# Laser Power

This is a continuation from the previous tutorial - laser oscillation.

In this tutorial, we consider the output power of a laser. Because the situation of a multimode laser can be quite complicated due to mode competition, we consider for simplicity only a homogeneously broadened, CW laser oscillating in a single longitudinal and transverse mode. Therefore, the parameters mentioned in this tutorial are not labeled with mode indices because all of these parameters are clearly associated with the only oscillating mode being considered.

The simple Fabry-Perot cavity that contains an isotropic gain medium with a filling factor $$\Gamma$$ as shown in Figure 11-5 is considered [refer to the resonant optical cavities tutorial]. To illustrate the general concepts, we first consider the situation when the gain medium is uniformly pumped so that the entire gain medium has a spatially independent gain coefficient $$g$$. We then consider at the end of this tutorial the case of optically pumped lasers, as also considered for the laser threshold in the laser oscillation tutorial, taking into account the longitudinal spatial dependence of the gain coefficient.

For the single oscillating mode of the Fabry-Perot laser considered here, the round-trip gain factor $$G$$ is that given by (11-52) [refer to the laser oscillation tutorial], and the cavity decay rate $$\gamma_\text{c}$$ defined by (11-24) [refer to the resonant optical cavities tutorial] is that given by (11-47) [refer to the refer to the resonant optical cavities tutorial].

Therefore,

$\tag{11-66}G^2=\exp(2\Gamma{gl}-\gamma_\text{c}T)$

Because $$G^2$$ is the net amplification factor of the intracavity field energy, or photon number, in a round-trip time $$T$$ of the laser cavity, we can define an intracavity energy growth rate, or intracavity photon growth rate, $$\Gamma\mathrm{g}$$, for the oscillating laser mode through the following relation:

$\tag{11-67}G^2=\exp[(\Gamma\mathrm{g}-\gamma_\text{c})T]$

for a laser containing a gain medium with a filling factor $$\Gamma$$.

We find, by comparing (11-66) with (11-67), that

$\tag{11-68}\mathrm{g}=\frac{2gl}{T}=\frac{cg}{\bar{n}}$

and, by comparing (11-47) [refer to the resonant optical cavities tutorial] with (11-56) [refer to the laser oscillation tutorial], that

$\tag{11-69}\gamma_\text{c}=\Gamma\frac{2g_\text{th}l}{T}=\Gamma\frac{cg_\text{th}}{\bar{n}}$

Note that while $$g$$ and $$g_\text{th}$$ are measured per meter, $$\mathrm{g}$$ and $$\gamma_\text{c}$$ are measured per second.

The relation in (11-68) translates the gain coefficient $$g$$ that characterizes space-dependent amplification of a laser field propagating through the intracavity gain medium into an intracavity energy growth rate $$\mathrm{g}$$ that characterizes time-dependent amplification of the energy in a laser mode by the gain medium.

The relation in (11-69) clearly indicates that the threshold intracavity energy growth rate for laser oscillation is the cavity decay rate:

$\tag{11-70}\Gamma\mathrm{g}_\text{th}=\gamma_\text{c}$

This relation can also be obtained by applying the threshold gain condition of $$G=1$$ given in (11-54) [refer to the laser oscillation tutorial] to the relation in (11-67).

It is easy to understand because for a laser mode to oscillate, the growth of intracavity photons in that mode through amplification by the gain medium has to at least match the decay of photons caused by all of the loss mechanisms combined.

Therefore, we shall call the energy growth rate $$\Gamma\mathrm{g}$$ and the cavity decay rate $$\gamma_\text{c}$$, both of which are specific to a laser mode, the gain parameter and the loss parameter, respectively, of the laser mode.

By using temporal growth and decay rates instead of spatial gain and loss coefficients to describe the characteristics of a laser, we are in effect moving from a spatially distributed description of the laser to a lumped-device description.

