# Optical Absorption and Amplification

This is a continuation from the previous tutorial - optical transitions for laser amplifiers.

Optical absorption results in attenuation of an optical field, while stimulated emission leads to amplification of an optical field.

To quantify the net effect of a resonant transition on the attenuation or amplification of an optical field, we consider the interaction of a monochromatic plane optical field at a frequency $$\nu$$ with a material that consists of electronic or atomic systems with population densities $$N_1$$ and $$N_2$$ in energy levels $$|1\rangle$$ and $$|2\rangle$$, respectively.

Because the spectral intensity distribution of the monochromatic plane optical field that has an intensity $$I$$ is simply $$I(\nu)=I\delta(\nu'-\nu)$$, the induced transition rates between energy levels $$|1\rangle$$ and $$|2\rangle$$ in this interaction are

$\tag{10-41}W_{21}=\frac{I}{h\nu}\sigma_\text{e}(\nu)\qquad\text{and}\qquad{W_{12}}=\frac{I}{h\nu}\sigma_\text{a}(\nu)$

The net power that is transferred from the optical field to the material is the difference between that absorbed by the material and that emitted due to stimulated emission:

\tag{10-42}\begin{align}\overline{W}_\text{p}&=h\nu{W}_{12}N_1-h\nu{W}_{21}N_2\\&=[N_1\sigma_\text{a}(\nu)-N_2\sigma_\text{e}(\nu)]I\end{align}

In the case when $$\overline{W}_\text{p}\gt0$$, there is net power absorption from the optical field by the medium due to resonant transitions between energy levels $$|1\rangle$$ and $$|2\rangle$$. The absorption coefficient is

$\tag{10-43}\alpha(\nu)=N_1\sigma_\text{a}(\nu)-N_2\sigma_\text{e}(\nu)=\left(N_1-\frac{g_1}{g_2}N_2\right)\sigma_\text{a}(\nu)$

In the case when $$\overline{W}_\text{p}\lt0$$, net power flows from the medium to the optical field, resulting in an amplification to the optical field with a gain coefficient given by

$\tag{10-44}g(\nu)=N_2\sigma_\text{e}(\nu)-N_1\sigma_\text{a}(\nu)=\left(N_2-\frac{g_2}{g_1}N_1\right)\sigma_\text{e}(\nu)$

The coefficients $$\alpha$$ and $$g$$ are quoted per meter, and are also often quoted per centimeter.

Note that $$\alpha(\nu)=\alpha(\omega)$$ and $$g(\nu)=g(\omega)$$ because $$\sigma(\nu)=\sigma(\omega)$$.

According to (10-41), both $$\sigma_\text{a}(\nu)$$ and $$\sigma_\text{e}(\nu)$$ are positive because $$W_{21}\ge0$$ and $$W_{12}\ge0$$ by definition. We then find that $$\alpha(\nu)\gt0$$ and $$g(\nu)\lt0$$ if $$N_1\gt(g_1/g_2)N_2$$, whereas $$g(\nu)\gt0$$ and $$\alpha(\nu)\lt0$$ if $$N_2\gt(g_2/g_1)N_1$$. Therefore, a material absorbs optical energy in its normal state of thermal equilibrium when the lower energy level is more populated than the upper energy level. In order to provide a net optical gain to the optical field, a material has to be in a nonequilibrium state of population inversion with the upper energy level more populated than the lower energy level.

Example 10-4

The upper and lower laser levels of the ruby laser are shown in Figure 10-6. The lower laser level $$|1\rangle$$ of the ruby laser is the ground state $$^4\text{A}_2$$, which has a degeneracy factor of $$g_1=4$$. The upper laser level $$|2\rangle$$ is the $$^2\text{E}$$ state, which consists of two closely spaced $$2\overline{\text{A}}$$ and $$\overline{\text{E}}$$ sublevels, each with a degeneracy factor of 2.

