# Laser Oscillation

This is a continuation from the precious tutorial - **resonant optical cavities**.

In the previous tutorial - resonant optical cavities, it is mentioned that a practical laser device can be constructed by placing an optical gain medium inside an optical resonator. The gain medium provides amplification to the intracavity optical field while the resonator provides optical feedback.

A laser is basically a coherent optical oscillator, and the basic function of an oscillator is to generate a coherent signal through resonant oscillation without an input signal.

Therefore, no external optical field is injected into the optical cavity for laser oscillation. The intracavity optical field has to grow from the field generated by spontaneous emission from the intracavity gain medium.

When steady-state oscillation is reached, the coherent laser field at any given location inside the cavity should become a constant with time in both phase and magnitude. In the model shown in Figure 11-2 [refer to the resonant optical cavities tutorial], the situation of steady-state laser oscillation requires that \(\mathbf{E}_\text{in}=0\) and \(\mathbf{E}_\text{c}=\text{constant}\ne0\). Therefore, from (11-6) [refer to the resonant optical cavities tutorial], the ** condition for steady-state laser oscillation** is

\[\tag{11-48}a=G\exp(\text{i}\varphi_\text{RT})=1\]

To illustrate the implications of this condition, we consider in the following the simple Fabry-Perot laser shown in Figure 11-5 [refer to the resonant optical cavities tutorial] that contains an isotropic gain medium with a filling factor \(\Gamma\).

The total permittivity of the gain medium, including the contribution of the resonant laser transition, is \(\epsilon_\text{res}=\epsilon+\epsilon_0\chi_\text{res}\) given in (11-37) [refer to the resonant optical cavities tutorial].

Therefore, the total complex propagation constant of the gain medium including the contribution of the resonant transition is

\[\tag{11-49}\begin{align}k_\text{t}&=\omega\mu_0^{1/2}(\epsilon+\epsilon_0\chi_\text{res})^{1/2}\\&=k+\Delta{k}_\text{res}-\text{i}\frac{g}{2}\end{align}\]

where

\[\tag{11-50}\Delta{k}_\text{res}\approx{k}\frac{\chi_\text{res}'}{2n^2}=\frac{\omega}{2nc}\chi_\text{res}'\]

\[\tag{11-51}g\approx-k\frac{\chi_\text{res}^"}{n^2}=-\frac{\omega}{nc}\chi_\text{res}^"\]

Here \(g\) is the gain coefficient of the laser medium associated with the laser transition identified in (10-55) [refer to the optical absorption and amplification tutorial], and \(\Delta{k}_\text{res}\) is the corresponding change in the optical wavenumber in the medium caused by the change in the refractive index associated with population changes in the resonant laser levels.

As discussed in the optical absorption and amplification tutorial, when population inversion is achieved, \(\chi_\text{res}^"\lt0\) so that the gain coefficient \(g\) has a positive value.

By replacing \(k\) for a cold medium with \(k_\text{t}\) for a pumped gain medium, we find that \(\bar{k}\) in (11-39) [refer to the resonant optical cavities tutorial] has to be replaced with \(\bar{k}+\Gamma\Delta{k}_\text{res}-\text{i}\Gamma{g/2}\) when an actively pumped laser cavity is considered.

We then find the mode-dependent round-trip gain factor

\[\tag{11-52}G_{mn}=R_1^{1/2}R_2^{1/2}\exp[(\Gamma{g}-\bar{\alpha}_{mn})l]\]

and mode-dependent round-trip phase shift

\[\tag{11-53}\varphi_{mn}^\text{RT}=2(\bar{k}+\Gamma\Delta{k}_\text{res})l+\zeta_{mn}^\text{RT}+\varphi_1+\varphi_2\]

Because both \(G_{mn}\) and \(\varphi_{mn}^\text{RT}\) are real parameters, the condition in (11-48) can be satisfied for a given laser mode to oscillate only if the *gain condition*

\[\tag{11-54}G_{mn}=1\]

and the *phase condition*

\[\tag{11-55}\varphi_{mn}^{RT}=2q\pi,\qquad{q=1,2,\ldots}\]

are simultaneously fulfilled.