In the lumped-device description, a laser mode is considered an integral entity with its spatial characteristics effectively integrated into the parameters $$\Gamma\mathrm{g}$$ and $$\gamma_\text{c}$$. The detailed spatial characteristics of the mode are irrelevant and are lost in this description.

Therefore, instead of the intensity of the oscillating laser field, we have to consider the intracavity photon density, $$S$$, of the oscillating laser mode. For a Fabry-Perot laser containing a gain medium with a filling factor $$\Gamma$$ so that the average refractive index inside the cavity is $$\bar{n}=\Gamma{n}+(1-\Gamma)n_0$$ as defined in (11-4) [refer to the resonant optical cavities tutorial], the average photon density in the cavity is

$\tag{11-71}S=\frac{\bar{n}I}{ch\nu}$

where $$I$$ is the spatially averaged intensity inside the laser cavity and $$h\nu$$ is the photon energy of the oscillating laser mode.

Because the gain parameter $$\mathrm{g}$$ is directly proportional to the gian coefficient $$g$$ of the gain medium, the relation between the unsaturated, small-signal gain parameter $$\mathrm{g}_0$$ and the saturated gain parameter $$\mathrm{g}$$ for a laser mode in the lumped-device description can be obtained by converting the relation between $$g_0$$ and $$g$$ discussed in the population inversion and optical gain tutorial through the relation in (11-68).

Therefore, for the gain parameter of a laser mode, we have

$\tag{11-72}\mathrm{g}=\frac{\mathrm{g_0}}{1+S/S_\text{sat}}$

where

$\tag{11-73}\mathrm{g}_0=\frac{cg_0}{\bar{n}}$

is the unsaturated gain parameter and

$\tag{11-74}S_\text{sat}=\frac{\bar{n}I_\text{sat}}{ch\nu}=\frac{\bar{n}}{c\tau_\text{s}\sigma_\text{e}}$

is the saturation photon density.

When a CW laser oscillates in the steady state, the value of $$\Gamma\mathrm{g}$$ for the oscillating mode is clamped at its threshold value of $$\gamma_\text{c}$$, just as the value of $$g$$ is clamped at $$g_\text{th}$$. Therefore, by setting $$\Gamma\mathrm{g}$$ to equal $$\gamma_\text{c}$$ with $$\mathrm{g}$$ given in (11-72), we find that the photon density for a CW laser mode in steady-state oscillation is

$\tag{11-75}S=\left(\frac{\Gamma\mathrm{g}_0}{\gamma_\text{c}}-1\right)S_\text{sat}=(r-1)S_\text{sat},\qquad\text{for }r\ge1$

The dimensionless pumping ratio, $$r$$, represents that a laser is pumped at $$r$$ times its threshold. It is defined as

$\tag{11-76}r=\frac{\Gamma\mathrm{g}_0}{\gamma_\text{c}}=\frac{g_0}{g_\text{th}}$

Note that (11-75) is valid only for $$r\ge1$$ when the laser oscillates because only then is the laser gain saturated. For $$r\lt1$$, the laser does not reach threshold. The laser cavity is then filled with spontaneous photons at a density that is small in comparison to the high density of coherent photons when the laser reaches threshold and oscillates.

From the photon density of the oscillating laser mode, we can easily find the following total intracavity energy contained in this mode:

$\tag{11-77}U_\text{mode}=\mathcal{V}_\text{mode}Sh\nu$

where $$\mathcal{V}_\text{mode}$$ is the volume of the oscillating mode.

The mode volume can be found by integrating the normalized intensity distribution of the mode over the three-dimensional space defined by the laser cavity. It is usually a fraction of the volume of the cavity.

The output power of the laser is simply the coherent optical energy emitted from the laser per second. Therefore, it is simply the product of the mode energy and the output-coupling rate, $$\gamma_\text{out}$$, of the cavity:

$\tag{11-78}P_\text{out}=U_\text{mode}\gamma_\text{out}=\mathcal{V}_\text{mode}Sh\nu\gamma_\text{out}=(r-1)\mathcal{V}_\text{mode}S_\text{sat}h\nu\gamma_\text{out}$

The output-coupling rate is also called the output-coupling loss parameter because it contributes to the total loss of a laser cavity and is a fraction of the total loss parameter $$\gamma_\text{c}$$.

