Calculating Stimulated-Emission Cross Section from Line Shape and Transition Parameters

Good to see you again. Last time, we separated homogeneous from inhomogeneous broadening and introduced the normalized line-shape function $g(\nu )$. The key result was that a transition has finite spectral width: an atom or ion does not interact equally strongly with light at every frequency.

This lesson turns that spectral profile into a quantitative transition parameter: the stimulated-emission cross section ${\sigma }_e(\nu )$. It is the bridge from microscopic radiative data—lifetime, spontaneous-emission rate, central wavelength, and linewidth—to the gain a laser medium can provide. In the next module, it will appear directly in rate equations and gain coefficients.

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The cross section: an effective interaction area

For a beam at frequency $\nu$, the photon flux is

$$\Phi (\nu )=\frac{I(\nu )}{h\nu },$$

where $I$ is intensity in $\mathrm{W}{\mathrm{m}}^{-2}$, and $\Phi$ has units of photons per unit area per unit time.

The stimulated-emission cross section is defined so that the stimulated-emission rate per excited emitter is

$$R_{\mathrm{st}}(\nu )={\sigma }_e(\nu )\Phi (\nu ).$$

Thus, ${\sigma }_e$ has units of area, usually ${\mathrm{m}}^2$ or $\mathrm{c}{\mathrm{m}}^2$. It is not literally the geometrical area of an ion or atom. Rather, it is an effective optical area: a large cross section means that a given photon flux is efficient at inducing stimulated emission.

Two points follow immediately:

1. The cross section is frequency dependent.
   It is largest near a strong part of the transition spectrum and small far from resonance.

2. The cross section is a property of the transition, not the inversion.
   Population inversion determines whether the medium produces net gain. The cross section determines how strongly an inverted emitter couples to light at a particular frequency.

For the two levels used throughout this course, the small-signal propagation law can be written

$$\frac{\partial I(\nu ,z)}{\partial z}=\left[N_2-\frac{g_2}{g_1}{N}_1\right]{\sigma }_e(\nu )I(\nu ,z).$$

The bracketed quantity is the degeneracy-corrected inversion density from the earlier lesson. The product of that density and the cross section is the frequency-dependent gain coefficient, which we will use extensively in Module 2.

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From Einstein coefficients to cross section

The Einstein $B_{21}$ coefficient describes the stimulated-emission probability in a radiation field. Once broadening is included, that interaction probability must be weighted by the normalized line shape.

The line shape obeys

$${\int }_{-\infty }^{\infty }{g}_{\nu }(\nu )d\nu =1.$$

The subscript is important: $g_{\nu }$ is normalized with respect to ordinary frequency $\nu$ in hertz. Therefore,

$$[g_{\nu }]=\mathrm{H}{\mathrm{z}}^{-1}=\mathrm{s}.$$

For a homogeneously broadened transition, equating the energy added through stimulated emission with the beam’s intensity increase gives the vacuum form

$${\sigma }_e(\omega )=\frac{{\pi }^2{c}^2}{{\omega }_0^2}{A}_{21}{g}_{\omega }(\omega ),$$

where:

• $A_{21}$ is the spontaneous-emission Einstein coefficient for the particular transition,
• ${\omega }_0$ is the center angular frequency,
• $g_{\omega }(\omega )$ is normalized over angular frequency.

This is often more convenient in ordinary frequency. Since

$$\omega =2\pi \nu ,$$

and the probability in corresponding frequency intervals must be the same,

$$g_{\omega }(\omega )=\frac{g_{\nu }(\nu )}{2\pi }.$$

Using ${\lambda }_0=c/{\nu }_0$, the vacuum result becomes

$$\boxed{{\sigma }_e(\nu )=\frac{{\lambda }_0^2}{8\pi }{A}_{21}{g}_{\nu }(\nu )}$$

For a bulk dielectric gain medium, the common Fuchtbauer-Ladenburg convention includes the refractive index $n$:

$$\boxed{{\sigma }_e(\nu )=\frac{{\lambda }_0^2}{8\pi n^2}{A}_{21}{g}_{\nu }(\nu )}$$

or, if the radiative lifetime for the relevant transition is ${\tau }_{21}=1/A_{21}$,

$$\boxed{{\sigma }_e(\nu )=\frac{{\lambda }_0^2}{8\pi n^2{\tau }_{21}}{g}_{\nu }(\nu ).}$$

Use $n=1$ for a vacuum or dilute-gas treatment unless a different convention is specified.