The 694.3 nm ruby laser transition takes place between the $$\overline{\text{E}}$$ sublevel, which has a degeneracy factor of $$g(\overline{\text{E}})=2$$, and the ground state $$^4\text{A}_2$$ with an emission cross section of $$\sigma_\text{e}^\text{line}=2.5\times10^{-24}\text{ m}^2$$ for the $$\mathbf{E}\perp{c}$$ polarization.

Find the peak value of the absorption cross section for an optical wave at 694.3 nm polarized with $$\mathbf{E}\perp{c}$$. At room temperature without pumping, what is the absorption coefficient at 694.3 nm of a ruby crystal that is doped with a Cr concentration of $$1.58\times10^{25}\text{ m}^{-3}$$?

To find $$\sigma_\text{a}$$, we use (10-37) [refer to the optical transitions for laser amplifiers tutorial] by taking $$\sigma_\text{e}=\sigma_\text{e}^\text{line}$$ for the $$\overline{\text{E}}$$ sublevel. Thus, we have

$\sigma_\text{a}=\frac{g(\overline{\text{E}})}{g_1}\sigma_\text{e}=\frac{2}{4}\times2.5\times10^{-24}\text{ m}^2=1.25\times10^{-24}\text{ m}^2$

At room temperature without pumping, the upper laser level is almost totally unpopulated because it is $$1.786\text{ eV}$$ above ground level. Virtually all of the Cr ions are in the ground level. Therefore, $$N_1=1.58\times10^{25}\text{ m}^{-3}$$ and $$N_2=0$$. Then, by using (10-43) we find that the absorption coefficient is

$\alpha=N_1\sigma_\text{a}=1.58\times10^{25}\times1.25\times10^{-24}\text{ m}^{-1}=19.75\text{ m}^{-1}$

Laser Level Splitting

In Example 10-4, we see that the upper laser level of the ruby laser consists of two closely spaced but clearly separate sublevels, corresponding to laser lines at 692.9 and 694.3 nm, respectively. The population $$N_2$$ in the upper laser level is split between these two sublevels. Such laser level splitting also occurs in most other lasers.

As a consequence, only a fractional population of $$\xi{N}_2$$ that resides in a particular sublevel of the upper laser level is directly responsible for a particular laser transition, whereas the remaining population of $$(1-\xi)N_2$$ that resides in other sublevels does not contribute to this transition.

While taking $$N_2$$ to be the total population of the upper laser level including all sublevels in a situation like this, the emission cross section $$\sigma_\text{e}$$ used in (10-43) and (10-44) is a cross section that is weighted as $$\sigma_\text{e}=\xi\sigma_\text{e}^\text{line}$$, where $$\sigma_\text{e}^\text{line}$$ is the emission cross section of the specific transition line for the population $$\xi{N_2}$$ in its sublevel.

The parameter $$\xi$$ of a given laser medium varies with many factors, including temperature, crystal quality, doping concentration, and the presence of codopants. This explains the variations in the measured values for the emission cross section of a laser medium, as seen in the values of $$\sigma_\text{e}$$ for ruby and Nd : YAG listed in Table (10-1) [refer to the optical transitions for laser amplifiers tutorial].

A similar effect also exists for the absorption cross section.

Example 10-5

For the ruby laser, the $$2\overline{\text{A}}$$ and $$\overline{\text{E}}$$ sublevels within upper laser level $$|2\rangle$$ of the $$^2\text{E}$$ state have an energy separation of $$\Delta{E}=29\text{ cm}^{-1}$$, which is $$\Delta{E}=3.6\text{ meV}$$ ($$1\text{ cm}^{-1}\equiv30\text{ GHz}\equiv124\text{ μeV})$$.

As discussed in Example 10-4, the laser transition between sublevel $$\overline{\text{E}}$$ and the ground state $$^4\text{A}_2$$ is the 694.3-nm line with $$\sigma_\text{e}^\text{line}=2.5\times10^{-24}\text{ m}^2$$ for $$\mathbf{E}\perp{c}$$ polarization.