One can indeed write $$\gamma_\text{c}=\gamma_\text{i}+\gamma_\text{out}$$, where $$\gamma_\text{i}$$ is the internal loss of the laser, which does not contribute to output coupling of the laser power.

As an example, for the Fabry-Perot laser with its $$\gamma_\text{c}$$ given by (11-47) [refer to the resonant optical cavities tutorial], we have the internal loss given by $$\gamma_\text{i}=c\bar{\alpha}_{mn}/\bar{n}$$ and the output coupling loss given by

$\tag{11-79}\gamma_\text{out}=-\frac{c}{\bar{n}l}\ln\sqrt{R_1R_2}$

In this case, $$\gamma_\text{out}$$ is the total output-coupling loss through both mirrors. Therefore, $$P_\text{out}$$ given in (11-78) is the total output power emitted through both mirrors.

For the power output through each mirror, we find that

$\tag{11-80}\gamma_{\text{out},1}=-\frac{c}{\bar{n}l}\ln\sqrt{R_1}\qquad\text{and}\qquad\gamma_{\text{out},2}=-\frac{c}{\bar{n}l}\ln\sqrt{R_2}$

and that

$\tag{11-81}P_{\text{out},1}=U_\text{mode}\gamma_{\text{out},1}=P_\text{out}\frac{\gamma_{\text{out},1}}{\gamma_\text{out}}\qquad\text{and}\qquad{P}_{\text{out},2}=U_\text{mode}\gamma_{\text{out},2}=P_\text{out}\frac{\gamma_{\text{out},2}}{\gamma_\text{out}}$

It is convenient to define the saturation output power as

$\tag{11-82}P_\text{out}^\text{sat}=\mathcal{V}_\text{mode}S_\text{sat}h\nu\gamma_\text{out}$

Using the definition of $$S_\text{sat}$$ in (11-74), it can be shown that

$\tag{11-83}P_\text{out}^\text{sat}=-P_\text{sat}\ln\sqrt{R_1R_2}$

where $$P_\text{sat}$$ is the saturation power of the gain medium found by integrating $$I_\text{sat}$$ over the cross-sectional area of the gain medium.

Combining (11-78) with (11-82), we can express the output laser power in terms of $$P_\text{out}^\text{sat}$$ as

$\tag{11-84}P_\text{out}=(r-1)P_\text{out}^\text{sat}$

Note that $$P_\text{out}^\text{sat}$$ is not the level at which the output power of a laser saturates. Its physical meaning can be easily seen from (11-83) and (11-84).

From (11-83), we find that the output power of a laser is $$P_\text{out}^\text{sat}$$ when the intracavity laser power is $$P_\text{sat}$$ of the gain medium. From (11-84), we find that $$P_\text{out}=P_\text{out}^\text{sat}$$ when $$r=2$$; in order words, a laser has an output power of $$P_\text{out}^\text{sat}$$ when its is pumped at twice its threshold level.

In order to express the output laser power explicitly as a function of the pump power, it is necessary to specify the pumping mechanism and the pumping geometry.

For this purpose, we consider longitudinal optical pumping with negligible transverse pump beam divergence but with a spatially varying gain coefficient $$g(z)$$ along the longitudinal axis of the gain medium, as is the case considered in the laser oscillation tutorial for the threshold pump power obtained in (11-61) [refer to the laser oscillation tutorial].

