Resonant Optical Cavities
This is a continuation from the previous tutorial - rare-earth ion-doped fiber amplifiers.
There are a wide variety of lasers, covering a spectral range from the soft X-ray to the far infrared, delivering output powers from microwatts to terawatts, operating from continuous wave to femtosecond pulses, and having spectral linewidths from just a few hertz to many terahertz.
The gain media utilized include plasma, free electrons, ions, atoms, molecules, gases, liquids, solids, and so on.
The sizes range from microscopic, of the order of \(10\text{ μm}^3\), to gigantic, of an entire building, to stellar, of astronomical dimensions.
An optical gain medium can amplify an optical field through stimulated emission. If the gain medium is sufficiently long, it is possible to generate laser light at one end of the medium through amplification of some initial optical field from spontaneous emission produced at the other end of the gain medium.
Astrophysical laser action in space has been found to occur naturally, for example at the deep ultraviolet wavelength of 250 nm from the star Eta Carinae, at the near-infrared H2 wavelength of 2.286 μm from the star NGC 7072, at the far-infrared wavelength of 169 μm in a disk of hydrogen gas surrounding the star MWC349 in the constellation Cygnus, and at the mid-infrared CO2 wavelength of 10.6 μm in the Martian atmosphere.
In a practical laser device, however, it is generally necessary to have certain positive optical feedback in addition to optical amplification provided by a gain medium. This requirement can be met by placing the gain medium in an optical resonator. The optical resonator provides selective feedback to the amplified optical field.
Lasers are indeed fascinating, but not all of them are of practical usefulness as photonic devices. In this tutorial, we discuss the characteristics of laser oscillators in general. Optical fiber lasers are specifically discussed in a later tutorial. Semiconductor lasers are arguably the most important lasers in the photonics industry. They are covered in great detail in another later tutorial.
Resonant Optical Cavities
One major characteristic of laser light is that it is highly collimated and spatially and temporally coherent. This characteristic is a direct consequence of the fact that laser oscillation takes place only along a longitudinal axis of an optical resonator, which can be either straight or folded.
The gain medium emits spontaneous photons in all directions, but only the radiation that propagates along the longitudinal axis within a small divergence angle defined by the resonator obtains sufficient regenerative amplification to reach the threshold for oscillation.
In order for the oscillating laser field in the longitudinal direction to be amplified most efficiently, any spontaneous photon emitted in a direction outside of this small angular range should not be allowed to compete for the gain.
For this reason, a functional laser oscillator is necessarily an open cavity with optical feedback only along the longitudinal axis. Most of the randomly directed spontaneous photons escape from the cavity through the open sides very quickly.
Only a very small fraction of them that happen to be emitted within the divergence angle of the laser field mix with the oscillating laser field to become the major incoherent noise source of the laser.
A laser cavity can take a variety of forms. Figure 11-1 shows the schematic structures of a few common laser cavities.
Though a laser cavity is always an open cavity with a clearly defined longitudinal axis, the axis can lie on a straight line, as in Figures 11.1(a) and (e), or it can be defined by a folded path, as in Figures 11.1(b), (c), and (d).
A linear cavity with two end mirrors, as in Figure 11.1(a), is known as a Fabry-Perot cavity because it takes the form of a Fabry-Perot interferometer.
A folded cavity can simply be a folded Fabry-Perot cavity with a standing oscillating field, as in Figure 11-1(b). A folded cavity can also be a non-Fabry-Perot ring cavity that supports two independent oscillating fields traveling in opposite directions, as in Figures 11.1(c) and (d).
The optical feedback in a Fabry-Perot cavity is provided simply by the two end mirrors perpendicular to the longitudinal axis, as in Figures 11.1(a) and (b).
In a ring cavity, it is provided by the circulation of the laser field along the ring path defined by mirrors, as in Figure 11.1(c), or by a fiber waveguide, as in Figure 11.1(d).
It can also be supplied by the distributed feedback of a distributed Bragg grating along the axis, as in Figure 11.1(e).
The cavity can also be constructed with an optical waveguide, as in the case of a semiconductor laser or a fiber laser.
In the following discussions, we take the longitudinal axis to define the \(z\) coordinate, and the transverse coordinates perpendicular to the longitudinal axis to be the \(x\) and \(y\) coordinates. In a folded cavity, the \(z\) axis is thus also folded along with the longitudinal optical path.