The following reading gives the energy-balance derivation and shows how the Einstein relations produce the cross section.

A useful physical interpretation of the result is

$${\sigma }_e(\nu )\propto {\lambda }_0^2{A}_{21}{g}_{\nu }(\nu ).$$

The cross section is large when the transition has:

• a long wavelength, through the factor ${\lambda }_0^2$;
• a large radiative transition rate $A_{21}$;
• a narrow spectral distribution, because concentrating a fixed total radiative strength into a smaller frequency interval raises $g_{\nu }$ near line center.

The final point is particularly important. A narrower line generally has a larger peak cross section, even if the integrated transition strength is unchanged.

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Line shape determines the spectral cross-section profile

Because

$${\sigma }_e(\nu )=C{g}_{\nu }(\nu ),$$

where $C$ is independent of frequency across a narrow optical line, the cross section has exactly the same frequency dependence as the line shape.

For a Lorentzian line of FWHM $\Delta {\nu }_{\mathrm{L}}$,

$$g_{\mathrm{L}}(\nu )=\frac{1}{\pi }\frac{\Delta {\nu }_{\mathrm{L}}/2}{(\nu -{\nu }_0{)}^2+{\left(\Delta {\nu }_{\mathrm{L}}/2\right)}^2}.$$

At line center,

$$g_{\mathrm{L}}({\nu }_0)=\frac{2}{\pi \Delta {\nu }_{\mathrm{L}}},$$

so the peak stimulated-emission cross section is

$$\boxed{{\sigma }_{e,\mathrm{peak}}^{\mathrm{L}}=\frac{{\lambda }_0^2}{4{\pi }^2{n}^2{\tau }_{21}\Delta {\nu }_{\mathrm{L}}}.}$$

For a Gaussian line of FWHM $\Delta {\nu }_{\mathrm{G}}$,

$$g_{\mathrm{G}}(\nu )=\frac{2\sqrt{\ln 2}}{\Delta {\nu }_{\mathrm{G}}\sqrt{\pi }}\exp \left[-4\ln 2{\left(\frac{\nu -{\nu }_0}{\Delta {\nu }_{\mathrm{G}}}\right)}^2\right].$$

Its peak value is

$$g_{\mathrm{G}}({\nu }_0)=\frac{2\sqrt{\ln 2}}{\Delta {\nu }_{\mathrm{G}}\sqrt{\pi }},$$

giving

$$\boxed{{\sigma }_{e,\mathrm{peak}}^{\mathrm{G}}=\frac{{\lambda }_0^2\sqrt{\ln 2}}{4{\pi }^{3/2}{n}^2{\tau }_{21}\Delta {\nu }_{\mathrm{G}}}.}$$

The Gaussian peak is about $1.48$ times the Lorentzian peak when both profiles have the same FWHM and the same integrated transition strength. The Lorentzian, however, retains much stronger far-off-resonance wings.

A compact comparison is useful:

Line-shape model	Typical broadening picture	Peak line shape	Peak cross-section scaling
Lorentzian	Homogeneous dephasing, lifetime or collision broadening	$\frac{2}{\pi \Delta \nu }$	${\sigma }_{\mathrm{peak}}\propto 1/\Delta \nu$
Gaussian	Doppler or static inhomogeneous broadening	$\frac{2\sqrt{\ln 2}}{\sqrt{\pi }\Delta \nu }$	${\sigma }_{\mathrm{peak}}\propto 1/\Delta \nu$
Voigt	Mixed Gaussian and Lorentzian broadening	Numerical evaluation usually required	Depends on both widths

The line shape is therefore not cosmetic fitting information. It specifies which frequency components of a laser field can efficiently extract stored excitation.

