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Coupled-wave analysis of nonlinear optical interactions

This is a continuation from the previous tutorial - nonlinear optical interactions.

The coupled-wave theory developed in the coupled-wave theory tutorial is used for analysis of nonlinear optical interactions in a homogeneous medium.

In nonlinear optical waveguides, coupled-mode theory can be used if there is no mixing between different optical frequencies, but a combination of coupled-wave and coupled-mode formalisms has to be used if there is frequency mixing in the interaction.

In applying coupled-wave or coupled-mode theory to the analysis of a nonlinear interaction, the perturbing polarization, generally expressed as \(\Delta\mathbf{P}\) in the coupled-wave theory tutorial, is identified as the characteristic nonlinear polarization, \(\mathbf{P}^{(n)}\), of the specific interaction.

A nonlinear optical interaction often takes place in an anisotropic crystal, as can be expected from the fact that \(\boldsymbol{\chi}^{(2)}\) vanishes identically in the bulk of an isotropic medium under the electric-dipole approximation. Even when a nonlinear interaction takes place in an isotropic medium, a longitudinal field component can sometimes be generated because of a field-dependent birefringence induced by a third-order nonlinear process such as the optical Kerr effect discussed above. For these reasons, the correct coupled-wave equation to be used, under the slowly varying amplitude approximation, for the analysis of nonlinear optical interactions is the one in (18) [refer to the coupled-wave theory tutorial]:


where \(\mathbf{P}^{(n)}_q\) is identified with \(\mathbf{P}^{(2)}_q\), or \(\mathbf{P}^{(2)}(\omega_q)\) of (9-19) [refer to the optical nonlinearity tutorial], for a second-order process and with \(\mathbf{P}^{(3)}_q\), or \(\mathbf{P}^{(3)}(\omega_q)\) of (9-20) [refer to the optical nonlinearity tutorial], for a third-order process and, according to (9-14) and (9-16) [refer to the optical nonlinearity tutorial], the field amplitude \(\pmb{\mathcal{E}}_q\) is defined by the following relation:


In most cases of interest, the amplitudes of all of the interacting waves vary along the same direction, which is designated the \(z\) direction. Then, the coupled-wave equation can be written as


Note that the propagation direction, which is the direction normal to the wavefront and is defined by the wavevector \(\mathbf{k}\), of each wave is not necessarily the same as the direction along which the field amplitude varies. In general, the nonlinear polarization \(\mathbf{P}_q^{(n)}\) may also have variations along other directions.

Except in some unusual cases, the longitudinal field components of the interacting optical waves are small and unimportant though they may exist. Then, \(\mathbf{P}^{(n)}_{q,\text{T}}\) in (9-52) and (9-54) can be replaced by \(\mathbf{P}^{(n)}_q\), further simplifying the coupled-wave equation. When this simplification is done, we can multiply both sides of (9-54) by the unit vector \(\hat{e}_q^{*}\) to write the coupled-wave equation as


In the analysis of a nonlinear optical interaction, a coupled-wave equation is written for each of the interacting waves. The nonlinear polarization on the right-hand side of each equation couples the equations for different waves, resulting in an array of coupled nonlinear equations.


Parametric interactions

To illustrate several important concepts using a concrete example, we consider the coupled equations that describe a parametric second-order interaction of three different frequencies \(\omega_1\), \(\omega_2\), and \(\omega_3\) with the relation \(\omega_3=\omega_1+\omega_2\). We also take the approximations that allow us to use (9-55). Using (9-19) [refer to the optical nonlinearity tutorial] and the intrinsic permutation symmetry, we find that




The full permutation symmetry is valid for the real \(\boldsymbol{\chi}^{(2)}\) that characterizes the parametric process. Therefore, we can define an effective nonlinear susceptibility as


Following the relation given in (9-34) [refer to the nonlinear optical susceptibilities tutorial], the effective \(d\) coefficient for this interaction is simply \(d_\text{eff}=\chi_\text{eff}/2\). We have the following coupled equations for a parametric second-order interaction:




where \(\Delta\mathbf{k}=\mathbf{k}_1+\mathbf{k}_2-\mathbf{k}_3=\Delta{k}\hat{z}\) is the phase mismatch. For linearly polarized waves, we have \(\hat{e}^*=\hat{e}\), and \(\chi^*_\text{eff}=\chi_\text{eff}\) is a real quantity. Otherwise, \(\chi_\text{eff}\) can be complex.