Note that both \(G_{mn}\) and \(\varphi_{mn}^\text{RT}\) are frequency dependent.

**Laser Threshold**

The condition in (11-54) implies that there are a threshold gain and a corresponding threshold pumping level for laser oscillation.

For the Fabry-Perot laser shown in Figure 11-5 [refer to the resonant optical cavities tutorial], which has a length \(l\) and contains a gain medium of a length \(l_\text{g}\) for a filling factor of \(\Gamma=l_\text{g}/l\), the ** threshold gain coefficient**, \(g_{mn}^\text{th}\), for the \(\text{TEM}_{mn}\) mode can be found from

\[\tag{11-56}\Gamma{g}_{mn}^\text{th}=\bar{\alpha}_{mn}-\frac{1}{l}\ln\sqrt{R_1R_2}\]

or

\[\tag{11-57}{g}_{mn}^\text{th}l_\text{g}=\bar{\alpha}_{mn}l-\ln\sqrt{R_1R_2}\]

Because the distributed loss \(\bar{\alpha}_{mn}\) is mode dependent, the threshold gain coefficient \(g_{mn}^\text{th}\) varies from one transverse mode to another.

In addition, the effective gain coefficient can be different for different transverse mode because different transverse modes have different field distribution patterns and thus overlap with the gain volume differently.

The transverse mode that has the lowest loss and the largest effective gain at any given pumping level reaches threshold first and starts oscillating at the lowest pumping level.

In a typical laser, the transverse mode that reaches threshold first is normally the fundamental mode.

Unless a frequency-selecting mechanism is placed in a laser to create a frequency-dependent loss that varies from one longitudinal mode to another, the threshold gain coefficient \(g_{mn}^\text{th}\) does not vary much among the \(mnq\) longitudinal modes that share the common \(mn\) transverse mode pattern.

It is possible, however, to introduce a frequency-selecting device to a laser cavity so that \(\bar{\alpha}_{mn}\) and, consequently, \(g_{mn}^\text{th}\) become highly frequency dependent for the purpose of frequency selection or frequency tuning of the laser output.

The power required to pump a laser to reach its threshold is called the ** threshold pump power**, \(P_\text{p}^\text{th}\). Because the threshold gain coefficient is mode dependent and frequency dependent, the threshold pump power is also mode dependent and frequency dependent.

The threshold pump power of a laser mode can be found by calculating the power required for the gain medium to have an unsaturated gain coefficient equal to the threshold gain coefficient of the mode: \(g_0=g_{mn}^\text{th}\), assuming uniform pumping throughout the medium.

For a quasi-two-level or three-level laser, there is also a ** transparency pump power**, \(P_\text{p}^\text{tr}\), corresponding to \(g_0=0\), assuming uniform pumping throughout the gain medium.

In the situation of nonuniform pumping, these conditions for reaching threshold and transparency have to be modified, as discussed below.

Clearly, \(P_\text{p}^\text{tr}\lt{P}_\text{p}^\text{th}\) by definition.

In a nonwaveguiding laser, the transverse cross section of the gain medium is normally larger than the cross-sectional area of a laser mode. In tis situation, it is not necessary to pump the entire gain medium, but only the volume of the gain medium seen by the laser mode.

Calculation of \(P_\text{p}^\text{th}\) depends on the specifics of the pump source and the pumping geometry. Nevertheless, if we consider a ** saturation pump power**, \(P_\text{p}^\text{sat}\), as the pump power required for the pumping rate to be \(W_\text{p}=1/\tau_2\) following the same concept of \(I_\text{p}^\text{sat}\) as defined in (10-85) [refer to the population inversion and optical gain tutorial], we can find \(P_\text{p}^\text{th}\) in terms of \(P_\text{p}^\text{sat}\).