In this situation, all of the results obtained so far in this tutorial are still applicable if we make the following substitution for the gain coefficient:

$\tag{11-85}g=\frac{1}{l_\text{g}}\displaystyle\int\limits_0^{l_\text{g}}g(z)\text{d}z=\frac{1}{\Gamma{l}}\int\limits_0^{l_\text{g}}g(z)\text{d}z$

Then, in the case when $$p=0$$ or $$p\ll1$$, the pumping ratio at an input pump power of $$P_\text{p}$$ can be expressed as

$\tag{11-86}r=\frac{\Gamma\mathrm{g}_0}{\gamma_\text{c}}=\frac{\displaystyle\int\limits_0^{l_\text{g}}g_0(z)\text{d}z}{g_\text{th}l_\text{g}}\approx\frac{\zeta_\text{p}P_\text{p}-\zeta_\text{p}^\text{tr}P_\text{p}^\text{tr}}{\zeta_\text{p}^\text{th}P_\text{p}^\text{th}-\zeta_\text{p}^\text{tr}P_\text{p}^\text{tr}}$

where $$P_\text{p}^\text{th}$$ and $$P_\text{p}^\text{tr}$$ are the threshold pump power and the transparency pump power of the laser found in (11-61) and (11-62) [refer to the laser oscillation tutorial], respectively, and $$\zeta_\text{p}$$, $$\zeta_\text{p}^\text{th}$$, and $$\zeta_\text{p}^\text{tr}$$ are the pump power utilization factors at the pumping levels of $$P_\text{p}$$, $$P_\text{p}^\text{th}$$, and $$P_\text{p}^\text{tr}$$, respectively. [$$\zeta_\text{p}$$ was defined in (10-107) in the laser amplifiers tutorial].

In the case of $$P_\text{p}\ll{P}_\text{p}^\text{sat}$$ when the absorption saturation of the pump is negligible, $$\zeta_\text{p}\approx\zeta_\text{p}^\text{th}\approx\zeta_\text{p}^\text{tr}$$.

In the presence of significant absorption saturation of the pump, however, we find that $$\zeta_\text{p}\lt\zeta_\text{p}^\text{th}\lt\zeta_\text{p}^\text{tr}$$ because $$P_\text{p}\gt{P}_\text{p}^\text{th}\gt{P}_\text{p}^\text{tr}$$.

By substituting (11-86) for $$r$$ in (11-84), we find the following relation between the output power of a laser and the power launched to pump the laser:

$\tag{11-87}P_\text{out}=\frac{\zeta_\text{p}P_\text{p}-\zeta_\text{p}^\text{th}P_\text{p}^\text{th}}{\zeta_\text{p}^\text{th}P_\text{p}^\text{th}-\zeta_\text{p}^\text{tr}P_\text{p}^\text{tr}}P_\text{out}^\text{sat}$

This relation is obtained by using (11-86) under the following assumptions:

1. $$p=0$$ or $$p\ll1$$
2. $$P_\text{out}^\text{sat}$$ is a constant throughout the gain medium
3. the intracavity laser photon density is relatively uniformly distributed

The relation in (11-87) works best in the situation where

1. the gain medium is a four-level or three-level system so that $$p=0$$
2. $$W_\text{p}\tau_2\ll1$$ so that $$\tau_\text{s}$$ and $$P_\text{out}^\text{sat}$$ are not spatially varying
3. $$R_1R_2$$ approaches unity so that the intracavity photon density distribution is quite uniform

A laser that satisfies these conditions is considered in Example 11-4 below.

Alternatively, by consideration of energy conservation, the output laser power can be found through the following relation:

$\tag{11-88}P_\text{out}=\eta_\text{p}\frac{\gamma_\text{out}}{\gamma_\text{c}}\frac{h\nu}{h\nu_\text{p}}(\zeta_\text{p}P_\text{p}-\zeta_\text{p}^\text{th}P_\text{p}^\text{th})=\eta_\text{p}\frac{\gamma_\text{out}}{\gamma_\text{c}}\frac{\lambda_\text{p}}{\lambda}(\zeta_\text{p}P_\text{p}-\zeta_\text{p}^\text{th}P_\text{p}^\text{th})$

where $$\eta_\text{p}$$ is the pump quantum efficiency defined in (10-84) [refer to the population inversion and optical gain tutorial], and $$\lambda$$ and $$\lambda_\text{p}$$ are the laser and pump wavelengths, respectively.

