\text{a}}{c}(\omega_\text{p}+\omega_\text{S})=\frac{2nv_\text{a}/c}{1+nv_\text{a}/c}\omega_\text{p}\approx\frac{2nv_\text{a}}{c}\omega$

where we have used the fact that $$\omega_\text{p}\approx\omega_\text{S}=\omega\gg\Omega_\text{B}$$, or

$\tag{9-185}f_\text{B}=\frac{\Omega_\text{B}}{2\pi}=\frac{2nv_\text{a}}{\lambda}$

With a pump beam in the optical spectral region, the Brillouin frequency $$f_\text{B}$$ falls in the hypersonic region, typically in the range of $$1-50\text{ GHz}$$ for a large variety of materials.

The Brillouin gain factor of a material can be expressed in a form similar to that of the Raman gain factor. For backward interaction at a Brillouin frequency $$\Omega_\text{B}$$, we have

$\tag{9-186}\tilde{g}_\text{B}=\tilde{g}_\text{B0}\frac{\gamma_\text{B}^2}{(\Omega-\Omega_\text{B})^2+\gamma_\text{B}^2}=\tilde{g}_\text{B0}\frac{\gamma_\text{B}^2}{(\omega_\text{p}-\omega_\text{S}-\Omega_\text{B})^2+\gamma_\text{B}^2}$

This Brillouin gain factor has a FWHM linewidth of $$\Delta\Omega_\text{B}=2\gamma_\text{B}$$, or $$\Delta{f}_\text{B}=\gamma_\text{B}/\pi$$, which is associated with a response time of $$\gamma_\text{B}^{-1}$$ for the acoustic excitation that is responsible for the Brillouin process. Because the Brillouin response time in a common material is typically on the order of nanoseconds, the Brillouin linewidth $$\Delta{f}_\text{B}$$ is typically in the range of 10 MHz to 1 GHz. Therefore, a Brillouin device does not respond efficiently to very short optical pulses, or to any optical waves that have spectral widths in the gigahertz range or above.

For a pump optical wave that has a Lorentzian spectral shape with a FWHM linewidth $$\Delta\nu_\text{p}$$, the peak Brillouin gain factor of a medium scales as

$\tag{9-187}\tilde{g}_\text{B0}=\frac{\Delta{f}_\text{B}}{\Delta{f}_\text{B}+\Delta\nu_\text{p}}\tilde{g}_\text{B0}^\text{max}$

where $$\tilde{g}_\text{B0}^\text{max}$$ is the peak Brillouin gain factor for an idealistic CW wave of zero linewidth with $$\Delta\nu_\text{p}=0$$. Clearly, when $$\Delta\nu_\text{p}\gg\Delta{f}_\text{B}$$, the peak Brillouin gain factor is greatly reduced.

The Brillouin gain factor has other characteristics that are different from those of the Raman gain factor due to the fact that the Brillouin frequency is dictated by the phase-matching condition of (9-180). We have seen in (9-184) that $$\Omega_\text{B}\propto\omega$$. In addition, $$\gamma_\text{B}\propto\omega^2$$, but $$\tilde{g}_\text{B0}$$ is independent of optical frequency.

For fused silica, $$f_\text{B}\approx(17.3/\lambda)\text{ GHz}$$ and $$\Delta{f}_\text{B}\approx(38.4/\lambda^2)\text{ MHz}$$, where $$\lambda$$ is in micrometers, and $$\tilde{g}_\text{B0}^\text{max}=4.5\times10^{-11}\text{ m W}^{-1}$$.

Many gases, such as $$\text{H}_2$$, $$\text{N}_2$$, $$\text{O}_2$$, $$\text{Ar}$$, and $$\text{Xe}$$, have useful Raman and Brillouin gains and frequency shifts for practical applications. A Gas for such applications is normally contained in a high-pressure cell, often called a Raman cell or a Brillouin cell depending on its intended application.

One significant difference between a gaseous medium and a liquid or solid medium is that $$\tilde{g}_\text{R0}$$, $$\Delta{f}_\text{R}$$, $$\tilde{g}_\text{B0}$$, and $$\Delta{f}_\text{B}$$ of a gaseous medium depend on the density of the molecules in the medium, which can be varied by varying the gas pressure in a cell of fixed length and volume. The value of $$\tilde{g}_\text{R0}$$ scales linearly with the density of the gas molecules at low pressures until it saturates at a certain pressure. In comparison, the value of $$\tilde{g}_\text{B0}$$ scales quadratically with the density of the gas molecules. Therefore, for a given gaseous medium, $$\tilde{g}_\text{R0}$$ can be larger than $$\tilde{g}_\text{B0}$$ at low pressures, but $$\tilde{g}_\text{B0}$$ eventually becomes larger than $$\tilde{g}_\text{R0}$$ at a sufficiently high pressure. In addition, $$\tilde{g}_\text{B0}$$ and $$\Delta{f}_\text{B}$$ also depend on temperature.