Example 9-6

Take \(\hat{x}\), \(\hat{y}\), and \(\hat{z}\) to be the principal axes of LiNbO3. Find the effective nonlinear susceptibilities for second-harmonic generation in LiNbO3 with

(a) a linearly \(x\)-polarized fundamental wave propagating in the \(z\) direction

(b) a linearly \(z\)-polarized fundamental wave propagating in the \(x\) direction.

The propagation direction of the second harmonic is in practice determined by many factors. Here we make it the same as that of the fundamental wave.


For second-harmonic generation, we have the degenerate case of \(\omega_1=\omega_2=\omega\) and \(\omega_3=2\omega\). From Table 9-3 [refer to the nonlinear optical susceptibilities tutorial], we find that the only nonvanishing second-order nonlinear susceptibility elements of LiNbO3 are \(d_{31}=d_{32}=d_{24}=d_{15}=-4.4\text{ pm V}^{-1}\), \(d_{22}=-d_{21}=-d_{16}=2.4\text{ pm V}^{-1}\), and \(d_{33}=-25.2\text{ pm V}^{-1}\). We know that \(\chi_{i\alpha}^{(2)}=2d_{i\alpha}\), according to (9-34) [refer to the nonlinear optical susceptibilities tutorial].

In case (a), we have \(\mathbf{k}_{2\omega}\parallel\mathbf{k}_\omega\parallel\hat{z}\) and \(\hat{e}_\omega=\hat{x}\). We then find from the nonvanishing elements of \(\boldsymbol{\chi}^{(2)}\) for LiNbO3 that \(\mathbf{P}^{(2)}(2\omega)\) has only two components in the \(y\) and \(z\) directions contributed by \(\chi_{21}^{(2)}\) and \(\chi_{31}^{(2)}\), respectively. However, because \(\mathbf{k}_{2\omega}\) is forced to be in the \(z\) direction, \(P^{(2)}_z\) is a longitudinal component that cannot contribute to the propagation of the second-harmonic wave. In this situation, \(\hat{e}_{2\omega}=\hat{y}\) because only the transverse component \(P_y^{(2)}\) is useful for generating the second-harmonic wave. Therefore,

\[\chi_\text{eff}=\hat{e}^*_{2\omega}\cdot\boldsymbol{\chi}^{(2)}:\hat{e}_{\omega}\hat{e}_{\omega}=\hat{y}\cdot\boldsymbol{\chi}^{(2)}:\hat{x}\hat{x}=\chi^{(2)}_{21}=2d_{21}=-4.8\text{ pm V}^{-1}\]

In case (b), we have \(\mathbf{k}_{2\omega}\parallel\mathbf{k}_\omega\parallel\hat{x}\) and \(\hat{e}_\omega=\hat{z}\). We find from the nonvanishing elements of \(\boldsymbol{\chi}^{(2)}\) for LiNbO3 that \(\mathbf{P}^{(2)}(2\omega)\) has only one component in the \(z\) direction contributed by \(\chi^{(2)}_{33}\). Therefore, \(\hat{e}_{2\omega}=\hat{z}\) and

\[\chi_\text{eff}=\hat{e}^*_{2\omega}\cdot\boldsymbol{\chi}^{(2)}:\hat{e}_{\omega}\hat{e}_{\omega}=\hat{z}\cdot\boldsymbol{\chi}^{(2)}:\hat{z}\hat{z}=\chi^{(2)}_{33}=2d_{33}=-50.4\text{ pm V}^{-1}\]

We see from this example that the value of \(\chi_\text{eff}\) can vary significantly depending on the polarization directions of the interacting waves, which in turn are constrained by the wave propagation directions.