This relation is quite general. It is not subject to the conditions that limit the applicability of (11-87). Under the conditions when (11-87) is valid, it can be shown that (11-87) yields exactly the same result as (11-88). When the conditions for (11-87) are not fully satisfied so that (11-87) no longer yield a reliable result, (11-88) can still be used to find the output laser power.

The relations in (11-87) and (11-88) state that the output power of a laser grows linearly with the pump power above threshold. It also indicates that the laser has zero output power before it reaches threshold.

Upon reaching the threshold, the optical output of the device also shows dramatic spectral narrowing that accompanies the start of laser oscillation. According to (11-65) [refer to the laser oscillation tutorial], the linewidth of an oscillating laser mode continues to narrow with increasing laser power as the laser is pumped higher and higher above threshold.

These are the unique characteristics that distinguish a laser from other types of light sources such as fluorescent light emitters and luminescent light sources.

However, a real laser does not have exactly such ideal characteristics, mainly because of the presence of spontaneous emission and nonlinearities in the gain medium.

Figure 11-10 below shows typical characteristics of the output power of a single-mode laser as a function of pump power.

On the one hand, the linear relation in (11-87) between $$P_\text{out}$$ and $$P_\text{p}$$ is a consequence of applying the linear relation between $$g_0$$ and $$P_\text{p}$$ to (11-76).

As discussed in the population inversion and optical gain tutorial, the linear relation between $$g_0$$ and $$P_\text{p}$$ derived from (10-88) [refer to the population inversion and optical gain tutorial] is itself an approximation near the transparency point of a gain medium. As the pump power increases to a sufficiently high level, the unsaturated gain coefficient of a medium cannot continue to increase linearly with pump power because of depletion of the ground-level population.

Therefore, we should expect the output power of a laser not to continue its linear dependence on pump power but to increase less than linearly with pump power at high pumping levels.

On the other hand, once the gain medium of a laser is pumped so that its upper laser level begins to be populated, it emits spontaneous photons regardless of whether the laser is oscillating or not.

Clearly, the output power of a laser pumped below threshold is not exactly zero because spontaneous power is already emitted from the laser before the laser reaches threshold.

Though this spontaneous power is incoherent and is generally small in a practical laser, it is significant for a laser below and right at the threshold. Above threshold, it is the major source of incoherent noise for the coherent field of the laser output.

Example 11-4

Find the pump power required for the Nd : YAG microchip laser described in Examples 11-1 to 11-3 [refer to the resonant optical cavities tutorial and the laser oscillation tutorial] to have an output power of 1 mW.

This laser satisfies the conditions required for the application of (11-87). We should first find $$P_\text{p}$$ through (11-84) and (11-86) in the spirit of (11-87). Then we show that the same result is obtained from (11-88).

The required pump power for a desired output power can be found using (11-84) to obtain the value of the pumping ratio $$r$$ for a given value of $$P_\text{out}$$. To use (11-84) for this purpose, we have to find $$P_\text{out}^\text{sat}$$ through the values of $$\mathcal{V}_\text{mode}$$, $$S_\text{sat}$$, and $$\gamma_\text{out}$$.

Because $$l=500\text{ μm}$$ and $$w=100\text{ μm}$$,

$\mathcal{V}_\text{mode}=\pi{w}^2l=\pi\times(100\times10^{-6})^2\times500\times10^{-6}\text{ m}^3=1.57\times10^{-11}\text{ m}^3$

Because $$s_\text{th}=P_\text{p}^\text{th}/P_\text{p}^\text{sat}=8.3\times10^{-4}\ll1$$ from Example 11-2 [refer to the laser oscillation tutorial], we expect $$s=W_\text{p}\tau_2=P_\text{p}/P_\text{p}^\text{sat}\ll1$$ so that $$\tau_\text{s}=\tau_2=240\text{ μs}$$ for the operating pump power range of this laser. With $$\bar{n}=n=1.82$$ for this laser, we then have