Because of absorption of the pump power by the gain medium, the pump power distribution in the pumped volume of a gain medium is often spatially nonuniform. The distribution of the pump power in a laser medium cannot be easily generalized because it is a function of many parameters specific to a particular pump source, a particular pumping geometry, an a given gain medium.

A case of common interest for solid-state lasers, however, is the longitudinal optical pumping considered in the laser amplifiers tutorial for laser amplifiers. In this situation, the laser threshold is reached when

\[\tag{11-58}\displaystyle\int\limits_0^{l_\text{g}}g_0(z)\text{d}z=g_\text{th}l_\text{g}\]

For single-pass optical pumping, as considered in the laser amplifiers tutorial, if transverse divergence of the pump beam is negligible, the integral of the unsaturated gain coefficient over a gain medium of a length \(l\) has the closed-formed solutions given in (10-106) and (10-108).

By taking (10-108) with \(l=l_\text{g}\) and using the condition in (11-58) for the laser threshold, we find that the threshold pump power of the laser can be expressed in terms of the pump power utilization factor \(\zeta_\text{p}\), which is defined in (10-107) [refer to the laser amplifiers tutorial], as

\[\tag{11-59}P_\text{p}^\text{th}=\begin{cases}\frac{1}{p}\frac{\exp\left[p\frac{\sigma_\text{a}N_\text{t}+g_\text{th}}{(\sigma_\text{e}+\sigma_\text{a})N_\text{t}}\alpha_\text{p}l_\text{g}\right]-1}{1-(1-\zeta_\text{p}^\text{th})\exp\left[p\frac{\sigma_\text{a}N_\text{t}+g_\text{th}}{(\sigma_\text{e}+\sigma_\text{a})N_\text{t}}\alpha_\text{p}l_\text{g}\right]}P_\text{p}^\text{sat},\qquad\text{for }p\ne0\\\frac{\sigma_\text{a}N_\text{t}+g_\text{th}}{(\sigma_\text{e}+\sigma_\text{a})N_\text{t}}\frac{\alpha_\text{p}l_\text{g}}{\zeta_\text{p}^\text{th}}P_\text{p}^\text{sat},\qquad\qquad\qquad\qquad\text{for }p=0\end{cases}\]

where \(\zeta_\text{p}^\text{th}\) is the pump power utilization factor at the laser threshold. It can be found by applying (10-106) [refer to the laser amplifiers tutorial] with \(l=l_\text{g}\) to the condition in (11-58) that

\[\tag{11-60}\zeta_\text{p}^\text{th}=1-\exp\left[-\frac{\sigma_\text{e}N_\text{t}-g_\text{th}}{(\sigma_\text{e}+\sigma_\text{a})N_\text{t}}\alpha_\text{p}l_\text{g}\right]\]

By plugging this relation for \(\zeta_\text{p}^\text{th}\) into (11-59), the threshold pump power can be explicitly expressed in terms of the laser parameters as

\[\tag{11-61}P_\text{p}^\text{th}=\begin{cases}\frac{1}{p}\frac{\exp\left[p\frac{\sigma_\text{a}N_\text{t}+g_\text{th}}{(\sigma_\text{e}+\sigma_\text{a})N_\text{t}}\alpha_\text{p}l_\text{g}\right]-1}{1-\exp\left[-\frac{(\sigma_\text{e}-p\sigma_\text{a})N_\text{t}-(1+p)g_\text{th}}{(\sigma_\text{e}+\sigma_\text{a})N_\text{t}}\alpha_\text{p}l_\text{g}\right]}P_\text{p}^\text{sat},\qquad\text{for }p\ne0\\\frac{\sigma_\text{a}N_\text{t}+g_\text{th}}{(\sigma_\text{e}+\sigma_\text{a})N_\text{t}}\frac{\alpha_\text{p}l_\text{g}}{1-\exp\left[-\frac{\sigma_\text{e}N_\text{t}-g_\text{th}}{(\sigma_\text{e}+\sigma_\text{a})N_\text{t}}\alpha_\text{p}l_\text{g}\right]}P_\text{p}^\text{sat},\qquad\text{for }p=0\end{cases}\]