In a ring cavity, an intracavity field completes one round trip by circulating inside the cavity in only one direction. The two contrapropagating fields that circulate in opposite directions in a ring cavity are independent of each other even when they have the same frequency.
In a Fabry-Perot cavity, an intracavity field has to travel the length of the cavity twice in opposite directions to complete a round trip.
The time it takes for an intracavity field to complete one round trip in the cavity is called the round-trip time, \(T\);
\[\tag{11-1}T=\frac{\text{round-trip optical path length}}{c}=\frac{l_\text{RT}}{c}\]
where the round-trip optical path length \(l_\text{RT}\) takes into account the refractive index of the medium inside the cavity.
A laser consists of at least a gain medium in a resonant laser cavity. The gain medium may fill up the entire length of the cavity, or it may occupy a fraction of the cavity length. For a gain medium of a length \(l_\text{g}\) in a laser cavity of a length \(l\), as shown in Figure 11-2, we can define an overlap factor between the gain medium and the laser mode intensity distribution as the ratio
\[\tag{11-2}\Gamma=\frac{\displaystyle\iiint\limits_{active}|\mathbf{E}|^2\text{d}x\text{d}y\text{d}z}{\displaystyle\int\limits_{-\infty}^\infty\int\limits_{-\infty}^\infty\int\limits_{-\infty}^\infty|\mathbf{E}|^2\text{d}x\text{d}y\text{d}z}\approx\frac{\mathcal{V}_\text{active}}{\mathcal{V}_\text{mode}}\approx\frac{l_\text{g}}{l}\]
This ratio is commonly known as the gain filling factor for a gain medium that takes up only a fraction of the length of the laser cavity but is related to the mode confinement factor in a waveguide laser, such as the fiber laser or the semiconductor laser.
When the gain medium fills up a laser cavity and covers the entire intracavity laser field distribution, \(\Gamma=1\); otherwise, \(\Gamma\lt1\). Take the refractive index of the gain medium to be \(n\) and that of the intracavity medium excluding the gain medium to be \(n_0\), then the round-trip optical path length can be expressed as
\[\tag{11-3}l_\text{RT}=\left\{\begin{array}{l}2\Gamma{nl}+2(1-\Gamma)n_0l=2\bar{n}l,\qquad\text{ for a linear cavity}\\\Gamma{nl}+(1-\Gamma)n_0l=\bar{n}l,\qquad\qquad\text{for a ring cavity}\end{array}\right.\]
where
\[\tag{11-4}\bar{n}=\Gamma{n}+(1-\Gamma)n_0\]
is the weighted average index of refraction throughout the laser cavity.
When optical elements other than a gain medium exist in a laser cavity, \(\bar{n}\) is still the weighted average index throughout the laser cavity with \(n_0\) being the weighted average index of the background medium and such optical elements.
Consider an intracavity filed, \(\mathbf{E}_\text{c}(z)\), at any point \(z\) on the longitudinal axis inside an optical cavity. When it completes a round trip back to position \(z\), it is modified by a complex amplification or attenuation factor \(a\) to become \(a\mathbf{E}_\text{c}(z)\). The factor \(a\) can be expressed generally as
\[\tag{11-5}a=G\exp(\text{i}\varphi_\text{RT})\]
where \(G\) is the round-trip gain factor for the field amplitude, equivalent to the power gain in a single pass through a linear Fabry-Perot cavity, and \(\varphi_\text{RT}\) is the round-trip phase shift for the intracavity field.
Both \(G\) and \(\varphi_\text{RT}\) have real values, and \(G\ge0\). If \(G\gt1\), the intracavity field is amplified. If \(G\lt1\), the intracavity field is attenuated.
Longitudinal Modes
We first consider the resonant characteristics of a passive optical cavity. A passive cavity cannot generate or amplify an optical field. In order to keep a resonant intracavity field in such a cavity, it is necessary to inject an input optical field, \(\mathbf{E}_\text{in}\), to the cavity constantly.