Example 9-18

For the optical wave at $$1.55\text{ μm}$$ wavelength propagating in a silica fiber as described in Example 9-17 above, what are the Brillouin frequency shift, the Brillouin linewidth, and the peak Brillouin gain factor if the optical wave has a linewidth of $$1\text{ MHz}$$? What is the peak Brillouin gain factor if the optical wave has a linewidth of $$100\text{ MHz}$$?

According to the characteristics of fused silica described above, the Brillouin frequency shift is $$f_\text{B}=(17.3/1.55)\text{ GHz}=11.16\text{ GHz}$$, and the Brillouin linewidth is $$\Delta{f}_\text{B}=(38.4/1.55^2)\text{ MHz}=15.98\text{ MHz}$$.

Though $$\tilde{g}_\text{B0}^\text{max}=4.5\times10^{-11}\text{ m W}^{-1}$$ for the silica fiber is quite independent of the optical wavelength, the peak Brillouin gain factor varies with the linewidth $$\Delta\nu_\text{p}$$ of the optical wave according to (9-187).

Therefore, the peak Brillouin gain factor is $$\tilde{g}_\text{B0}=(15.98/(15.98+1))\times4.5\times10^{-11}\text{ m W}^{-1}\approx4.23\times10^{-11}\text{ m W}^{-1}$$ if the optical wave has a narrow linewidth of $$\Delta\nu_\text{p}=1\text{ MHz}$$.

If the linewidth of the optical wave is increased to $$\Delta\nu_\text{p}=100\text{ MHz}$$, the peak Brillouin gain is reduced to $$\tilde{g}_\text{B0}=(15.98/(15.98+100))\times4.5\times10^{-11}\text{ m W}^{-1}\approx6.2\times10^{-12}\text{ m W}^{-1}$$. Further increase of the linewidth of the optical wave will further reduce the peak Brillouin gain.

Raman amplifiers

The Raman gain can be utilized to amplify an optical signal at a Stokes frequency $$\omega_\text{S}$$ through the process of stimulated Raman scattering by choosing a proper pump wave at the frequency $$\omega_\text{p}=\omega_\text{S}+\Omega$$ with a frequency shift $$\Omega$$ that is within the Raman gain spectrum, ideally at the gain-peak frequency $$\Omega_\text{R}$$.

Because stimulated Raman scattering has the same gain factor in forward and backward directions, the pump wave and the Stokes signal wave can propagate either codirectionally, as shown in Figure 9-32(a), or contradirectionally, as shown in Figure 9-32(b), in a Raman amplifier.

However, though the gain factor is the same for the two configurations, a Raman amplifier with a contradirectional configuration would have little or no efficiency for short optical pulses because of the short interaction length between the pump and the Stokes pules that propagate in opposite directions. Here we consider only Raman amplification in a codirectional configuration. The general formulation and characteristics for Raman amplification in a contradirectional configuration are similar to those for Brillouin amplification discussed later.

Following (9-78) and (9-79) [refer to the coupled-wave analysis of nonlinear optical interactions tutorial] and allowing for the existence of linear absorption loss in a medium, we have the following coupled equations for Raman amplification in a codirectional configuration:

$\tag{9-188}\frac{\text{d}I_\text{S}}{\text{d}z}+\alpha_\text{S}I_\text{S}=\tilde{g}_\text{R}I_\text{S}I_\text{p}$

$\tag{9-189}\frac{\text{d}I_\text{p}}{\text{d}z}+\alpha_\text{p}I_\text{p}=-\frac{\omega_\text{p}}{\omega_\text{S}}\tilde{g}_\text{R}I_\text{S}I_\text{p}$

with given values of $$I_\text{p}(0)$$ and $$I_\text{S}(0)$$ at the input end, $$z=0$$, of the amplifier as the initial conditions. The parameters $$\alpha_\text{S}$$ and $$\alpha_\text{p}$$ are the linear absorption coefficients of the medium at the Stokes and pump frequencies, respectively.

The coupled equations in (9-188) and (9-189) have an exact analytical solution when $$\alpha_\text{p}=\alpha_\text{S}$$. In the amplification of a weak signal when the depletion of the pump intensity due to Raman interaction can be neglected, a simple approximate solution can be obtained by ignoring the term on the right-hand side of (9-189). Then, for a Raman amplifier of length $$l$$, we have the signal at the output end of the amplifier given by

$\tag{9-190}I_\text{S}(l)=I_\text{S}(0)\exp(g_\text{R}l_\text{eff}-\alpha_\text{S}l)$

where $$g_\text{R}$$ is the Raman gain coefficient defined as

$\tag{9-191}g_\text{R}=\tilde{g}_\text{R}I_\text{p}(0)=-\frac{3\omega_\text{S}\mu_0}{n_{\text{S},z}n_{\text{p},z}}\chi_\text{R}''I_\text{p}(0)$

and $$l_\text{eff}$$ is the effective interaction length of Raman amplification given by

$\tag{9-192}l_\text{eff}=\frac{1-\text{e}^{-\alpha_\text{p}l}}{\alpha_\text{p}}$

Note that the Raman gain coefficient increases with pump intensity. Such dependence on an optical intensity is characteristic of an optical gain that is contributed by a nonlinear optical process.