The transparency condition is such that the integral of the unsaturated gain coefficient over the length of the gain medium is zero. Therefore, by replacing \(g_\text{th}\) with \(0\) in (11-61), the transparency pump power of a laser gain medium can be found:

\[\tag{11-62}P_\text{p}^\text{tr}=\begin{cases}\frac{1}{p}\frac{\exp\{[p\sigma_\text{a}/(\sigma_\text{e}+\sigma_\text{a})]\alpha_\text{p}l_\text{g}\}-1}{1-\exp\{-[(\sigma_\text{e}-p\sigma_\text{a})/(\sigma_\text{e}+\sigma_\text{a})]\alpha_\text{p}l_\text{g}\}}P_\text{p}^\text{sat},\qquad\text{for }p\ne0\\\frac{\sigma_\text{a}}{(\sigma_\text{e}+\sigma_\text{a})}\frac{\alpha_\text{p}l_\text{g}}{1-\exp\{-[\sigma_\text{e}/(\sigma_\text{e}+\sigma_\text{a})]\alpha_\text{p}l_\text{g}\}}P_\text{p}^\text{sat},\qquad\text{for }p=0\end{cases}\]

The relation obtained in (11-61) for the threshold pump power \(P_\text{p}^\text{th}\) and that obtained in (11-62) for the transparency pump power \(P_\text{p}^\text{tr}\) are generally valid at all pumping levels for single-pass longitudinal optical pumping. They are applicable no matter whether there is significant absorption saturation of the pump power in the gain medium or not.

They are not valid for multiple-pass longitudinal optical pumping, however.

The absorption saturation of the pump power is negligible under the condition that \(s_\text{th}=P_\text{p}^\text{th}/P_\text{p}^\text{sat}\ll1\). In this situation, the pump power decays exponentially along the longitudinal pumping axis in each pass through the gain medium. Then, a closed-form solution of \(P_\text{p}^\text{th}\) that has a common form for both single-pass and multiple-pass longitudinal pumping arrangements can be found in terms of the pump power utilization factor \(\zeta_\text{p}^\text{th}\).

In a single-pass arrangement under the condition that \(s_\text{th}\ll1\), \(\zeta_\text{p}^\text{th}\approx1-\text{e}^{-\alpha_\text{p}l_\text{g}}\), assuming no reflection of the pump beam at the pump input surface of the gain medium.

In a multiple-pass situation, however, \(\zeta_\text{p}^\text{th}\) has to be properly evaluated to account for the total pump power absorbed by the gain medium in all passes.

**Example 11-2**

The Nd : YAG microchip laser considered in Example 11-1 [refer to the resonant optical cavities tutorial] is pumped through a multimode fiber of 200 μm core diameter in the same manner as that for the amplifier described in Example 10-9 [refer to the laser amplifiers tutorial]. Both the pump and the laser spots have circular \(\text{TEM}_{00}\) mode profiles of 100 μm radius. Relevant parameters, from Example 10-9 [refer to the laser amplifiers tutorial], are \(N_\text{t}=1.52\times10^{26}\text{ m}^{-3}\) for the Nd concentration, \(\sigma_\text{a}^\text{p}=3.0\times10^{-24}\text{ m}^2\) with a pump quantum efficiency of \(\eta_\text{p}=80\%\) for the pump at 808 nm, \(\sigma_\text{e}=3.1\times10^{-23}\text{ m}^2\) with \(\tau_2=240\text{ μs}\) for the laser transition at 1.064 μm.

(a) Find the threshold gain coefficient for the laser.

(b) Find the threshold pump power of the laser.