As shown in Figure 11-2, the forward traveling component of the intracavity field at location \(z_1\) just inside the cavity next to the injection point is the sum of the transmitted input field and the fraction of the intracavity field returning after one round trip through the cavity:
\[\tag{11-6}\mathbf{E}_\text{c}(z_1)=t_\text{in}\mathbf{E}_\text{in}+a\mathbf{E}_\text{c}(z_1)\]
where \(t_\text{in}\) is the complex transmission coefficient for the input field. We find that
\[\tag{11-7}\mathbf{E}_\text{c}(z_1)=\frac{t_\text{in}}{1-a}\mathbf{E}_\text{in}\]
The transmitted output field, \(\mathbf{E}_\text{out}\), is proportional to the intracavity field: \(\mathbf{E}_\text{out}\propto\mathbf{E}_\text{c}(z_1)\). Therefore, the output intensity is proportional to the input intensity through the following relationship:
\[\tag{11-8}I_\text{out}\propto\frac{I_\text{in}}{|1-a|^2}=\frac{I_\text{in}}{(1-G)^2+4G\sin^2(\varphi_\text{RT}/2)}\]
The proportionality constant of this relationship depends on the transmission coefficient of the output port and the amount of intracavity absorption over the distance from point \(z_1\) to the output point. The transmittance of the cavity is \(T_\text{c}=I_\text{out}/I_\text{in}\), which is scaled by the value of this proportionality constant.
For our discussions in the following, however, this proportionality constant is irrelevant. Therefore, we only have to consider the following normalized transmittance of the passive cavity:
\[\tag{11-9}\hat{T}_\text{c}=\frac{1}{1+[4G/(1-G)^2]\sin^2(\varphi_\text{RT}/2)}=\frac{1}{1+[(4/G)/(1-1/G)^2]\sin^2(\varphi_\text{RT}/2)}\]
which is obtained by normalizing \(T_\text{c}\) to its peak value.
In Figure 11-3, \(\hat{T}_\text{c}\) is plotted as a function of \(\varphi_\text{RT}\) for a few different values of \(G\). We find that the spectral shape for a gain of \(G\) is the same as that for a gain of \(1/G\).
At any given input field intensity, the intracavity field intensity of a passive cavity is proportional to \(\hat{T}_\text{c}\) because the transmitted field is directly proportional to the intracavity field. Therefore, resonances of the cavity occur at the peaks of \(\hat{T}_\text{c}\) where the intracavity intensity reaches a maximum with respect to a constant input field intensity.
As can be seen from Figure 11-3, the resonance condition of the cavity is that the round-trip phase shift is an integral multiple of \(2\pi\):
\[\tag{11-10}\varphi_\text{RT}=2q\pi,\qquad{q}=1,2,\dots\]
From (11-9) and Figure 11-3, we find that the separation between two neighboring resonance peaks of \(\hat{T}_\text{c}\) is
\[\tag{11-11}\Delta\varphi_\text{L}=2\pi\]
and that the FWHM of each resonance peak of the cavity is
\[\tag{11-12}\Delta\varphi_\text{c}=2\frac{1-G}{G^{1/2}}\]
The finesse, \(F\), of the cavity is the ratio of the separation to the FWHM of the peaks:
\[\tag{11-13}F=\frac{\Delta\varphi_\text{L}}{\Delta\varphi_\text{c}}=\frac{\pi{G}^{1/2}}{1-G}\]
In the simplest situation where the optical field is a plane wave at a frequency \(\omega\), the round-trip phase shift can be generally expressed as
\[\tag{11-14}\varphi_\text{RT}=\frac{\omega}{c}l_\text{RT}+\varphi_\text{local}\]
where the first term on the right-hand side is the phase shift contributed by the propagation of the optical field over an optical path length of \(l_\text{RT}\), and the second term, \(\varphi_\text{local}\), is the sum of all the localized, and usually fixed, phase shifts such as those caused by reflection from the mirrors of a cavity.
In the case when the frequency of the input field is fixed, the resonance condition given in (11-10) can be satisfied by varying the cavity path length \(l_\text{RT}\), either by varying the physical length of the cavity or by varying the refractive index of the intracavity medium, or both. The optical cavity then functions as an optical interferometer, which is used to measure the frequency and the spectral width of an optical wave accurately.
When both the optical path length and the localized phase shifts are fixed, as is typically the case in a laser resonator, the resonance condition of \(\varphi_\text{RT}=2q\pi\) is satisfied only if the optical frequency satisfies
\[\tag{11-15}\omega_\text{q}=\frac{c}{l_\text{RT}}(2q\pi-\varphi_\text{local})\]
or
\[\tag{11-16}\nu_\text{q}=\frac{c}{l_\text{RT}}\left(q-\frac{\varphi_\text{local}}{2\pi}\right)\]
These discrete resonance frequencies are the longitudinal mode frequencies of the optical resonator because they are defined by the resonance condition of the round-trip phase shift along the longitudinal axis of the cavity.