$S_\text{sat}=\frac{\bar{n}}{c\tau_\text{s}\sigma_\text{e}}=\frac{1.82}{3\times10^8\times240\times10^{-6}\times3.1\times10^{-23}}\text{ m}^{-3}=8.15\times10^{17}\text{ m}^{-3}$

Because $$R_1=100\%$$ and $$R_2=99.7\%$$, we have

$\gamma_\text{out}=\gamma_{\text{out},2}=-\frac{c}{\bar{n}l}\ln\sqrt{R_2}=-\frac{3\times10^8}{1.82\times500\times10^{-6}}\times\ln\sqrt{0.997}\text{ s}^{-1}=4.95\times10^8\text{ s}^{-1}$

We then find that

\begin{align}P_\text{out}^\text{sat}&=\mathcal{V}_\text{mode}S_\text{sat}h\nu\gamma_\text{out}\\&=1.57\times10^{-11}\times8.15\times10^{17}\times1.165\times1.6\times10^{-19}\times4.95\times10^8\text{ W}\\&=1.18\text{ mW}\end{align}

We can then use (11-84) to find that

$P_\text{out}=1.18(r-1)\text{ mW}$

For an output of $$P_\text{out}=1\text{ mW}$$, we find that the required pumping ratio is

$r=1+\frac{1}{1.18}=1.85$

Because this laser is a four-level laser, we have $$P_\text{p}^\text{tr}=0$$. We already found from Example 11-2 [refer to the laser oscillation tutorial] that $$P_\text{p}^\text{th}=11.1\text{ mW}$$ and that the absorption saturation of the pump is negligible. Therefore, $$\zeta_\text{p}\approx\zeta_\text{p}^\text{th}\approx0.204$$. Using these parameters, the required pump power is found from (11-86) to be

$P_\text{p}=\frac{\zeta_\text{p}^\text{th}}{\zeta_\text{p}}rP_\text{p}^\text{th}\approx{r}P_\text{p}^\text{th}=1.85\times11.1\text{ mW}=20.5\text{ mW}$

Alternatively, we can find $$P_\text{p}$$ using (11-88). For this purpose, we have, from Example 11-2 [refer to the laser oscillation tutorial], the following parameters: $$\eta_\text{p}=0.8$$, $$\lambda=1.064\text{ μm}$$, $$\lambda_\text{p}=808\text{ nm}$$, and $$g_\text{th}=3.5\text{ m}^{-1}$$. We also find that

$\gamma_\text{c}=\Gamma\frac{cg_\text{th}}{\bar{n}}=1\times\frac{3\times10^8\times3.5}{1.82}\text{ s}^{-1}=5.77\times10^8\text{ s}^{-1}$

For $$P_\text{out}=1\text{ mW}$$ and $$P_\text{p}^\text{th}=11.1\text{ mW}$$, we find from (11-88) that

\begin{align}P_\text{p}&=\frac{1}{\zeta_\text{p}\eta_\text{p}}\frac{\gamma_\text{c}}{\gamma_\text{out}}\frac{\lambda}{\lambda_\text{p}}P_\text{out}+\frac{\zeta_\text{p}^\text{th}}{\zeta_\text{p}}P_\text{p}^\text{th}\\&=\frac{1}{0.204\times0.8}\times\frac{5.77\times10^8}{4.95\times10^8}\times\frac{1.064\times10^{-6}}{808\times10^{-9}}\times1\text{ mW}+\frac{0.204}{0.204}\times11.1\text{ mW}\\&=20.5\text{ mW}\end{align}

We see that the same result is obtained.