Note that though \(\mathbf{k}_1\), \(\mathbf{k}_2\), and \(\mathbf{k}_3\) individually may not be parallel to \(\hat{z}\), the phase mismatch \(\Delta\mathbf{k}\) has to be parallel to \(\hat{z}\) if the field amplitudes are to vary only along the \(z\) direction. This fact is required mathematically in (9-60) - (9-62) because \(\mathcal{E}_1\), \(\mathcal{E}_2\), and \(\mathcal{E}_3\) are all functions of \(z\) only.

Physically, the boundary conditions, which are dictated by Maxwell's equations, at the surface of a nonlinear crystal where the input waves enter the crystal require that the tangential component, but not the normal component, of \(\mathbf{k}_1+\mathbf{k}_2\) be equal to that of \(\mathbf{k}_3\) for an interaction defined by the relation \(\omega_3=\omega_1+\omega_2\).

Therefore, any phase mismatch \(\Delta\mathbf{k}\) occurs only in the direction normal to the input surface of the nonlinear crystal, as illustrated in Figure 9-8(a) below.  


Figure 9-8. (a) In a parametric interaction, the boundary conditions at the input surface of a nonlinear crystal require that the phase mismatch \(\Delta\mathbf{k}\), as well as the \(z\) direction, along which the field amplitudes vary, be normal to the input surface. (b) The wavefront, defined by the plane of constant phase, of each wave is normal to its wavevector, but the plane of constant field amplitude is parallel to the input surface and is normal to \(\hat{z}\). (c) Periodic variation of the intensity of a nonlinearly generated wave in the presence of a phase mismatch.


This condition can always be satisfied because only one or two of the interacting waves are provided at the input and only their wavevectors are initially given. For example, in sum-frequency generation, \(\mathbf{k}_1\) and \(\mathbf{k}_2\) are determined by the propagation directions of the input waves at \(\omega_1\) and \(\omega_2\), respectively. The propagation direction, \(\mathbf{k}_3\), of the generated sum-frequency wave is then determined by two conditions: (1) its magnitude, \(k_3=\omega_3n_3/c\), is determined by the dispersion and birefringence of the nonlinear crystal; (2) its projection on the crystal surface has to be equal to the projection of \(\mathbf{k}_1+\mathbf{k}_2\) on the crystal surface, as Figure 9-8(a) shows. Because \(\Delta\mathbf{k}=\Delta{k}\hat{z}\), the \(z\) direction, along which the field amplitudes vary, is normal to the input surface of the nonlinear crystal. Figure 9-8(b) shows the fact that though the Poynting vector of each wave may not line up with \(\hat{z}\), its magnitude varies only along the \(z\) direction.

The intensity of a wave at a frequency \(\omega_q\), projected on the plane of constant amplitude that is normal to the \(z\) direction, is given by


where \(\pmb{S}_q\) is the Poynting vector, \(n_{q,z}=ck_{q,z}/\omega_q=n_q\text{cos}\theta_q\), and \(\theta_q\) is the angle between \(\mathbf{k}_q\) and \(\hat{z}\). In a birefringent crystal, a possible walk-off between the vectors \(\pmb{S}_q\) and \(\mathbf{k}_q\) is neglected by taking the approximation in (9-63). We find that (9-60) - (9-62) lead to






\(\varphi_\chi\) is the phase of \(\chi_\text{eff}\) defined as \(\chi_\text{eff}=|\chi_\text{eff}|\text{e}^{\text{i}\varphi_\chi}\), and \(\varphi_1\), \(\varphi_2\), and \(\varphi_3\) are the phases of \(\mathcal{E}_1\), \(\mathcal{E}_2\), and \(\mathcal{E}_3\), respectively, defined as \(\mathcal{E}_q=|\mathcal{E}_q|\text{e}^{\text{i}\varphi_q}\).

The total intensity in the three interacting waves is \(\mathit{I}=\mathit{I}_1+\mathit{I}_2+\mathit{I}_3\). Using the relation \(\omega_3=\omega_1+\omega_2\), we find from (9-64) - (9-66) that


Consequently, the total optical energy is conserved in a parametric process, as is expected. In addition, we also find that


Therefore, every time a photon at \(\omega_3\) is annihilated, one photon at \(\omega_1\) and another at \(\omega_2\) are generated simultaneously, and vice versa. The relations in (9-68) and (9-69) are known as the Manley-Rowe relations.