The 692.9-nm transition between sublevel $$2\overline{\text{A}}$$ and the ground state $$^4\text{A}_2$$ has a similar cross section.

(a)  What is the population distribution at $$300\text{ K}$$ between the two sublevels in the upper laser level? What is the weighted emission cross section $$\sigma_\text{e}$$ for the 694.3-nm transition?

(b) What is $$\sigma_\text{e}$$ for the 694.3-nm transition at $$77\text{ K}$$?

(a)  For this question, we need to use (10-26) [refer to the optical transitions for laser amplifiers tutorial]. In thermal equilibrium at temperature $$T$$, the population ratio of the atoms in the upper and the lower levels follows the Boltzmann distribution.

At $$T=300\text{ K}$$, $$k_\text{B}T=25.9\text{ meV}$$. Because the $$2\overline{\text{A}}$$ state lies above the $$\overline{\text{E}}$$ state by an energy difference of $$\Delta{E}=3.6\text{ meV}$$ and the two states have the same degeneracy factor of 2, we have

$\frac{N(2\overline{\text{A}})}{N(\overline{\text{E}})}=\frac{g(2\overline{\text{A}})}{g(\overline{\text{E}})}\exp(-\Delta{E}/k_\text{B}T)=\frac{2}{2}\times\text{e}^{-3.6/25.9}=0.87$

Therefore, the fraction of the $$N_2$$ population in sublevel $$\overline{\text{E}}$$ is

$\xi=\frac{N(\overline{\text{E}})}{N(2\overline{\text{A}})+N(\overline{\text{E}})}=\frac{1}{0.87+1}=0.535$

This means that only $$53.5\%$$ of population $$N_2$$ in the upper laser level $$^2\text{E}$$ state contributes directly to the 694.3-nm transition. Thus the weighted emission cross section for the 694.3-nm transition is

$\sigma_\text{e}=\xi\sigma_\text{e}^\text{line}=0.535\times2.5\times10^{-24}\text{ m}^2=1.34\times10^{-24}\text{ m}^2$

(b) At $$T=77\text{ K}$$, $$k_\text{B}T=6.64\text{ meV}$$. Then

$\frac{N(2\overline{\text{A}})}{N(\overline{\text{E}})}=\frac{g(2\overline{\text{A}})}{g(\overline{\text{E}})}\exp(-\Delta{E}/k_\text{B}T)=\frac{2}{2}\times\text{e}^{-3.6/6.64}=0.58$

and $$\xi=1/(0.58+1)=0.632$$. Therefore, $$63.2\%$$ of population $$N_2$$ in the upper laser level now contributes to the 694.3-nm transition with a weighted emission cross section of

$\sigma_\text{e}=\xi\sigma_\text{e}^\text{line}=0.632\times2.5\times10^{-24}\text{ m}^2=1.58\times10^{-24}\text{ m}^2$

If the temperature is further lowered, $$\sigma_\text{e}$$ for the 694.3-nm transition will further increase toward its maximum value of $$2.5\times10^{-24}\text{ m}^2$$ as the $$\overline{\text{E}}$$ sublevel takes up a larger fraction of the total population in the upper laser level.

In many systems, the degenerate states in each laser level are split not into clearly separate sublevels but into very closely spaced sublevels that form a small quasi-continuous energy band, as shown in Figure 10-7.

In a molecular gas medium such as CO2, for example, transition levels $$|1\rangle$$ and $$|2\rangle$$ are defined by the vibrational states of the CO2 molecule, each of which consists of many closely spaced rotational sublevels.

In laser dyes, transition levels $$|1\rangle$$ and $$|2\rangle$$ are electronic states. Due to the vibrational and rotational motions of the dye molecules, the electronic states are split into vibrational sublevels, which are further split into finer structures of rotational sublevels.