**(a)**

This laser has a filling factor of \(\Gamma=1\). It also has \(\bar{\alpha}=0.5\text{ m}^{-1}\), \(l=500\text{ μm}\), \(R_1=100\%\), and \(R_2=99.7\%\), as given in Example 11-1 [refer to the resonant optical cavities tutorial].

Therefore, according to (11-56), the threshold gain coefficient is

\[g_\text{th}=\frac{1}{\Gamma}\left(\bar{\alpha}-\frac{1}{l}\ln\sqrt{R_1R_2}\right)=\frac{1}{1}\left(0.5-\frac{1}{500\times10^{-6}}\ln\sqrt{0.997}\right)\text{ m}^{-1}=3.5\text{ m}^{-1}\]

**(b)**

The laser is a four-level system with \(\sigma_\text{a}=0\) and \(p=0\). The threshold pump power can be found directly by using (11-61) for \(p=0\). However, it is instructive to find \(\zeta_\text{p}^\text{th}\) through (11-60) first and then use (11-59) to find \(P_\text{p}^\text{th}\).

Because \(\alpha_\text{p}=\sigma_\text{a}^\text{p}N_\text{t}=456\text{ m}^{-1}\) and \(l_\text{g}=500\text{ μm}\), we find that \(\alpha_\text{p}l_\text{g}=0.228\). Therefore, for this single-pass pumping arrangement, we have

\[\zeta_\text{p}^\text{th}=1-\exp\left(-\frac{3.1\times10^{-23}\times1.52\times10^{26}-3.5}{3.1\times10^{-23}\times1.52\times10^{26}}\times0.228\right)=0.2037\]

Because the pump spot size and all other pump parameters are the same as those used for the amplifier in Example 10-9 [refer to the laser amplifiers tutorial], we find from Example 10-9 that \(P_\text{p}^\text{sat}=13.4\text{ W}\). Then, using (11-59) for \(p=0\), we find the following threshold pump power:

\[P_\text{p}^\text{th}=\frac{3.5}{3.1\times10^{-23}\times1.52\times10^{26}}\times\frac{0.228}{0.2037}\times13.4\text{ W}=11.1\text{ mW}\]

By comparing \(\zeta_\text{p}^\text{th}=0.2037\) found above to \(1-\text{e}^{-\alpha_\text{p}l_\text{g}}=1-\text{e}^{-0.228}=0.2039\), we find that the pump power decays along the longitudinal axis of the gain medium almost exponentially.

This characteristic indicates that there is almost no absorption saturation of the pump power, which can be understood from the fact that \(g_\text{th}/\sigma_\text{e}N_\text{t}=7.4\times10^{-4}\ll1\) and \(P_\text{p}^\text{th}/P_\text{p}^\text{sat}=8.3\times10^{-4}\ll1\).

The pump power is not fully utilized in this single-pass pumping arrangement because only \(20.4\%\) of the input power is absorbed by the gain medium. The laser threshold can be lowered by taking a multiple-pass arrangement or by properly increasing the length of the gain medium to increase the utilization fraction of the pump power.

**Mode Pulling**

Comparing (11-53) for an active Fabry-Perot laser with (11-41) [refer to the resonant optical cavities tutorial] for its cold cavity, we find that the round-trip phase shift for a field in a laser cavity is a function of \(\chi_\text{res}'\) through its dependence on \(\Delta{k}_\text{res}\). Consequently, the longitudinal mode frequencies \(\omega_{mnq}\) at which a laser oscillates are not exactly the same as the longitudinal mode frequencies \(\omega_{mnq}^\text{c}\) given in (11-42) for the cold Fabry-Perot cavity [refer to the resonant optical cavities tutorial].