The coupled equations and the Manley-Rowe relations formulated above apply to all parametric second-order interactions that involve three different optical frequencies. We see from (9-64) - (9-66) that optical energy is converted from \(\omega_3\) to \(\omega_1\) and \(\omega_2\) if \(\text{sin}\varphi\gt0\), whereas it is converted from \(\omega_1\) and \(\omega_2\) to \(\omega_3\) if \(\text{sin}\varphi\lt0\). If \(\varphi_\chi\), \(\varphi_1\), \(\varphi_2\), and \(\varphi_3\) are fixed or vary slowly with \(z\), as is normally the case, then \(\text{sin}\varphi\) changes sign periodically with \(z\) because of the phase mismatch \(\Delta{k}\). This periodic change of sign in \(\text{sin}\varphi\) results in periodic reversal of a parametric process. Therefore, in the presence of a phase mismatch, the maximum interaction length a given frequency-conversion process can take without a reversal of the process is the coherence length:


From the above discussions and an examination of (9-64) - (9-66), we can see that the intensity of a wave generated by a parametric nonlinear process in the presence of a finite phase mismatch varies periodically along the direction normal to the input surface of the nonlinear crystal with a half period of \(l_\text{coh}\), as illustrated in Figure 9-8(c) above. The intensities of other waves in the interaction also vary with the same period along the \(z\) direction. Therefore, an interaction length larger than \(l_\text{coh}\) is not useful and can even be detrimental. Clearly, phase matching is very important for an efficient parametric interaction.


Example 9-7

With a fundamental wave at \(\lambda\) = 1.10 μm, find the coherence length for each of the two cases of second-harmonic generation in LiNbO3 discussed in Example 9-6 above. At room temperature, LiNbO3 has \(n_\text{o}\) = 2.2319 and \(n_\text{e}\) = 2.1536 at 1.10 μm wavelength and \(n_\text{o}\) = 2.3168 and \(n_\text{e}\) = 2.2260 at 550 nm wavelength.

Because \(\mathbf{k}_{2\omega}\parallel\mathbf{k}_\omega\), we have \(\Delta{k}=2k_\omega-k_{2\omega}=4\pi(n_\omega-n_{2\omega})/\lambda\).

In case (a), we have \(n_\omega=n_\text{o}(\omega)=2.2319\) and \(n_{2\omega}=n_\text{o}(2\omega)=2.3168\) because \(\hat{e}_\omega=\hat{x}\) and \(\hat{e}_{2\omega}=\hat{y}\). Then

\[l_\text{coh}=\frac{\pi}{|\Delta{k}|}=\frac{\lambda}{4\times|n_\omega^\text{o}-n_{2\omega}^\text{o}|}=\frac{1.1}{4\times0.0849}\text{ }\mu\text{m}=3.24\text{ }\mu\text{m}\]

In case (b), we have \(n_\omega=n_\text{e}(\omega)=2.1536\) and \(n_{2\omega}=n_\text{e}(2\omega)=2.2260\) because \(\hat{e}_\omega=\hat{z}\) and \(\hat{e}_{2\omega}=\hat{z}\). Then

\[l_\text{coh}=\frac{\pi}{|\Delta{k}|}=\frac{\lambda}{4\times|n_\omega^\text{e}-n_{2\omega}^\text{e}|}=\frac{1.1}{4\times0.0724}\text{ }\mu\text{m}=3.80\text{ }\mu\text{m}\]

We see from this example that the coherence length is very small for both cases. Clearly, the interaction would not be efficient. The reason for this undesirable situation is that we have arbitrarily chosen in Example 9-6 some convenient propagation and polarization directions for the optical waves involved in the second-harmonic generation process without any consideration of the requirement for phase matching. These two examples together illustrate that it is possible to obtain a decent value of \(|\chi_\text{eff}|\), while the interaction is still very inefficient because of phase mismatch. Methods for phase matching are discussed in following tutorials. Phase-matched second-harmonic generation processes in LiNbO3 with properly chosen propagation and polarization directions of the optical waves are demonstrated in examples in following tutorials as well.