The frequency spacing, \(\Delta\nu_\text{L}\), between two neighboring longitudinal modes is known as the free spectral range of the optical resonator.
The FWHM of a longitudinal mode spectral peak is \(\Delta\nu_\text{c}\). If the values of \(l_\text{RT}\) and \(\varphi_\text{local}\) are independent of frequency, then \(\Delta\nu_\text{L}\propto\Delta\varphi_\text{L}\) and \(\Delta\nu_\text{c}\propto\Delta\varphi_\text{c}\).
Therefore, the finesse of the resonator is the ratio of the free spectral range to the longitudinal mode width:
\[\tag{11-17}F=\frac{\Delta\varphi_\text{L}}{\Delta\varphi_\text{c}}=\frac{\Delta\nu_\text{L}}{\Delta\nu_\text{c}}\]
From (11-16), we find that
\[\tag{11-18}\Delta\nu_\text{L}=\nu_\text{q+1}-\nu_\text{q}=\frac{c}{l_\text{RT}}=\frac{1}{T}\]
The longitudinal mode width can be expressed as
\[\tag{11-19}\Delta\nu_\text{c}=\frac{\Delta\nu_\text{L}}{F}=\frac{1-G}{\pi{G}^{1/2}}\Delta\nu_\text{L}\]
Transverse Modes
Any realistic optical cavity has a finite transverse cross-sectional area. Therefore, the resonant optical field inside an optical cavity cannot be a plane wave.
Indeed, there exist certain normal modes for the transverse field distribution in a given optical cavity. Such transverse field patterns are known as the transverse modes of a cavity.
A transverse mode of an optical cavity is a stable transverse field pattern that reproduces itself after each round-trip pass in the cavity, except that it might be amplified or attenuated in magnitude and shifted in phase.
The transverse modes of an optical cavity are defined by the transverse boundary conditions that are imposed by the transverse cross-sectional index profile of the cavity.
For a cavity that utilizes an optical waveguide for lateral confinement of the optical field, the transverse modes are clearly the waveguide modes, such as the TE and TM modes of a slab waveguide or the TE, TM, HE, and EH modes of a cylindrical fiber waveguide.
For a nonwaveguiding cavity, the transverse modes are the TEM fields determined by the shapes and sizes of the end mirrors of the cavity, as well as by the properties of the medium and any other optical components inside the cavity. The Gaussian modes discussed in the Gaussian beam tutorial are an important set of such unguided TEM modes.
In an optical cavity that supports multiple transverse modes, the round-trip phase shift is generally a function of the transverse mode indices \(m\) and \(n\). Therefore, the resonance condition can be explicitly written as
\[\tag{11-20}\varphi_{mn}^\text{RT}=2q\pi\]
As a result, the resonance frequencies of the cavity are \(\omega_{mnq}\), or \(\nu_{mnq}\), which are functions of both longitudinal and transverse mode indices.
For a given longitudinal mode index \(q\), multiple resonance frequencies associated with different transverse modes can exist, as illustrated schematically in Figure 11-4.
In a cavity that consists of an optical waveguide, the propagation constant \(\beta_{mn}(\omega)\) is a function of the waveguide mode. If the physical length of the waveguide cavity is \(l\), the effective round-trip optical path length of a waveguide is
\[\tag{11-21}l_{mn}^\text{RT}=\left\{\begin{array}{l}2\frac{c}{\omega}\beta_{mn}(\omega)l,\qquad\text{for a linear cavity}\\\frac{c}{\omega}\beta_{mn}(\omega)l,\qquad\text{ for a ring cavity}\end{array}\right.\]
The round-trip optical path length, \(l_{mn}^\text{RT}\), generally varies from one mode to another due to modal dispersion of the waveguide. In addition, the localized phase shift can also be mode dependent. Therefore, instead of \(\omega_q\) given by (11-15) for a plane wave, the resonance frequencies \(\omega_{mnq}\) for a waveguide cavity are the solutions of the following resonance condition:
\[\tag{11-22}\varphi_{mn}^\text{RT}=\frac{\omega}{c}l_{mn}^\text{RT}+\varphi_{mn}^\text{local}=2q\pi\]
In a nonwaveguiding cavity, the propagation constant, \(k\), is a property of the medium only and is not mode dependent. However, a mode-dependent on-axis phase variation \(\xi_{mn}(z)\) does exist, which is given in (140) for a Hermite-Gaussian mode as discussed in the Gaussian beam tutorial.