The amplification factor, or the Raman amplifier gain, for the Stokes signal in the case of negligible pump depletion is then

$\tag{9-193}G_\text{R}=\frac{I_\text{S}(l)}{I_\text{S}(0)\exp(-\alpha_\text{S}l)}=\exp(g_\text{R}l_\text{eff})=\exp[\tilde{g}_\text{R}I_\text{p}(0)l_\text{eff}]$

For a given Raman amplifier with a fixed length, the amplifier gain can be controlled by varying the pump intensity.

Example 9-19

An optical fiber that has the Raman gain characteristics described in Example 9-17 is used as a fiber Raman amplifier for codirectional amplification of an optical signal at $$\lambda_\text{S}=1.55\text{ μm}$$. The input power of the signal is $$-15\text{ dBm}$$, and the desired output power is $$0\text{ dBm}$$. The fiber has an absorption coefficient of $$\alpha=0.2\text{ dB km}^{-1}$$ at this signal wavelength and a length of $$l=25\text{ km}$$. Its core has an effective cross-sectional area of $$\mathcal{A}_\text{eff}=5\times10^{-11}\text{ m}^2$$. What is the pump wavelength for the largest Raman gain? What is the required pump power if the absorption coefficient at the pump wavelength is the same as that at the signal wavelength?

Because the Raman gain spectrum of an optical fiber is very broad, it is in general only necessary to pick a pump wavelength so that the signal wavelength falls within the Raman gain spectral range of the pump. However, to have the largest Raman gain, we need to choose a pump wavelength properly so that the Raman gain peak appears at the signal wavelength. From Example 9-17, we have $$f_\text{R}/c=460\text{ cm}^{-1}=4.6\times10^4\text{ m}^{-1}$$ at the peak of the Raman spectrum. Therefore, the pump wavelength for the largest Raman gain is

$\lambda_\text{p}=\left(\frac{1}{1.55\times10^{-6}}+4.6\times10^4\right)^{-1}\text{ m}=1.4468\text{ μm}$

The peak Raman gain factor at $$\lambda_\text{S}=1.55\text{ μm}$$ is

$\tilde{g}_\text{R}=\frac{1\times10^{-13}}{1.55}\text{ m W}^{-1}=6.45\times10^{-14}\text{ m W}^{-1}$

With $$\alpha=0.2\text{ dB km}^{-1}=0.046\text{ km}^{-1}$$ and $$l=25\text{ km}$$, we have

$l_\text{eff}=\frac{1-\text{e}^{-0.046\times25} }{0.046}\text{ km}=14.86\text{ km}$

Because $$P_\text{S}^\text{in}=I_\text{S}(0)\mathcal{A}_\text{eff}$$ and $$P_\text{S}^\text{out}=I_\text{S}(l)\mathcal{A}_\text{eff}$$, we find from (9-193) that the required Raman amplifier gain in decibels is

$\tag{9-194}G_\text{R}=P_\text{S}^\text{out}\text{(dBm)}-P_\text{S}^\text{in}\text{(dBm)}+\alpha\text{(dB km}^{-1}\text{)}l\text{(km)}=20\text{ dB}$

Therefore, $$G_\text{R}=20\text{ dB}=100$$. Identifying the pump power $$P_\text{p}=I_\text{p}(0)\mathcal{A}_\text{eff}$$ and using (9-193), we find the following required pump power:

$\tag{9-195}P_\text{p}=\frac{\mathcal{A}_\text{eff}}{l_\text{eff}}\frac{\ln G_\text{R}}{\tilde{g}_\text{R}}=\frac{5\times10^{-11}\times\ln100}{14.86\times10^3\times6.45\times10^{-14}}\text{ W}=240\text{ mW}$

Note that in using (9-193) to obtain (9-194) and (9-195), we have implicitly assumed that depletion of the pump power due to its conversion to the signal power is negligible. This assumption is clearly valid here because the pump power obtained under such an assumption is 240 mW while the output signal is only 0 dBm, which is 1 mW. Stimulated Brillouin scattering has to be suppressed in a Raman amplifier because it can deplete the pump power for Raman amplification. Because the Brillouin gain factor decreases with the linewidth of the pump, an optical source of a linewidth that is large enough to suppress stimulated Brillouin scattering is normally used for pumping a Raman amplifier. Multimode semiconductor lasers can serve such a purpose for pumping fiber Raman amplifiers.