For a parametric interaction among linearly polarized waves in a homogeneous bulk crystal, \(\varphi_\chi=0\) or \(\pi\), depending on the sign of \(\chi_\text{eff}\). Then, \(\varphi=\varphi_1+\varphi_2-\varphi_3\) or \(\varphi=\pi+\varphi_1+\varphi_2-\varphi_3\) in the case of perfect phase matching. In this situation, it is possible for a frequency-conversion process to continue over the entire length of a crystal. Which parametric process occurs is determined completely by the value of \(\varphi\). For sum-frequency generation, we need \(\varphi=-\pi/2\) so that optical energy is converted most efficiently in the direction \(\omega_1+\omega_2\rightarrow\omega_3\). For difference-frequency generation, optical parametric amplification, or optical parametric generation, \(\varphi=\pi/2\) is needed to have the highest efficiency for the conversion of optical energy in the direction \(\omega_3\rightarrow\omega_1+\omega_2\).

In real experimental settings, a desired process is controlled by the input conditions. Normally only one or two waves in a parametric three-wave interaction are supplied at the input; therefore, only one or two phases are set, and at least one phase is arbitrary. Consider the situation where \(\chi_\text{eff}\) is real and positive so that \(\varphi_\chi=0\). If the input waves are at \(\omega_1\) and \(\omega_2\) and the phase-matching condition \(\mathbf{k}_3=\mathbf{k}_1+\mathbf{k}_2\) is satisfied, sum-frequency generation occurs with the generation of a wave at \(\omega_3\) that automatically picks a phase of \(\varphi_3=\varphi_1+\varphi_2+\pi/2\). If the same phase-matching condition is satisfied but the input waves are at \(\omega_3\) and \(\omega_2\), a wave at \(\omega_1\) is generated with a phase of \(\varphi_1=\varphi_3-\varphi_2+\pi/2\), resulting in difference-frequency generation, or optical parametric amplification in the case when the amplification of the signal at \(\omega_2\) is the objective. If only a wave at \(\omega_3\) is supplied at the input, optical parametric generation occurs with \(\varphi_1+\varphi_2=\varphi_3+\pi/2\). In this situation, the values of \(\omega_1\) and \(\omega_2\) are determined by the phase-matching condition subject to the condition that \(\omega_3=\omega_1+\omega_2\).

An interesting question is whether it is possible for other parametric processes, such as sum-frequency generation for \(\omega_1+\omega_3\) and difference-frequency generation for \(\omega_1-\omega_2\), and so on, to occur once all three waves at \(\omega_1\), \(\omega_2\), and \(\omega_3\) exist in a crystal, say, through a sum-frequency generation process of \(\omega_1+\omega_2\rightarrow\omega_3\). From the above discussions, it is clear that any parametric process can occur if it (1) has a nonvanishing \(\chi_\text{eff}\), (2) is phase matched, and (3) has the correct initial value of the phase \(\varphi\). It is thus possible to have simultaneous multiple parametric processes if all of them satisfy the required conditions. In normal situations, however, it is highly unusual for two or more different processes to occur in a single experimental arrangement because of the difficulty of satisfying their respective phase-matching conditions all at once.


Nonparametric interactions

When writing the coupled-wave equations for any nonlinear process, it is important to clearly understand the properties of the nonlinear susceptibility that characterizes the process under consideration first. For a parametric process, the full permutation symmetry is valid for the susceptibility; this fact is used in defining the effective susceptibility given in (9-59).

In general, the susceptibility for a nonparametric process does not have the full permutation symmetry because its imaginary parts is significant for the process.

The susceptibilities for different nonparametric processes generally have different permutation properties because they are related to different resonant transitions in the material.