The overall efficiency of a laser, known as the power conversion efficiency, is

$\tag{11-89}\eta_\text{c}=\frac{P_\text{out}}{P_\text{p}}$

The linear dependence of the laser output power on the pump power indicated by (11-87) leads to the concept of the differential power conversion efficiency, also known as the slope efficiency, of a laser, defined as

$\tag{11-90}\eta_\text{s}=\frac{\text{d}P_\text{out}}{\text{d}P_\text{p}}=\frac{\zeta_\text{p}P_\text{out}^\text{sat}}{\zeta_\text{p}^\text{th}P_\text{p}^\text{th}-\zeta_\text{p}^\text{tr}P_\text{p}^\text{tr}}=\frac{\zeta_\text{p}P_\text{out}}{\zeta_\text{p}P_\text{p}-\zeta_\text{p}^\text{th}P_\text{p}^\text{th}}=\eta_\text{p}\zeta_\text{p}\frac{\gamma_\text{out}}{\gamma_\text{c}}\frac{\lambda_\text{p}}{\lambda}$

Referring to the power characteristics of the laser shown in Figure 11-10, the threshold of a given laser can usually be lowered by increasing the finesse, thus lowering the values of $$\gamma_\text{c}$$ and $$\gamma_\text{out}$$, of the laser cavity, but only at the expense of reducing the differential power conversion efficiency of the laser.

Clearly, in the linear region of the laser power characteristics where the relation given in (11-87) is valid, $$\eta_\text{s}$$ is a constant that is independent of the operating point of the laser. In contrast, $$\eta_\text{c}$$ increases with pump power, but $$\eta_\text{c}$$ is always smaller than $$\eta_\text{s}$$ in the linear region. At high pumping levels where the laser output power does not increase linearly with pump power, $$\eta_\text{s}$$ is no longer independent of the operating point. It can even become smaller than $$\eta_\text{c}$$ in some unfavorable situations.

The quantum efficiency of a laser oscillator is defined differently from that of a laser amplifier. The external quantum efficiency, $$\eta_\text{e}$$, also known as the differential quantum efficiency, measures the efficiency of converting pump photons or pump electrons above threshold into laser photons at the laser output.

Furthermore, an internal quantum efficiency, $$\eta_\text{i}$$, can be defined to measure the efficiency of converting the pump photons or the pump electrons above threshold into intracavity laser photons.

They are defined through the following relation:

$\tag{11-91}\eta_\text{e}=\frac{\Phi_\text{out}}{\zeta_\text{p}\Phi_\text{p}-\zeta_\text{p}^\text{th}\Phi_\text{p}^\text{th}}\qquad\text{and}\qquad\eta_\text{i}=\frac{\gamma_\text{c}}{\gamma_\text{out}}\eta_\text{e}$

where $$\Phi_\text{out}=P_\text{out}/h\nu$$ is the output photon flux of the laser, $$\Phi_\text{p}$$ is the pump photon flux, in the case of optical pumping, or the pump electron flux, in the case of electrical pumping, $$\Phi_\text{p}^\text{th}$$ is the threshold pump photon or electron flux, and $$\zeta_\text{p}$$ is the pump power utilization factor. In the case of optical pumping, $$\Phi_\text{p}=P_\text{p}/h\nu_\text{p}$$.

Then, the external quantum efficiency is directly related to the slope efficiency and the pump quantum efficiency as

$\tag{11-92}\eta_\text{e}=\frac{\lambda}{\lambda_\text{p}}\frac{\eta_\text{s}}{\zeta_\text{p}}=\eta_\text{p}\frac{\gamma_\text{out}}{\gamma_\text{c}}$

From (11-91) and (11-92), we find that $$\eta_\text{i}=\eta_\text{p}$$.

Because $$\gamma_\text{out}\lt\gamma_\text{c}$$, the external quantum efficiency $$\eta_\text{e}$$ is smaller than the internal quantum efficiency $$\eta_\text{i}$$ for a typical laser.

This reflects the fact that, because of the presence of losses in the laser cavity other than the output coupling loss, not all photons generated inside a laser cavity contribute to the output of the laser.

Furthermore, the internal quantum efficiency $$\eta_\text{i}$$ is equal to the pump quantum efficiency $$\eta_\text{p}$$, reflecting the fact that after a laser is pumped above threshold, every additional atom excited to the upper laser level results in the contribution of one photon in the oscillating laser mode through stimulated emission.