Raman generators

When there is a pump optical wave in a medium that has a Raman susceptibility, spontaneous Raman scattering that generates incoherent Stokes and anti-Stokes emission in all directions always occurs. In a Raman amplifier where a coherent input signal is amplified, such incoherent spontaneous emission contributes to the noise in the amplifier. In the absence of an input signal, however, the ubiquitous spontaneous Raman emission can be the seed for the generation of a Stokes or anti-Stokes wave through stimulated amplification under the right conditions. A Raman generator is normally used for the generation of the Stokes wave at the down-shifted Stokes frequency of $$\omega_\text{S}=\omega_\text{p}-\Omega_\text{R}$$ with the medium initially unexcited. A Raman generator can simply be a Raman amplifier without an input signal but with a pump of sufficient intensity for significant power conversion from the pump frequency to the Stokes frequency in a single pass through the medium.

In a Raman generator, the Stokes wave grows from stimulated amplification of the spontaneous Stokes emission. Because spontaneous Stokes emission occurs along the entire length of the generator, the total Stokes power at the output is the result of the cumulative amplification of all spontaneous Stokes emission over the length of the generator.

A detailed analysis that takes into account such cumulative amplification can be carried out. For forward Raman interaction, the net result is equivalent to treating the generator as an amplifier with the injection of an effective Stokes signal $$I_\text{S}^\text{eff}(0)$$ at $$z=0$$ while ignoring all of the spontaneous Stokes emission in the generator.

For backward interaction, it is equivalent to injection of an effective Stokes signal $$I_\text{S}^\text{eff}(l)$$ at $$z=l$$ while ignoring all of the spontaneous Stokes emission.

The values of the effective signals depend on the Raman characteristics, particularly $$\tilde{g}_\text{R0}$$ and $$\Delta{f}_\text{R}$$, of the medium, as well as on the pump intensity. Besides, due to the difference between the forward and the backward interactions in the geometric relation of the pump and the Stokes waves, the effective signal $$I_\text{S}^\text{eff}(l)$$ for backward interaction is significantly smaller than the effective signal $$I_\text{S}^\text{eff}(0)$$ for forward interaction at a given pump intensity in a given medium.

This difference leads to a higher threshold for backward Raman generation than that for forward Raman generation. As a result, only a Stokes wave in the forward direction is generated in a Raman generator. No backward generation occurs.

Because significant power conversion from the pump wave to the Stokes wave is desired in the application of a Raman generator, pump depletion cannot be neglected. If we assume for simplicity that $$\alpha_\text{S}=\alpha_\text{p}$$ and consider the fact that $$I_\text{p}(0)\gg{I}_\text{S}^\text{eff}(0)$$ for a Raman generator, the complete solution for the coupled equations in (9-188) and (9-189) leads to the relation

$\tag{9-196}\frac{I_\text{S}(l)}{I_\text{p}(l)}\approx\frac{I_\text{S}^\text{eff}(0)}{I_\text{p}(0)}\exp[\tilde{g}_\text{R}I_\text{p}(0)l_\text{eff}]=\frac{I_\text{S}^\text{eff}(0)}{I_\text{p}(0)}G_\text{R}$

The threshold os a Raman generator can be defined as the condition for $$I_\text{S}(l)=I_\text{p}(l)$$. Then, the following threshold amplification factor is obtained at the Raman threshold:

$\tag{9-197}G_\text{R}^\text{th}=\exp\left[\tilde{g}_\text{R}I_\text{p}^\text{th}(0)l_\text{eff}\right]=\frac{I_\text{p}^\text{th}(0)}{I_\text{S,th}^\text{eff}(0)}$

The physical meaning of this relation is that the Raman threshold is reached when stimulated amplification of the spontaneous Stokes emission brings the Stokes intensity at the output to the same level as that of the pump intensity. Because the value of $$I_\text{S}^\text{eff}(0)$$ depends on the characteristics of the medium, the value of $$G_\text{R}^\text{th}$$ is also a function of the characteristics of the medium. Therefore, the threshold pump intensity for forward Raman Stokes generation in a medium of length $$l$$ can be calculated using the following relation:

$\tag{9-198}I_\text{p}^\text{th}(0)\approx\frac{\ln{G}_\text{R}^\text{th}}{\tilde{g}_\text{R}l_\text{eff}}$

For example, $$\ln{G}_\text{R}^\text{th}\approx16$$ for forward Raman Stokes generation in single-mode silica fibers. For backward Raman Stokes generation in a single-mode silica fiber, $$\ln{G}_\text{R}^\text{th}\approx20$$. Backward Raman Stokes generation normally does not occur because it has a much higher threshold than forward generation.