For a concise visual review of why the line shape multiplies all Einstein interaction rates, watch this segment.

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A reliable calculation procedure

When given a transition wavelength, refractive index, radiative lifetime, and normalized line shape, proceed in this order.

1. Choose the spectral variable.
   Determine whether the supplied line shape is normalized in $\nu$, $\omega$, or $\lambda$. Use a formula written for that same variable.

2. Check normalization and units.
   For example,

   $$\int g_{\nu }(\nu )d\nu =1.$$

   A line shape in $\mathrm{H}{\mathrm{z}}^{-1}$ cannot be inserted directly into a formula requiring a line shape in $\mathrm{ra}{\mathrm{d}}^{-1}\mathrm{s}$, or vice versa.

3. Convert the linewidth if necessary.
   Near a central wavelength ${\lambda }_0$, a narrow wavelength bandwidth $\Delta \lambda$ corresponds approximately to

   $$\Delta \nu \simeq \frac{c\Delta \lambda }{{\lambda }_0^2}.$$

4. Evaluate $g_{\nu }$ at the requested frequency.
   At line center, use the peak expression for the relevant line shape. At finite detuning, substitute directly into the Lorentzian or Gaussian formula.

5. Use the radiative rate for the specified transition.
   If a branching fraction ${\beta }_{21}$ of the total radiative decay enters the laser transition, then

   $$A_{21}=\frac{{\beta }_{21}}{{\tau }_{\mathrm{rad}}}.$$

   Do not automatically use an observed total lifetime if nonradiative decay is significant; that lifetime does not by itself equal $1/A_{21}$.

6. Report sensible units.
   Solid-state laser cross sections are commonly reported in $\mathrm{c}{\mathrm{m}}^2$, with

   $$1{\mathrm{m}}^2=10^4\mathrm{c}{\mathrm{m}}^2.$$

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Worked example: a Lorentzian solid-state transition

Consider a laser material with:

$${\lambda }_0=1030\mathrm{nm},$$ $$n=1.85,$$ $${\tau }_{\mathrm{rad}}=950\mu \mathrm{s},$$

and a single Lorentzian emission line of wavelength FWHM

$$\Delta \lambda =5.0\mathrm{nm}.$$

Assume all radiative decay belongs to the transition, so that

$$A_{21}=\frac{1}{{\tau }_{\mathrm{rad}}}.$$

1. Convert the linewidth

Using the narrowband conversion,

$$\Delta {\nu }_{\mathrm{L}}\simeq \frac{c\Delta \lambda }{{\lambda }_0^2},$$

we obtain

$$\Delta {\nu }_{\mathrm{L}}\simeq \frac{(3.00\times 10^8)(5.0\times 10^{-9})}{(1.030\times 10^{-6}{)}^2}.$$

Thus,

$$\Delta {\nu }_{\mathrm{L}}\simeq 1.41\times 10^{12}\mathrm{Hz}.$$

2. Evaluate the normalized Lorentzian at line center

$$g_{\mathrm{L}}({\nu }_0)=\frac{2}{\pi \Delta {\nu }_{\mathrm{L}}}=\frac{2}{\pi (1.41\times 10^{12})}.$$

Therefore,

$$g_{\mathrm{L}}({\nu }_0)\simeq 4.52\times 10^{-13}\mathrm{H}{\mathrm{z}}^{-1}.$$

3. Calculate the peak cross section

Using

$${\sigma }_{e,\mathrm{peak}}=\frac{{\lambda }_0^2}{8\pi n^2{\tau }_{\mathrm{rad}}}{g}_{\mathrm{L}}({\nu }_0),$$

we find

$${\sigma }_{e,\mathrm{peak}}=\frac{(1.030\times 10^{-6}{)}^2}{8\pi (1.85{)}^2(950\times 10^{-6})}(4.52\times 10^{-13}).$$