Example 11-5

Find the power conversion efficiency, the slope efficiency, and the external and internal quantum efficiencies of the Nd : YAG microchip laser described in Example 11-4 operating at an output power of 1 mW.

From Example 11-4, we find that $$P_\text{p}=20.5\text{ mW}$$ for $$P_\text{out}=1\text{ mW}$$. The power conversion efficiency in this operating condition is

$\eta_\text{c}=\frac{P_\text{out}}{P_\text{p}}=\frac{1}{20.5}=4.9\%$

This laser has a threshold pump power of $$P_\text{p}^\text{th}=11.1\text{ mW}$$ found in Example 11-2 [refer to the laser oscillation tutorial]. Also from Example 11-2, we know that $$\zeta_\text{p}\approx\zeta_\text{p}^\text{th}\approx0.204$$ because of negligible absorption saturation of the pump. The slope efficiency can then be found from (11-90) to be

$\eta_\text{s}=\frac{\zeta_\text{p}P_\text{out}}{\zeta_\text{p}P_\text{p}-\zeta_\text{p}^\text{th}P_\text{p}^\text{th}}\approx\frac{P_\text{out}}{P_\text{p}-P_\text{p}^\text{th}}=\frac{1}{20.5-11.1}=10.6\%$

With $$\lambda=1.064\text{ μm}$$ and $$\lambda_\text{p}=808\text{ nm}$$, the external quantum efficiency is thus found from (11-92) to be

$\eta_\text{e}=\frac{\lambda}{\lambda_\text{p}}\frac{\eta_\text{s}}{\zeta_\text{p}}=\frac{1.064\times10^{-6}}{808\times10^{-9}}\times\frac{10.6\%}{0.204}=68.4\%$

For this laser, we have $$\gamma_\text{c}=5.78\times10^8\text{ s}^{-1}$$ from Example 11-1 [refer to the resonant optical cavities tutorial] and $$\gamma_\text{out}=4.95\times10^8\text{ s}^{-1}$$ from Example 11-4 above. The internal quantum efficiency can then be found by using (11-91) to be

$\eta_\text{i}=\frac{\gamma_\text{c}}{\gamma_\text{out}}\eta_\text{e}=\frac{5.78\times10^8}{4.95\times10^8}\times68.4\%=80\%$

We see that, as expected, $$\eta_\text{i}$$ is the same as $$\eta_\text{p}$$, which is $$80\%$$ as given in Example 11-2 [refer to the laser oscillation tutorial].

This laser has a power conversion efficiency of $$4.9\%$$ compared to a slope efficiency of $$10.6\%$$. Compared to the high quantum efficiencies of $$\eta_\text{e}=68.4\%$$ and $$\eta_\text{i}=80\%$$, these power efficiencies are relatively low.

The power conversion efficiency can be increased by operating the laser at a higher pumping level, but it cannot exceed the slope efficiency, which is a constant before nonlinearity saturates the laser output at a significantly high pumping level.

The reason for $$\eta_\text{c}$$ to be always smaller than $$\eta_\text{s}$$ before saturation is that a laser has to overcome its threshold before it starts to oscillate.

The reason for the low power efficiencies in this example is that only $$20.4\%$$ of the pump power is absorbed by the gain medium because the pump beam passes through the gain medium in only one single pass. Therefore, close to $$80\%$$ of the input pump power simply passes through the gain medium without being utilized.

Both $$\eta_\text{c}$$ and $$\eta_\text{s}$$ for this laser can be increased by taking a multiple-pass arrangement or by properly increasing the length of the gain medium to increase the utilization factor $$\zeta_\text{p}$$ of the pump power.

However, quantum efficiencies are not increased by such steps. Indeed, if the cavity parameters are changed, the external quantum efficiency $$\eta_\text{e}$$ might even be reduced while the slope efficiency $$\eta_\text{s}$$ is increased.

The next tutorial covers the topic of pulsed lasers.

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