The total on-axis phase variation for the \(\text{TEM}_{mn}\) Gaussian mode is \(\varphi_{mn}(z)=kz+\xi_{mn}(z)\), which includes the mode-independent phase shift \(kz\) and the mode-dependent phase shift \(\xi_{mn}(z)\). Consequently, the resonance condition for a Gaussian mode is a modification of that for a plane wave by adding the round-trip contribution of the mode-dependent phase shift:
\[\tag{11-23}\varphi_{mn}^\text{RT}=\frac{\omega}{c}l_\text{RT}+\xi_{mn}^\text{RT}+\varphi_{mn}^\text{local}=2q\pi\]
where the localized phase shift can, in general, be mode dependent.
Cavity Lifetime and Quality Factor
Here we consider some important parameters of a passive optical cavity with no optical gain. Such a passive optical cavity with \(\chi_\text{res}=0\), thus \(g=0\), is also known as a cold cavity. To be specific, we identify the round-trip gain factor for the field amplitude in a cold cavity as \(G_\text{c}\), or as \(G_{mn}^\text{c}\) for the transverse mode \(mn\).
Because there is no optical gain in a cold cavity, \(G_\text{c}\lt1\). Any optical field that initially exists in the cavity gradually decays as it circulates inside the cavity. Because the field amplitude is attenuated by a factor \(G_\text{c}\) per round trip, the intensity and thus the number of intracavity photons are attenuated by a factor of \(G_\text{c}^2\) per round trip. We can define a photon lifetime, also called cavity lifetime, \(\tau_\text{c}\), and a cavity decay rate, \(\gamma_\text{c}\), for a cold cavity through the following relation:
\[\tag{11-24}G_\text{c}^2=\text{e}^{-T/\tau_\text{c}}=\text{e}^{-\gamma_\text{c}T}\]
Therefore,
\[\tag{11-25}\tau_\text{c}=-\frac{T}{2\ln{G_\text{c}}}\]
The cavity decay rate is the decay rate of the optical energy stored in a cavity and is given by
\[\tag{11-26}\gamma_\text{c}=\frac{1}{\tau_\text{c}}=-\frac{2}{T}\ln{G_\text{c}}\]
In general, the value of \(G_\text{c}\) for a given cavity is mode dependent. Usually, the fundamental transverse mode has the lowest loss because its field distribution is most transversely concentrated toward the center of the cavity defined by the longitudinal axis.
As the order of a mode increases, its loss in the cavity increases due to the increased diffraction loss associated with transverse spreading of its field distribution.
Consequently, both \(\tau_\text{c}\) and \(\gamma_\text{c}\) are also mode dependent: \(\tau_{mnq}^\text{c}\) and \(\gamma_{mnq}^\text{c}\). Unless a specific mode-discriminating mechanism is introduced in a cavity, either intentionally or unintentionally, the fundamental mode generally has the largest value of \(\tau_\text{c}\) and the lowest value of \(\gamma_\text{c}\).
The quality factor, \(Q\), of a resonator is generally defined as the ratio of the resonance frequency and the energy damping rate of the resonator:
\[\tag{11-27}Q=\omega_\text{res}\left(\frac{\text{energy stored in the resonator}}{\text{average power dissipation}}\right)=\frac{\omega_\text{res}}{\gamma}\]
where \(\omega_\text{res}\) is the resonance frequency of the resonator and \(\gamma\) is the energy decay rate of the resonator.
Therefore, the quality factor of a cold cavity is
\[\tag{11-28}Q=\frac{\omega_q}{\gamma_\text{c}}=\omega_q\tau_\text{c}\]
where \(\omega_q\) is the longitudinal mode frequency.