In the case when $$\alpha_\text{S}=\alpha_\text{p}=\alpha$$, a simple relation for calculating the conversion efficiency of a Raman generator can be obtained by assuming that $$I_\text{p}(0)\gg{I}_\text{S}^\text{eff}(0)=I_\text{S,th}^\text{eff}(0)$$ for any pump intensity. Then, the Raman conversion efficiency from the pump to the Stokes is found to be

$\tag{9-199}\eta_\text{R}=\frac{I_\text{S}(l)}{I_\text{p}(0)}=\frac{\omega_\text{S}}{\omega_\text{p}}\frac{1}{1+(\omega_\text{S}/\omega_\text{p})r(G_\text{R}^\text{th})^{1-r}}\text{e}^{-\alpha{l}}$

where $$r=I_\text{p}(0)/I_\text{p}^\text{th}(0)$$ is the pump ratio with respect to the threshold pump intensity.

The threshold of a Raman generator is very sharp. Below the threshold, $$\eta_\text{R}$$ quickly approaches zero, but it quickly approaches its maximum value of $$(\omega_\text{S}/\omega_\text{p})\text{e}^{-\alpha{l}}$$ above the threshold.

Therefore, (9-199) can be used to find the Raman conversion efficiency quite accurately for any value of $$r$$ irrespective of the assumption used in obtaining it.

By using (9-199) to calculate the Raman Stokes generation in a single-mode silica fiber, it is found that a reduction in the pump intensity by 1 dB below the threshold reduces the output Stokes intensity by more than 10 dB, but an increase in the pump intensity by 1 dB above the threshold causes the conversion from the pump to the Stokes to be more than 98% complete.

If the pump intensity is many times above the threshold, complete conversion of power from the pump to the Stokes occurs within a very short distance from the input end. This first Stokes wave at the frequency $$\omega_\text{S1}=\omega_\text{p}-\Omega_\text{R}$$ can then serve as a pump to generate the second Stokes wave at the frequency $$\omega_\text{S2}=\omega_\text{S1}-\Omega_\text{R}=\omega_\text{p}-2\Omega_\text{R}$$. This cascading process continues until the waves reach the end of the generator.

Therefore, with proper choices of generator length and pump intensity, a high-order Stokes wave can be generated at a frequency that is down-shifted from the pump frequency by an integral multiple of the Raman frequency.

However, such complete power conversion from the pump to the first Stokes and from a low-order Stokes to a high-order Stokes is possible only for CW waves or very long optical pulses.

For short optical pulses, power conversion from one order to another is normally not complete due to temporal walk-off between the interacting pulses of different wavelengths caused by group-velocity dispersion in the medium, but generation of multiple Stokes orders is still possible with high-intensity pulses.

Sometimes, in addition to the Stokes wave, an anti-Stokes wave at the up-shifted anti-Stokes frequency of $$\omega_\text{AS}=\omega_\text{p}+\Omega_\text{R}$$ can also be generated through Stokes-anti-Stokes coupling and/or parametric four-wave mixing with the pump if the required phase-matching conditions for such parametric processes are satisfied.

In practical applications, however, a Raman generator is normally used as a nonparametric frequency converter to convert the optical power at a high-frequency pump wave to a low-frequency Stokes wave.

Example 9-20

The fiber Raman amplifier described in Example 9-19 can be used as a fiber Raman generator for a Stokes signal at $$\lambda_\text{S}=1.55\text{ μm}$$ without an input signal at this wavelength by raising the pump power at $$\lambda_\text{p}=1.4468\text{ μm}$$. Find the threshold pump power for this fiber Raman generator.

Identifying $$P_\text{p}^\text{th}=I_\text{p}^\text{th}(0)A_\text{eff}$$ and using $$\ln{G}_\text{R}^\text{th}\approx16$$, we have, from (9-198),

$\tag{9-200}P_\text{p}^\text{th}=\frac{A_\text{eff}}{l_\text{eff}}\frac{\ln{G}_\text{R}^\text{th}}{\tilde{g}_\text{R}}=\frac{16A_\text{eff}}{\tilde{g}_\text{R}l_\text{eff}}$

for the threshold pump power of a fiber Raman generator. Using the parameters obtained in Example 9-19, we find that

$P_\text{p}^\text{th}=\frac{16\times5\times10^{-11}}{6.45\times10^{-14}\times14.86\times10^3}\text{ W}=835\text{ mW}$

When the pump power is below $$P_\text{p}^\text{th}$$, very little power is converted to the Stokes signal in a Raman generator. When the pump power exceeds $$P_\text{p}^\text{th}$$ at a certain level, it is completely converted to the Stokes. If the pump power continues to increase, the power is converted to a successively higher order of Stokes at the output.

Brillouin amplifiers

The Brillouin gain in a medium can also be utilized to amplify an optical signal at a frequency that is down-shifted from the pump frequency by an amount equal to the Brillouin frequency.

Due to the fundamental differences between the Raman and the Brillouin processes discussed above, the Brillouin amplifiers have several characteristics that are very different from those of the Raman amplifiers.