For example, the susceptibility, \(\boldsymbol{\chi}^{(3)}(\omega_\text{S}+\omega_\text{p}-\omega_\text{p})\), for the Stokes process of stimulated Raman scattering and the susceptibility, \(\boldsymbol{\chi}^{(3)}(\omega_1=\omega_1+\omega_2-\omega_2)\), for two-photon absorption look the same, but they have different microscopic forms and thus very different properties because the former is resonant at \(\omega_\text{p}-\omega_\text{S}\) while the latter is resonant at \(\omega_1+\omega_2\).

If \(\omega_\text{p}\), \(\omega_\text{S}\), and \(\omega_\text{p}+\omega_\text{S}\) are all far from any resonant transition frequencies while \(\omega_\text{p}-\omega_\text{S}\) is in resonance with a transition in the material, the following property applies to the Raman susceptbility:


In contrast, the susceptibility for two-photon absorption has the following property:


if \(\omega_1\), \(\omega_2\), and \(|\omega_1-\omega_2|\) are all far from any resonant transition frequencies while \(\omega_1+\omega_2\) is in resonance with a transition.

The difference between the relations in (9-71) and (9-72) is significant because the imaginary parts of these susceptibilities are responsible for the nonlinear processes under consideration.

The coupled-wave equations for the process of stimulated Raman scattering are considered. Using (9-20) [refer to the optical nonlinearity tutorial] and the intrinsic permutation symmetry, we can write



Applying the relation in (9-71), we can define an effective Raman susceptibility:


The coupled-wave equations for stimulated Raman scattering are



By comparing these two equations to the three equations given in (9-60) - (9-62) for the parametric second-order interaction, we see clearly that the nonparametric process of stimulated Raman scattering is automatically phase matched, as is discussed in the preceding section.

The relation in (9-63) can be used to transform (9-76) and (9-77) into



We find that the total light intensity, \(\mathit{I}=\mathit{I}_\text{S}+\mathit{I}_\text{p}\), varies as


Therefore, optical energy is not conserved in the nonparametric Raman process because there is energy exchange with the material due to the resonant transition at the frequency \(\Omega=\omega_\text{p}-\omega_\text{S}\). Nevertheless, one Stokes photon is created for every pump photon that is annihilated. Therefore, in the absence of other loss mechanisms, we still have the following Manley-Rowe relation:


We see from (9-78) and (9-79) that the direction of energy flow in the Raman process is determined by the sign of \(\chi_\text{R}''\), which depends on the state of the material. If the material is in the ground state of the Raman transition, the imaginary part of \(\boldsymbol{\chi}^{(3)}(\omega_\text{S}=\omega_\text{S}+\omega_\text{p}-\omega_\text{p})\) is negative, resulting in \(\chi_\text{R}''\lt0\) according to (9-75). In this situation, energy is converted from the pump wave to the Stokes wave. We also see from (9-80) that in a Stokes process, there is a net loss in the total optical intensity. The energy corresponding to this loss is absorbed by the material in making the Stokes Raman transition from the ground state to the excited state. In case the excited state of the Raman transition is more populated than the ground state, the imaginary part of \(\boldsymbol{\chi}^{(3)}(\omega_\text{S}=\omega_\text{S}+\omega_\text{p}-\omega_\text{p})\) for \(\Omega=\omega_\text{p}-\omega_\text{S}\) becomes positive. Then \(\chi_\text{R}''\gt0\). In this situation, the anti-Stokes process occurs with the wave at \(\omega_\text{S}\) acting as the pump wave and the wave at \(\omega_\text{p}\) acting as the anti-Stokes wave. In the anti-Stokes process, the total optical intensity has a net gain corresponding to the energy released by the material in making the anti-Stokes Raman transition from the excited state to the ground state.

From the above discussions, we see that the characteristics of a nonparametric process are completely determined by the state of the material. Phase matching for the interacting optical waves is automatically satisfied in a nonparametric process because any difference in the momenta of the interacting photons can be absorbed by the material if there is energy exchange between the optical field and the medium. For the same reason, the phase relationship among the interacting waves, which determines the direction of frequency conversion in a parametric process, plays no role in a nonparametric process. 


The next part continues with the phase-matching for nonlinear optical processes tutorial.

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