For a low-loss, high-\(Q\) cavity, \(G_\text{c}\) is not much less than unity, and it can be easily shown by using (11-19) and (11-24) that
\[\tag{11-29}\Delta\nu_\text{c}\approx\frac{1}{2\pi\tau_\text{c}}=\frac{\gamma_\text{c}}{2\pi}\]
and
\[\tag{11-30}Q\approx\frac{\nu_q}{\Delta\nu_\text{c}}\]
Note that though it is not explicitly spelled out in (11-28) and (11-30), the quality factor is a function of not only the longitudinal-mode index \(q\) but also the transverse-mode indices \(m\) and \(n\): \(Q=Q_{mnq}\). To be precise, (11-28) should be written as
\[\tag{11-31}Q_{mnq}=\frac{\omega_{mnq}}{\gamma_{mnq}^\text{c}}=\omega_{mnq}\tau_{mnq}^\text{c}\]
For an optical cavity, the dependence of \(Q_{mnq}\) on the longitudinal-mode index \(q\) is generally negligible because \(q\) is a very large value except in the case of a very short microcavity.
However, the dependence of \(Q_{mnq}\) on the transverse-mode indices \(m\) and \(n\) cannot be ignored. Indeed, \(Q_{00q}\) for the fundamental transverse mode is generally larger than \(Q_{mnq}\) for any high-order transverse mode because the fundamental transverse mode generally has the lowest loss.
Fabry-Perot Cavity
The most common laser cavity is a Fabry-Perot cavity consisting of two end mirrors and an optical gain medium, shown in Figure 11-5.
The radii of curvature of the left and right mirrors are \(\mathcal{R}_1\) and \(\mathcal{R}_2\), respectively. The sign of the radius of curvature is taken to be positive for a concave mirror and negative for a convex mirror. For the cavity shown in Figure 11-15, which is formed with two concave mirrors, \(\mathcal{R}_1\gt0\) and \(\mathcal{R}_2\gt0\).
Most of the important features of a nonwaveguiding Fabry-Perot laser cavity can be obtained by applying the following simple concept: for the cavity to be a stable cavity in which a Gaussian mode can establish, the radii of curvature of both end mirrors have to match the wavefront curvatures of the Gaussian mode at the surfaces of the mirrors: \(\mathcal{R}(z_1)=-\mathcal{R}_1\) and \(\mathcal{R}(z_2)=\mathcal{R}_2\), where \(z_1\) and \(z_2\) are, respectively, the coordinates of the left and right mirrors measured from the location of the Gaussian beam waist.
Based on this concept, we have from (136) [refer to the Gaussian beam tutorial] the following two relations:
\[\tag{11-32}z_1+\frac{z_\text{R}^2}{z_1}=-\mathcal{R}_1\qquad\text{and}\qquad{z_2}+\frac{z_\text{R}^2}{z_2}=\mathcal{R}_2\]
From these relations, we find that
\[\tag{11-33}z_\text{R}^2=\frac{l(\mathcal{R}_1-l)(\mathcal{R}_2-l)(\mathcal{R}_1+\mathcal{R}_2-l)}{(\mathcal{R}_1+\mathcal{R}_2-2l)^2}\]
where \(l=z_2-z_1\) is the length of the cavity defined by the separation between the mirrors.
Given the values of \(\mathcal{R}_1\), \(\mathcal{R}_2\), and \(l\), stable Gaussian modes exist for the cavity if both relations in (11-32) can be satisfied with a positive, real parameter \(z_\text{R}\gt0\) for a finite, positive spot size \(w_0\) according to (134) [refer to the Gaussian beam tutorial]. Then, the cavity is stable.
If the relations in (11-32) cannot be simultaneously satisfied with a positive, real value for \(z_\text{R}\), then the cavity is unstable because no stable Gaussian mode can be established in the cavity.
Application of this concept yields the following stable criterion for a Fabry-Perot cavity:
\[\tag{11-34}0\le\left(1-\frac{1}{\mathcal{R}_1}\right)\left(1-\frac{1}{\mathcal{R}_2}\right)\le1\]
In a stable resonator cavity, the mode-dependent on-axis phase shift in a single pass through the cavity from the left to the right mirror for the \(\text{TEM}_{mn}\) Hermite-Gaussian mode is simply \(\zeta_{mn}(z_2)-\zeta_{mn}(z_1)\). Therefore, the round-trip mode-dependent on-axis phase shift is
\[\tag{11-35}\zeta_{mn}^\text{RT}=2[\zeta_{mn}(z_2)-\zeta_{mn}(z_1)]\]
With some modifications, the same concept can be used to find the characteristics and stability criterion of a cavity with multiple mirrors, such as a folded Fabry-Perot cavity or a ring cavity.