First, only the contradirectional configuration shown in Figure 9-32(b) is acceptable for a Brillouin amplifier because there is no forward Brillouin scattering.

Second, the Brillouin linewidth is relatively narrow. Therefore, a Brillouin amplifier is useful only for the amplification of narrow-band signals, whereas a Raman amplifier can be used for broadband signals or short-pulse signals because of the large Raman linewidth.

Third, the peak Brillouin gain factor, $$\tilde{g}_\text{B0}$$, of a solid or liquid medium, or a high-pressure gaseous medium, is usually much larger than the peak Raman gain factor, $$\tilde{g}_\text{R0}$$, of the same. medium. Therefore, a Brillouin amplifier usually requires a much lower pump intensity than what a Raman amplifier needs to have the same amplification factor for the signal.

Because of the contradirectional configuration of a Brillouin amplifier, the signal propagates in the $$-z$$ direction while the pump propagates in the $$z$$ direction. Therefore, Brillouin amplification is described by the following coupled equations:

$\tag{9-201}-\frac{\text{d}I_\text{S}}{\text{d}z}+\alpha_\text{S}I_\text{S}=\tilde{g}_\text{B}I_\text{S}I_\text{p}$

$\tag{9-202}\frac{\text{d}I_\text{p}}{\text{d}z}+\alpha_\text{p}I_\text{p}=-\frac{\omega_\text{p}}{\omega_\text{S}}\tilde{g}_\text{B}I_\text{S}I_\text{p}$

with the input pump intensity $$I_\text{p}(0)$$ at $$z=0$$ and the input signal intensity $$I_\text{S}(l)$$ at $$z=l$$ given as the boundary conditions.

The exact solution for this backward amplification differs from that for the forward amplification. It can be found when $$\alpha_\text{S}=\alpha_\text{p}=0$$. In the application of an optical amplifier for the amplification of a weak signal, however, there is little pump depletion due to nonlinear Brillouin interaction. Then, the right-hand side of (9-202) can be ignored to obtain the following solution for the output intensity of the signal at $$z=0$$:

$\tag{9-203}I_\text{S}(0)=I_\text{S}(l)\exp(g_\text{B}l_\text{eff}-\alpha_\text{S}l)$

where $$l_\text{eff}$$ is the effective interaction length of the same form as that defined in (9-192) and $$g_\text{B}$$ is the Brillouin gain coefficient defined as

$\tag{9-204}g_\text{B}=\tilde{g}_\text{B}I_\text{p}(0)$

Therefore, in the case of negligible pump depletion, the amplification factor of a Brillouin amplifier, or the Brillouin amplifier gain, is

$\tag{9-205}G_\text{B}=\frac{I_\text{S}(0)}{I_\text{S}(l)\exp(-\alpha_\text{S}l)}=\exp(g_\text{B}l_\text{eff})=\exp[\tilde{g}_\text{B}I_\text{p}(0)l_\text{eff}]$

Example 9-21

If the fiber Raman amplifier described in Example 9-19 is turned into a Brillouin amplifier for the same input signal and the same desired output signal, what should the pump wavelength be? If the pump wave has a linewidth of 100 MHz, what is the required pump power?

From Example 9-18, we know that $$f_\text{B}=11.16\text{ GHz}$$. Therefore, $$f_\text{R}/c=37.2\text{ m}^{-1}$$, and the pump wavelength is

$\lambda_\text{p}=\left(\frac{1}{1.55\times10^{-6}}+37.2\right)^{-1}\text{ m}=1.5499\text{ μm}$

The pump wavelength is very close to the signal wavelength because of the small Brillouin frequency shift. We find from Example 9-18 that the peak Brillouin gain for this amplifier is $$\tilde{g}_\text{B}=6.2\times10^{-12}\text{ m W}^{-1}$$ because the pump has a linewidth of 100 MHz. Because a Brillouin amplifier functions only in the contradirectional configuration, we identify $$P_\text{S}^\text{in}=I_\text{S}(l)\mathcal{A}_\text{eff}$$ and $$P_\text{S}^\text{out}=I_\text{S}(0)\mathcal{A}_\text{eff}$$. Then we find from (9-205) that the required Brillouin amplifier gain in decibels is

$\tag{9-206}G_\text{B}=P_\text{S}^\text{out}(\text{dBm})-P_\text{S}^\text{in}(\text{dBm})+\alpha(\text{dB km}^{-1})l(\text{km})=20\text{ dB}$

which is the same as the Raman amplifier gain in Example 9-19. Therefore, $$G_\text{B}=100$$. From (9-205) we find by identifying the pump power as $$P_\text{p}=I_\text{p}(0)\mathcal{A}_\text{eff}$$ that

$\tag{9-207}P_\text{p}=\frac{\mathcal{A}_\text{eff}}{l_\text{eff}}\frac{\ln{G}_\text{B}}{\tilde{g}_\text{B}}=\frac{5\times10^{-11}\times\ln100}{14.86\times10^3\times6.2\times10^{-12}}\text{ W}=2.5\text{ mW}$

By comparing (9-195) with (9-207), we find that for the same amplifier gain, $$G_\text{B}=G_\text{R}$$, the pump power required for a Brillouin amplifier is scaled from that for a Raman amplifier by a factor of $$P_\text{p}^\text{B}/P_\text{p}^\text{R}=\tilde{g}_\text{R}/\tilde{g}_\text{B}$$. Because $$\tilde{g}_\text{B}\gg\tilde{g}_\text{R}$$ by about two orders of magnitude in this example, the pump power is reduced by as much.