Using (11-53) and (11-55), we find that the longitudinal mode frequencies of a Fabry-Perot laser are related to those of its cold cavity by

\[\tag{11-63}\omega_{mnq}=\omega_{mnq}^\text{c}\left(1+\frac{\chi_\text{res}'}{2n\bar{n}}\right)^{-1}\approx\omega_{mnq}^\text{c}\left(1-\frac{\chi_\text{res}'}{2n\bar{n}}\right)\]

Clearly, the mode frequencies \(\omega_{mnq}\) at which a laser oscillates differ from the cold cavity mode frequencies because they vary with the resonant susceptibility, which depends on the level of population inversion in the gain medium.

This dependence of oscillating laser mode frequencies on the population inversion in the gain medium is caused by the fact that the refractive index and the gain of the medium are intimately connected to each other, as is dictated by the Kramers-Kronig relation [refer to the material dispersion tutorial].

This effect causes a frequency shift of

\[\tag{11-64}\delta\omega_{mnq}=\omega_{mnq}-\omega_{mnq}^\text{c}\approx-\frac{\chi_\text{res}'}{2n\bar{n}}\omega_{mnq}^\text{c}\]

for the oscillation frequency of mode \(mnq\).

Because of the frequency dependence of \(\chi_\text{res}'\), the dependence of this frequency shift on \(\chi_\text{res}'\) results in the ** mode-pulling** effect demonstrated in Figure 11-7. Near the resonant transition frequency, \(\omega_{21}\), of the gain medium, \(\chi_\text{res}\) is highly dispersive.

In the presence of population inversion, \(\chi_\text{res}^"(\omega)\lt0\) for either \(\omega\lt\omega_{21}\) or \(\omega\gt\omega_{21}\), but \(\chi_\text{res}'(\omega)\lt0\) for \(\omega\lt\omega_{21}\) and \(\chi_\text{res}'(\omega)\gt0\) for \(\omega\gt\omega_{21}\). As a result, \(\omega_{mnq}\gt\omega_{mnq}^\text{c}\) for \(\omega_{mnq}^\text{c}\lt\omega_{21}\) and \(\omega_{mnq}\lt\omega_{mnq}^\text{c}\) for \(\omega_{mnq}^\text{c}\gt\omega_{21}\).

Therefore, in comparison to the resonance frequencies of the cold cavity, the oscillating mode frequencies of a laser are pulled toward the transition frequency of the gain medium.

In addition, the longitudinal modes belonging to a common transverse mode are no longer equally spaced in frequency. In a laser of relatively high gain and large dispersion, such as in a semiconductor laser, this can result in a large variation in the frequency spacing among the oscillating modes.

Because of the frequency dependence of the gain coefficient \(g\) due to the frequency dependence of \(\chi_\text{res}^"\), different longitudinal modes not only experience different values of refractive index but also see different values of gain coefficient, as also illustrated in Figure 11-7. A longitudinal mode whose frequency is close to the gain peak at the transition resonance frequency has a higher gain than one whose frequency is far away from the gain peak.

**Oscillating Laser Modes**

Because of the frequency dependence of the gain coefficient, the net gain, \(g-g_{mn}^\text{th}\), of a laser is always frequency dependent and varies among different transverse modes and among different longitudinal modes no matter whether the threshold gain coefficient \(g_{mn}^\text{th}\) for any given transverse mode is frequency dependent or not.

At a low pumping level before the laser starts oscillating, the net gain is negative for all laser modes. As the pumping level increases, the mode that reaches its threshold first will start oscillating.

Once a laser starts oscillating in one mode, whether any other longitudinal or transverse modes have the opportunity to oscillate through further increase of the pumping level is a complicated issue of mode interaction and competition that depends on a variety of parameters, including the properties of the gain medium, the structure of the laser, the pumping geometry, the optical nonlinearity in the system, and the operating condition of the laser.

Here we only discuss some basic concepts in the situation of steady-state oscillation of a CW laser. Interaction and competition among laser modes are more complicated when a laser is pulsed than when it is in CW operation. Therefore, some of the conclusions obtained here may not be valid for a pulsed laser.