In dielectric solid-state media doped with transition-metal or rare-earth ions, such as Ti : sapphire, Nd : glass, and Er-doped glass fiber, transition levels $$|1\rangle$$ and $$|2\rangle$$ are the electronic energy levels of the dopant ions. The degeneracies in such energy levels are contributed by the angular momentum states of the dopant ion. Because a dopant ion is embedded in a host solid-state medium, the electric fields of the neighboring atoms in the host medium cause some or all of its degenerate angular momentum states within a given energy level to split into a band of sublevels due to the Stark effect. Interaction with phonons in the lattice of the host medium can further broaden the energy band in each level.

Within the band of a transition level, the population at a higher sublevel can relax to a lower sublevel very quickly through nonradiative processes.

In CO2 and laser dyes, such relaxation takes place through collisions among the molecules. In a solid-state medium that is doped with transition-metal or rare-earth ions, relaxation takes place through interaction of the ions with the phonons, i.e., the lattice vibrations, of the host material.

These are thermal processes whose efficiency depends on temperature.

Because the sublevels within the band of a transition level are very closely spaced in energy, at room temperature relaxation among them takes place in a time much shorter than that between different transition levels.

As a result, before any optical transition begins, the sublevels within each transition level are generally thermalized to be in thermal equilibrium with the medium. This thermalization leads to a Boltzmann population distribution among the sublevels within the band of each transition level.

Within a band, the lower sublevels are more populated than the higher sublevels. Consequently, $$N_1$$ is not evenly distributed among the $$g_1$$ states of level $$|1\rangle$$, and $$N_2$$ is not evenly distributed among the $$g_2$$ states of level $$|2\rangle$$, as illustrated also in Figure 10-7.

Because of this nonuniform population distribution, absorption occurs with a higher probability from a lower sublevel in the band of level $$|1\rangle$$ to a higher sublevel in the band of level $$|2\rangle$$, whereas emission is more likely to take place from a low-lying sublevel in level $$|2\rangle$$ to a high-lying sublevel in level $$|1\rangle$$.

The consequences are:

1. The absorption and emission spectra associated with the same pair of transition levels $$|1\rangle$$ and $$|2\rangle$$ that consist of split sublevels are generally not identical. They have different shapes and widths, and both vary with temperature.

The absorption spectrum is generally shifted to the side of shorter wavelengths, corresponding to higher photon energies, with respect to the emission spectrum. As an example, Figure 10-8 shows the spectra of the absorption and emission cross sections of Ti : sapphire at room temperature.

2. The relation in (10-36) [refer to the optical transitions for laser amplifiers tutorial] is still valid but $$\hat{g}(\nu)$$ now represents the normalized emission spectral lineshape, which is different from the absorption lineshape. Therefore, the relation in (10-37) [refer to the optical transitions for laser amplifiers tutorial] is no longer valid. Instead, $$\sigma_\text{a}(\nu)$$ and $$\sigma_\text{e}(\nu)$$ satisfy the following general relation:

$\tag{10-45}\frac{1}{\tau_\text{sp}}=\frac{8\pi{n^2}}{c^2}\displaystyle\int\limits_0^\infty\nu^2\sigma_\text{e}(\nu)\text{d}\nu=\frac{8\pi{n^2}}{c^2}\frac{g_1}{g_2}\int\limits_0^\infty\nu^2\sigma_\text{a}(\nu)\text{d}\nu$

The validity of this relation is based on the assumption that all components in either of the two levels are equally populated or all transitions between the two levels have equal probability regardless of the components involved.