Note that in using (9-205) to obtain (9-206) and (9-207), we have implicitly assumed that depletion of the pump power due to its conversion to the signal power is negligible. This assumption is not really valid here because the pump power obtained under such an assumption is 2.5 mW but the output signal is 1 mW. A more detailed analysis with the effect of pump depletion taken into consideration is required to obtain the accurate result.

Brillouin generators

Similarly to the situation in a Raman generator, the emission from spontaneous Brillouin scattering can also seed the generation of a Brillouin Stokes frequency in the presence of a pump above a Brillouin threshold but in the absence of an input Stokes signal.

Besides the fundamental differences in terms of the frequency shift and the generation efficiency, an important difference between a Brillouin generator and a Raman generator is that the Brillouin Stokes wave is generated only in the backward direction but the Raman Stokes is generated only in the forward direction.

As discussed above, for backward generation, the net result of the cumulative backward amplification of spontaneous emission over the entire length of interaction is equivalent to the injection of an effective backward-propagating Stokes signal $$I_\text{S}^\text{eff}(l)$$ at $$z=l$$.

Considering the physical implication of the threshold amplification factor given in (9-197) for a Raman generator, the Brillouin threshold can be defined as the condition in which stimulated Brillouin amplification of the spontaneous Brillouin Stokes emission brings the Stokes intensity to the level of the pump intensity.

Because the effective Stokes signal at $$z=l$$ is related to the pump intensity at $$z=l$$, we then have the following threshold amplification factor for a Brillouin generator:

$\tag{9-208}G_\text{B}^\text{th}=\exp\left[\tilde{g}_\text{B}I_\text{p}^\text{th}(0)l_\text{eff}\right]=\frac{I_\text{p}^\text{th}(l)}{I_\text{S,th}^\text{eff}(l)}$

where $$I_\text{p}^\text{th}(0)$$ and $$I_\text{p}^\text{th}(l)$$ are the input pump intensity at $$z=0$$ and the remaining pump intensity at $$z=l$$, respectively, at the threshold of the Brillouin generator.

The value of $$G_\text{B}^\text{th}$$ is a function of the characteristics of the medium and is generally larger than that of $$G_\text{R}^\text{th}$$ for the same medium, primarily because of the fact that the Brillouin Stokes is generated in the backward direction.

Therefore, the threshold pump intensity for Brillouin Stokes generation in a medium of length $$l$$ is

$\tag{9-209}I_\text{p}^\text{th}(0)\approx\frac{\ln{G}_\text{B}^\text{th}}{\tilde{g}_\text{B}l_\text{eff}}$

For example, $$\ln{G}_\text{B}^\text{th}\approx21$$ for Brillouin Stokes generation in single-mode silica fibers.

The conversion efficiency of a Brillouin generator from the pump to the Stokes is measured in terms of an intensity reflectivity defined as

$\tag{9-210}R_\text{B}=\frac{I_\text{S}(0)}{I_\text{p}(0)}$

In the case when $$\alpha_\text{S}=\alpha_\text{p}=0$$, the value of $$R_\text{B}$$ can be found from the following transcendental relation:

$\tag{9-211}R_\text{B}=\left(G_\text{B}^\text{th}\right)^{r(1-R_\text{B})-1}$

under the approximation that $$\omega_\text{p}\approx\omega_\text{S}$$ for Brillouin scattering in the optical region, where $$r=I_\text{p}(0)/I_\text{p}^\text{th}(0)$$.

Because of the large value of $$G_\text{B}^\text{th}$$, the relation in (9-211) indicates a sharp threshold for Brillouin generation. Below the Brillouin threshold, $$r\lt1$$, and $$R_\text{B}$$ quickly approaches zero. Above the threshold, $$R_\text{B}$$ varies with pump intensity approximately as

$\tag{9-212}R_\text{B}\approx1-\frac{1}{r}=1-\frac{I_\text{p}^\text{th}(0)}{I_\text{p}(0)},\qquad\text{for }r\gt1$

This relation leads to the important conclusion that

$\tag{9-213}I_\text{p}(l)=I_\text{p}^\text{th}(0)\qquad\text{if}\qquad{I_\text{p}(0)}\gt{I}_\text{p}^\text{th}(0)$

Therefore, when the input pump intensity exceeds the threshold pump intensity of a lossless Brillouin generator, the transmitted intensity is clamped at the level of the threshold pump intensity. The excess above the threshold is converted to the Stokes frequency and is reflected back to the input end.