The gain condition in (11-54) implies that once a given laser mode is oscillating in steady state, the gain that is available to this mode does not increase with increased pumping above the threshold pumping level because \(G_{mn}\) for a laser mode has to be kept at unity for steady-state oscillation.

Thus the effective gain coefficient for an oscillating mode is "clamped" at the threshold level of the mode so long as the pumping level is kept at or above threshold.

The mechanism for holding down the gain coefficient at the threshold level is the effect of gain saturation discussed in the population inversion and optical gain tutorial. An increase in the pumping level above threshold only increases the field intensity for the oscillating mode in the cavity, but the gain coefficient is saturated at the threshold value by the high intensity of the intracavity laser field.

The fact that the gain of a laser mode oscillating in the steady state is saturated at the threshold value has a significant effect on the mode characteristics of a CW laser.

When the gain medium of a laser is homogeneously broadened, all modes that occupy the same spatial gain region compete for the gain from the population inversion in the same group of active atoms.

When the mode that first reaches threshold starts oscillating, the entire gain curve supported by this group of atoms saturates. Because this oscillating mode is normally the one that has a longitudinal mode frequency closest to the gain peak and a transverse mode pattern with the lowest loss, the gain curve is saturated in such a manner that its value at this longitudinal mode frequency is clamped at the threshold value of the transverse mode that has the lowest threshold gain coefficient among all transverse modes.

If the gain peak does not happen to coincide with this mode frequency, it still lies above the threshold when the gain curve is saturated, as shown in Figure 11-8.

Nevertheless, all other longitudinal modes belonging to this transverse mode have frequencies away from the gain peak. Therefore, even with increased pumping, they do not have sufficient gain to reach threshold because the entire gain curve shared by these modes is saturated, as illustrated in Figure 11-8.

Other transverse modes that are supported solely by this group of saturated, homogeneously broadened atoms do not have the opportunity to oscillate, either because the gain curve is saturated below their threshold levels.

Nevertheless, as different transverse modes have different spatial field distributions, a high-order transverse mode may draw its gain from a gain region outside of the region saturated by a low-order mode. Therefore, when the pumping level is increased, a high-order transverse mode may still reach its relatively high threshold for oscillation after a low-order transverse mode of a low threshold already oscillates.

Consequently, for a homogeneously broadened CW laser in a steady-state oscillation, only one among all of the longitudinal modes belonging to the same transverse mode will oscillate, but it is possible for more than one transverse mode to oscillate simultaneously at a high pumping level.

Note that this conclusion does not hold true for a pulsed laser. It is possible for multiple longitudinal modes all belonging to the same transverse mode to oscillate simultaneously in a pulsed laser even when the gain medium is homogeneously broadened.

In a laser containing an inhomogeneously broadened gain medium, there are different groups of active atoms in the same spatial region. Each group saturates independently.

Two modes occupying the same spatial gain region do not compete for the same group of atoms if the separation of their frequencies is larger than the homogeneous linewidth of each group of atoms.

When one longitudinal mode reaches threshold and oscillates, only the gain coefficient around its frequency is saturated, the gain coefficient at other frequencies continues to increase with increased pumping. As the pumping level increases, other longitudinal modes will reach threshold and oscillate successively.

As a result, at a sufficiently high pumping level, multiple longitudinal modes belonging to the same transverse mode can oscillate simultaneously. The saturation of the gain coefficient around each of the frequencies of these oscillating modes, but not across the entire gain curve, creates the effect of ** spectral hole burning** in the gain curve of an inhomogeneously broadened laser medium, as illustrated in Figure 11-9.

Different transverse modes also saturate independently in an inhomogeneously broadened medium if their longitudinal mode frequencies are sufficiently separated. Therefore, an inhomogeneously broadened laser can also oscillate in multiple transverse modes.