Hence,

$${\sigma }_{e,\mathrm{peak}}\simeq 5.9\times 10^{-24}{\mathrm{m}}^2.$$

Converting units,

$$\boxed{{\sigma }_{e,\mathrm{peak}}\simeq 5.9\times 10^{-20}\mathrm{c}{\mathrm{m}}^2.}$$

That is a plausible magnitude for a rare-earth solid-state gain transition. For perspective, a typical quoted peak emission cross section for Nd:YAG near $1064\mathrm{nm}$ is on the order of $10^{-19}\mathrm{c}{\mathrm{m}}^2$, with the precise value depending on polarization, temperature, host, and data convention.

At a detuning of one half-width,

$$\left|\nu -{\nu }_0\right|=\frac{\Delta {\nu }_{\mathrm{L}}}{2},$$

the Lorentzian denominator doubles. Therefore,

$${\sigma }_e=\frac{1}{2}{\sigma }_{e,\mathrm{peak}}.$$

This is simply the definition of the FWHM expressed as a cross-section statement.

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The wavelength-spectrum trap

Experimental fluorescence data are often plotted as a function of wavelength, but the fundamental cross-section relation above is written using a line shape normalized in frequency. These are not interchangeable without a Jacobian.

Probability conservation requires

$$g_{\nu }(\nu )d\nu =g_{\lambda }(\lambda )\left|d\lambda \right|.$$

Since

$$\nu =\frac{c}{\lambda },$$

we have

$$\left|\frac{d\lambda }{d\nu }\right|=\frac{{\lambda }^2}{c}.$$

Thus,

$$\boxed{g_{\nu }(\nu )=g_{\lambda }(\lambda )\frac{{\lambda }^2}{c}.}$$

This matters most for broad spectra. A curve that looks symmetric in wavelength is not exactly symmetric in frequency, because wavelength and frequency have a nonlinear relationship.

If a measured fluorescence spectrum $S_{\lambda }(\lambda )$ is a spectral power distribution, the frequency-normalized line shape can be constructed as

$$g_{\nu }(\nu )=\frac{{\lambda }^3{S}_{\lambda }(\lambda )}{c\int \lambda S_{\lambda }(\lambda )d\lambda }.$$

Substituting this into the bulk-medium cross-section expression gives the frequently used wavelength-domain Fuchtbauer-Ladenburg form:

$$\boxed{{\sigma }_e(\lambda )=\frac{{\lambda }^5}{8\pi n^{2}c{\tau }_{\mathrm{rad}}}\frac{S_{\lambda }(\lambda )}{\int \lambda S_{\lambda }(\lambda )d\lambda }.}$$

For a narrow line, it is often simpler and sufficiently accurate to convert its FWHM to frequency and use the Lorentzian or Gaussian peak formula. For a broad rare-earth fluorescence band, use the fully wavelength-aware expression instead.

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Takeaways

The stimulated-emission cross section converts photon flux into a stimulated-emission rate:

$$R_{\mathrm{st}}={\sigma }_e\Phi .$$

Its spectral dependence is determined entirely by the normalized line shape:

$${\sigma }_e(\nu )\propto g_{\nu }(\nu ).$$

For a transition in a dielectric medium, the principal working formula is

$$\boxed{{\sigma }_e(\nu )=\frac{{\lambda }_0^2}{8\pi n^2}{A}_{21}{g}_{\nu }(\nu )=\frac{{\lambda }_0^2}{8\pi n^2{\tau }_{21}}{g}_{\nu }(\nu ).}$$

A narrower transition concentrates its fixed spectral strength and therefore has a larger peak cross section. Lorentzian and Gaussian profiles both give peak cross sections proportional to inverse linewidth, but they predict different behavior away from line center.

Finally, always match the line-shape variable to the formula: a profile normalized per hertz, per radian per second, and per metre of wavelength carries different numerical values.

Next, we will relate linewidth to coherence time and coherence length, completing the connection between a transition’s spectral profile and the temporal persistence of its optical phase.