Because the experimentally measured spectra of emission and absorption cross sections are normally expressed as functions of wavelength rather than frequency, we can convert the relation in (10-45) into the following useful relation in terms of wavelength:

$\tag{10-46}\frac{1}{\tau_\text{sp}}=8\pi{n^2}c\displaystyle\int\limits_0^\infty\frac{\sigma_\text{e}(\lambda)}{\lambda^4}\text{d}\lambda=8\pi{n^2}c\frac{g_1}{g_2}\int\limits_0^\infty\frac{\sigma_\text{a}(\lambda)}{\lambda^4}\text{d}\lambda$

This relation can be used to determine the spontaneous lifetime $$\tau_\text{sp}$$ by integrating the experimentally measured absorption or emission cross section as a function of wavelength.

Note that $$\sigma(\lambda)=\sigma(\nu)$$ for $$\lambda=c/\nu$$.

3. A detailed relation between $$\sigma_\text{e}(\nu)$$ and $$\sigma_\text{a}(\nu)$$ is known as the McCumber relation:

$\tag{10-47}\sigma_\text{e}(\nu)=\sigma_\text{a}(\nu)\exp\left(\frac{h\nu_\text{c}-h\nu}{k_\text{B}T}\right)$

where $$\nu_\text{c}$$ is the optical frequency at which the absorption and emission cross sections are equal: $$\sigma_\text{e}(\nu_\text{c})=\sigma_\text{a}(\nu_\text{c})$$.

In terms of wavelength, the McCumber relation can be expressed as

$\tag{10-48}\sigma_\text{e}(\lambda)=\sigma_\text{a}(\lambda)\exp\left[\frac{hc}{k_\text{B}T}\left(\frac{1}{\lambda_\text{c}}-\frac{1}{\lambda}\right)\right]$

where $$\lambda_\text{c}=c/\nu_\text{c}$$ for which $$\sigma_\text{e}(\lambda_\text{c})=\sigma_\text{a}(\lambda_\text{c})$$.

The photon energy $$h\nu_\text{c}$$ corresponds to the temperature-dependent excitation energy that is equivalent to the free energy required to move one atom from the lower level $$|1\rangle$$ to the upper level $$|2\rangle$$.

According to (10-47), the spectra of $$\sigma_\text{e}(\nu)$$ and $$\sigma_\text{a}(\nu)$$ associated with the transition between two energy levels $$|1\rangle$$ and $$|2\rangle$$ cross at only one frequency $$\nu_\text{c}$$.

The McCumber relation is generally applicable because it does not depend on the assumption that is required for the validity of (10-45). The only assumption needed is that the sublevels within either level $$|1\rangle$$ and $$|2\rangle$$ reach thermal equilibrium in a time shorter than the lifetime of each energy level.

4. The first parts of the relations in (10-43) and (10-44) for the absorption and emission coefficients, respectively, are still valid, but not the second parts:

$\tag{10-49}\alpha(\nu)=N_1\sigma_\text{a}(\nu)-N_2\sigma_\text{e}(\nu)\ne\left(N_1-\frac{g_1}{g_2}N_2\right)\sigma_\text{a}(\nu)$

and

$\tag{10-50}g(\nu)=N_2\sigma_\text{e}(\nu)-N_1\sigma_\text{a}(\nu)\ne\left(N_2-\frac{g_2}{g_1}N_1\right)\sigma_\text{e}(\nu)$

Example 10-6

Find the spontaneous lifetime $$\tau_\text{sp}$$ for the laser transition of Ti : sapphire from the spectrum of its emission cross section shown in Figure 10-8.

According to (10-46), $$\tau_\text{sp}$$ can be found from $$\sigma_\text{e}(\lambda)$$ by integrating $$\sigma_\text{e}(\lambda)/\lambda^4$$ over the entire spectrum.

In applying (10-46) to the spectra shown in Figure 10-8, however, we have to account for the difference between the emission spectra for different polarizations.

With respect to the unique crystal axis $$c$$ of the uniaxial sapphire, there are three polarization modes, one parallel to $$c$$ and two perpendicular to $$c$$. As a consequence, the spontaneous emission resulting from the radiative transition that defines $$\tau_\text{sp}$$ for Ti : sapphire has a 1 : 2 ratio between the $$\mathbf{E}\parallel{c}$$ and $$\mathbf{E}\perp{c}$$ polarized emissions.