This characteristic allows very efficient Brillouin generation, but it also sets a very important limitation on the level of optical power that can be transmitted through an optical system. In particular, in a fiber-optic transmission system, the generation of the Brillouin Stokes in the optical fiber severely limits the transmission power level of the system.

Example 9-22

The fiber Brillouin amplifier described in Example 9-21 becomes a fiber Brillouin generator for a Stokes signal at $$\lambda_\text{S}=1.55\text{ μm}$$ without an input signal at this wavelength if the pump power at $$\lambda_\text{p}=1.5499\text{ μm}$$ is raised above a threshold level. Find the threshold pump power for this fiber Brillouin generator if the linewidth of the pump is 100 MHz. What is the threshold pump power if the linewidth of the pump is only 1 MHz?

Identifying $$P_\text{p}^\text{th}=I_\text{p}^\text{th}(0)\mathcal{A}_\text{eff}$$ and using $$\ln{G}_\text{B}^\text{th}\approx21$$, we have, from (9-209),

$\tag{9-214}P_\text{p}^\text{th}=\frac{\mathcal{A}_\text{eff}}{l_\text{eff}}\frac{\ln{G}_\text{B}^\text{th}}{\tilde{g}_\text{B}}=\frac{21\mathcal{A}_\text{eff}}{\tilde{g}_\text{B}l_\text{eff}}$

for the threshold pump power of a fiber Brillouin generator. Using the parameters obtained in Example 9-21 with $$\tilde{g}_\text{B}=6.2\times10^{-12}\text{ m W}^{-1}$$ for a pump wave of 100 MHz linewidth, we find that

$P_\text{p}^\text{th}=\frac{21\times5\times10^{-11}}{6.2\times10^{-12}\times14.86\times10^3}\text{ W}=11.4\text{ mW}$

If the pump has a narrow linewidth of only 1 MHz, we have $$\tilde{g}_\text{B}=4.23\times10^{-11}\text{ m W}^{-1}$$ from Example 9-18. Then the threshold pump power is reduced to

$P_\text{p}^\text{th}=\frac{21\times5\times10^{-11}}{4.23\times10^{-11}\times14.86\times10^3}\text{ W}=1.67\text{ mW}$

The Brillouin threshold pump power can be increased substantially if the linewidth of the pump is large. Because the power that remains in the pump is clamped to the Brillouin threshold, with the rest reflected back to the input end, suppressing the Brillouin Stokes generation by sufficiently increasing the Brillouin threshold is essential for the operation of a Raman amplifier, as discussed in Example 9-19, as well as for the operation of a Raman generator.

Because Raman and Brillouin gains exist in the same medium and both Raman Stokes and Brillouin Stokes can grow from spontaneous emission, these two processes compete with each other for the same pump power source. The one that has a lower threshold pump intensity quickly monopolizes the pump power and prohibits the other from occurring. Because $$\tilde{g}_\text{B0}$$ is usually much larger than $$\tilde{g}_\text{R0}$$ in the same medium, Brillouin generation usually dominates although $$G_\text{B}^\text{th}$$ is larger than $$G_\text{R}^\text{th}$$.

However, because the Brillouin gain has a very narrow linewidth, the threshold pump intensity for Brillouin generation increases very quickly when the pump wave has a linewidth exceeding $$\Delta{f}_\text{B}$$. Therefore, Brillouin generation dominates only when the pump has a narrow linewidth, whereas Raman generation dominates when the linewidth of the pump is larger than the Brillouin linewidth.

The pump power required for a Raman amplifier is generally lower than the threshold pump power of a Raman generator, and that for a Brillouin amplifier is lower than the threshold of a Brillouin generator.

However, because the Brillouin gain factor can be a few orders of magnitude higher than the Raman gain factor, stimulated Brillouin scattering can easily occur well below the power required for a Raman amplifier. The consequences of stimulated Brillouin scattering in a Raman amplifier include significant reduction of the Raman gain by depletion of the pump power, generation of noise, and distortion of the signal waveform. It is therefore necessary to suppress stimulated Brillouin scattering in a Raman amplifier by, for example, using a pump of a sufficiently broad linewidth.

Besides the amplifiers and generators discussed above, the Raman gain can also be utilized to construct a Raman laser by placing such a gain medium in an optical oscillator that is in resonance with the Raman Stokes frequency. Similarly, a Brillouin laser can also be constructed by placing a Brillouin gain medium in an optical oscillator that is in resonance with the Brillouin Stokes frequency.

The next part continues with the Nonlinear Optical Interactions in Waveguides tutorial