The linewidth of an oscillating laser mode is still described by (11-19) [refer to the resonant optical cavities tutorial]. From this relation, we see that in practice the round-trip field gain factor \(G\) of a laser in steady-state oscillation cannot be exactly equal to unity because the laser linewidth cannot be zero, due to the existence of spontaneous emission.

In reality, in steady-state oscillation the value of \(G\) is slightly less than unity, with the small difference made up by spontaneous emission.

Clearly, the linewidth of an oscillating laser mode is determined by the amount of spontaneous emission that is channeled into the laser mode. Therefore, (11-19) [refer to the resonant optical cavities tutorial] is not very useful for calculating the linewidth of a laser mode in steady-state oscillation without knowing the exact value of \(G\) in the presence of spontaneous emission.

Instead, a detailed analysis taking into account spontaneous emission yields the following ** Schawlow-Townes relation** for the linewidth of a laser mode in terms of the laser parameters:

\[\tag{11-65}\Delta\nu_\text{ST}=\frac{2\pi{h}\nu(\Delta\nu_\text{c})^2}{P_\text{out}}N_\text{sp}=\frac{h\nu}{2\pi\tau_\text{c}^2P_\text{out}}N_\text{sp}\]

where \(P_\text{out}\) is the output power of the laser mode being considered and \(N_\text{sp}\) is the spontaneous emission factor defined in (10-114) [refer to the laser amplifiers tutorial].

The effect of spontaneous emission on the linewidth of an oscillating laser mode enters the relation in (11-65) through the population densities of the upper and the lower laser levels in the form of the spontaneous emission factor.

Because \(N_\text{sp}\ge1\), the ultimate lower limit of the laser linewidth, which is known as the Schawlow-Townes limits, is that given in (11-65) with \(N_\text{sp}=1\).

**Example 11-3**

The Nd : YAG crystal used for the microchip laser described in Examples 11-1 [refer to the resonant optical cavities tutorial] and 11-2 has a spontaneous linewidth of 150 GHz at the laser wavelength of 1.064 μm.

(a) How many longitudinal modes will oscillate when the laser operates at room temperature?

(b) What is the linewidth of an oscillating laser mode when the laser has an output power of 1 mW?

**(a)**

Only one longitudinal mode will oscillate above the laser threshold, for two reasons. First, Nd : YAG is predominantly homogeneously broadened at room temperature. Second, according to Example 11-1 [refer to the resonant optical cavities tutorial], the longitudinal mode spacing for this microchip laser is \(\Delta\nu_\text{L}=164.8\text{ GHz}\), which is larger than the entire Nd : YAG linewidth of 150 GHz.

**(b)**

The laser photon energy is \(h\nu=(1.2398/1.064)\text{ eV}=1.165\text{ eV}\). From Example 11-1 [refer to the resonant optical cavities tutorial], \(\Delta\nu_\text{c}=91.91\text{ MHz}\). Because this laser is a four-level system, \(N_\text{sp}=1\). From (11-65), we find the following Schawlow-Townes linewidth for the oscillating laser mode:

\[\Delta\nu_\text{ST}=\frac{2\pi\times1.165\times1.6\times10^{-19}\times(91.91\times10^6)^2}{1\times10^{-3}}\times1\text{ Hz}=9.9\text{ Hz}\]

Compared to the longitudinal mode width of 91.91 MHz for the cold cavity, this oscillating mode width is nearly seven orders of magnitude smaller. This linewidth-narrowing effect is caused by the coherent nature of the stimulated emission and is a fundamental feature of lasers.

Note, however, that the Schawlow-Townes linewidth is only the theoretical lower limit of an oscillating laser mode. In practice, the linewidth of a laser is often broadened far above this limit by other mechanisms, such as fluctuations in the pump power and temperature, mechanical vibrations, and electronic noise from the circuit supporting the operation of the laser. The Schawlow-Townes limit can be approached only by making every effort to eliminate all external effects that broaden the laser linewidth.

The next tutorial covers the topic of **laser power**.