Therefore, for a uniaxial crystal such as Ti : sapphire, (10-46) has to be modified as

$\tag{10-51}\displaystyle\int\limits_0^\infty\frac{\sigma_\text{e}^\parallel(\lambda)+2\sigma_\text{e}^\perp(\lambda)}{3\lambda^4}\text{d}\lambda=\frac{g_1}{g_2}\int\limits_0^\infty\frac{\sigma_\text{a}^\parallel(\lambda)+2\sigma_\text{a}^\perp(\lambda)}{3\lambda^4}\text{d}\lambda=\frac{1}{8\pi{n^2}c\tau_\text{sp}}$

Using the $$\sigma_\text{e}^\parallel(\lambda)$$ and $$\sigma_\text{e}^\perp(\lambda)$$ spectra shown in Figure 10-8, we find that

$\displaystyle\int\limits_0^\infty\frac{\sigma_\text{e}^\parallel(\lambda)+2\sigma_\text{e}^\perp(\lambda)}{3\lambda^4}\text{d}\lambda=1.1\times10^{-5}\text{ m}^{-1}$

Using (10-51) and $$n=1.76$$ for Ti : sapphire, we find that

$\tau_\text{sp}=\frac{1}{8\pi\times1.76^2\times3\times10^8\times1.1\times10^{-5}}\text{ s}=3.89\text{ μs}$

Resonant Optical Susceptibility

The macroscopic optical properties of a medium are characterized by its electric susceptibility. As seen in the material dispersion tutorial, resonances in a medium contribute to the dispersion in the susceptibility of the medium. Clearly, the optical properties of a material are functions of the resonant optical transitions between the energy levels of the electrons in the material.

From the viewpoint of the macroscopic optical properties of a medium, interaction between an optical field and a medium is characterized by the polarization induced by the optical field in the medium.

The power exchange between the optical field and the medium is given by (30) [refer to the optical fields and Maxwell's equations tutorial].

For resonant interaction of an isotropic medium with a monochromatic plane optical field at a frequency $$\omega=2\pi\nu$$, we have

$\pmb{E}(t)=\mathbf{E}\text{e}^{-\text{i}\omega{t}}+\mathbf{E}^*\text{e}^{\text{i}\omega{t}}$

and

$\pmb{P}_\text{res}(t)=\epsilon_0(\chi_\text{res}(\omega)\mathbf{E}\text{e}^{-\text{i}\omega{t}}+\chi_\text{res}^*(\omega)\mathbf{E}^*\text{e}^{\text{i}\omega{t}})$

where $$\pmb{P}_\text{res}$$ is the polarization contributed by the resonant transitions and $$\chi_\text{res}$$ is the resonant susceptibility.

Using (30) [refer to the optical fields and Maxwell's equations tutorial], we find that the time-averaged power density absorbed by the medium is

$\tag{10-52}\overline{W}_\text{p}=2\omega\epsilon_0\chi_\text{res}^"(\omega)|\mathbf{E}|^2=\frac{\omega}{nc}\chi_\text{res}^"(\omega)I$

By identifying (10-52) with (10-42), we find that the imaginary part of the susceptibility contributed by the resonant transitions between energy levels $$|1\rangle$$ and $$|2\rangle$$ is

$\tag{10-53}\chi_\text{res}^"(\omega)=\frac{nc}{\omega}[N_1\sigma_\text{a}(\omega)-N_2\sigma_\text{e}(\omega)]$

The real part $$\chi_\text{res}'(\omega)$$ of the resonant susceptibility can be found through the Kramers-Kronig relations given in (177) [refer to the material dispersion tutorial].

As discussed in the propagation in an isotropic medium tutorial and the material dispersion tutorial, a medium has an optical loss if $$\chi^"\gt0$$, and it has an optical gain if $$\chi^"\lt0$$.

It is also clear from (10-52) that there is a net power loss from the optical field to the medium if $$\chi_\text{res}^"\gt0$$, but there is a net power gain for the optical field if $$\chi_\text{res}^"\lt0$$.

By comparing (10-53) with (10-43) and (10-44), we find that the medium has an absorption coefficient given by

$\tag{10-54}\alpha(\omega)=\frac{\omega}{nc}\chi_\text{res}^"(\omega)$

in the case of normal population distribution when $$\chi_\text{res}^"\gt0$$, whereas it has a gain coefficient given by

$\tag{10-55}g(\omega)=-\frac{\omega}{nc}\chi_\text{res}^"(\omega)$

in the case of population inversion when $$\chi_\text{res}^"\lt0$$.

Note that the material susceptibility characterizes the response of a material to the excitation of an electromagnetic field. Therefore, the resonant susceptibility $$\chi_\text{res}$$ accounts for only the contributions from the induced processes of absorption and stimulated emission, but not that from the process of spontaneous emission.

The resonant susceptibility contributed by the induced transitions between two energy levels is proportional to the population difference between the two levels, but the power density of the optical radiation due to spontaneous emission is a function of the population density in the upper energy level alone.

By taking $$\Delta\mathbf{P}=\mathbf{P}_\text{res}$$, the behavior of an optical field propagating in the presence of resonant transitions can be formulated with the coupled-wave theory discussed in the coupled-wave theory tutorial, if the medium is spatially homogeneous, or with the coupled-mode theory discussed in the coupled-mode theory tutorial, if the medium has waveguiding structures.

Here we consider the simplest situation involving a monochromatic wave at a frequency $$\omega$$ that propagates along the $$z$$ direction in a spatially homogeneous, isotropic medium with a resonant susceptibility $$\chi_\text{res}$$.

Then, the index $$q$$ in the coupled-wave equation expressed in (13) [refer to the coupled-wave theory tutorial] can be dropped:

$\tag{10-56}\frac{\text{d}\boldsymbol{\mathcal{E}}(z)}{\text{d}z}=\frac{\text{i}\omega^2\mu_0}{2k}\mathbf{P}_\text{res}(z)\text{e}^{-\text{i}kz}$

where

$\tag{10-57}\mathbf{P}_\text{res}(z)=\epsilon_0\chi_\text{res}(\omega)\mathbf{E}(z)=\epsilon_0(\chi_\text{res}'+\text{i}\chi_\text{res}^")\boldsymbol{\mathcal{E}}(z)\text{e}^{\text{i}kz}$

Substitution of (10-57) in (10-56) yields

$\tag{10-58}\frac{\text{d}\boldsymbol{\mathcal{E}}}{\text{d}z}=\text{i}\frac{\omega}{2nc}\chi_\text{res}'\boldsymbol{\mathcal{E}}-\frac{\omega}{2nc}\chi_\text{res}^"\boldsymbol{\mathcal{E}}$

We see from this equation that, as the optical field propagates, not only is its amplitude varied by the resonant susceptibility, but its phase is modified as well.

When the phase information of the optical wave is of no interest, we can take $$\boldsymbol{\mathcal{E}}^*\cdot(10-58)+\text{c.c.}$$ to find the evolution of the intensity of the optical wave as it propagates through the medium.

Using the relations in (10-54) and (10-55), we find that

$\tag{10-59}\frac{\text{d}I}{\text{d}z}=-\alpha{I}$

in the case of optical attenuation when $$\chi_\text{res}^"\gt0$$, and

$\tag{10-60}\frac{\text{d}I}{\text{d}z}=gI$

in the case of optical amplification when $$\chi_\text{res}^"\lt0$$.

Clearly, the coefficients $$\alpha$$ and $$g$$ respectively characterize the attenuation and growth of the optical intensity per unit length traveled by the optical wave in a medium.

The next tutorial covers population inversion and optical gain