We consider a cavity that contains an isotropic gain medium with a filling factor \(\Gamma\). The surfaces of the gain medium are antireflection coated so that there is no reflection inside the cavity other than the reflection at the two end mirrors. If the gain medium fills up the entire cavity, we simply make \(\Gamma=1\) in the results obtained below.
The Fabry-Perot cavity has a physical length \(l\) between the two end mirrors. The field amplitude reflection coefficients are \(r_1\) and \(r_2\) for the left and right mirrors, respectively. They are generally complex to account for the phase changes on reflection, \(\varphi_1\) and \(\varphi_2\), respectively, and can be written as
\[\tag{11-36}r_1=R_1^{1/2}\text{e}^{\text{i}\varphi_1},\qquad{r_2}=R_2^{1/2}\text{e}^{\text{i}\varphi_2}\]
where \(R_1\) and \(R_2\) are the intensity reflectivities of the left and right mirrors, respectively.
The dielectric property of the intracavity gain medium contains the permittivity of the background material plus a contribution from the resonant susceptibility, \(\chi_\text{res}(\omega)\), that characterizes the laser transition.
To identify the effect of each contribution clearly, it is instructive to express the permittivity of the gain medium explicitly, including the contribution of the resonant laser transition, as
\[\tag{11-37}\epsilon_\text{res}(\omega)=\epsilon(\omega)+\epsilon_0\chi_\text{res}(\omega)\]
where \(\epsilon(\omega)=n^2\) is the background permittivity of the gain medium excluding the contribution of the resonant laser transition.
In a cold cavity, \(\chi_\text{res}=0\). Therefore, the weighted average of the propagation constant for the intracavity field in a cold cavity is
\[\tag{11-38}\bar{k}=\frac{\bar{n}\omega}{c}=\Gamma{k}+(1-\Gamma)k_0\]
where \(k=n\omega/c\) is the propagation constant in the gain medium and \(k_0=n_0\omega/c\) is that in the surrounding background medium. The round-trip optical path length in this cavity is \(l_\text{RT}=2\bar{n}l\).
Usually there is an intracavity background loss contributed by a variety of different mechanisms, such as scattering or absorption, that are irrelevant to the laser transition. In addition, mode-dependent diffraction losses exist for the intracavity optical field due to the finite sizes of the end mirrors.
The combined effect of these losses can be accounted for by taking a spatially averaged, mode-dependent loss coefficient, \(\bar{\alpha}_{mn}\), so that the effective propagation constant is complex with a mode-dependent imaginary part: \(\bar{k}+\text{i}\bar{\alpha}_{mn}/2\). This loss is known as the distributed loss of the laser cavity mode. In general, \(\bar{\alpha}\ll\bar{k}\) for any practical gain medium.
Based on (11-5), by following a mode field over one round trip in the cavity, we find that
\[\tag{11-39}a=r_1r_2\exp(\text{i}2\bar{k}l-\bar{\alpha}_{mn}l+\text{i}\zeta_{mn}^\text{RT})\]
for the \(\text{TEM}_{mn}\) Hermite-Gaussian mode.
Therefore, both the round-trip gain factor and the round-trip phase shift are mode dependent:
\[\tag{11-40}G_{mn}^\text{c}=R_1^{1/2}R_2^{1/2}\exp(-\bar{\alpha}_{mn}l)\]
and
\[\tag{11-41}\varphi_{mn}^\text{RT}=2\bar{k}l+\zeta_{mn}^\text{RT}+\varphi_1+\varphi_2\]
Using (11-41) for the resonance condition in (11-20), we find the following resonance frequencies of the cold Fabry-Perot cavity:
\[\tag{11-42}\omega_{mnq}^\text{c}=\frac{c}{2\bar{n}l}(2q\pi-\zeta_{mn}^\text{RT}-\varphi_1-\varphi_2)\]
and \(\nu_{mnq}^\text{c}=\omega_{mnq}^\text{c}/2\pi\), where the superscript \(\text{c}\) indicates the fact that the frequencies are associated with a cold cavity with \(\chi_\text{res}=0\).
These frequencies are clearly functions of the transvers-mode indices because of the mode-dependent phase shift \(\zeta_{mn}^\text{RT}\). However, because \(\zeta_{mn}^\text{RT}\) is not a function of the longitudinal-mode index \(q\), the frequency separation between different longitudinal modes of the same transverse mode group is a constant:
\[\tag{11-43}\Delta\nu_\text{L}=\nu_{mn,q+1}^\text{c}-\nu_{mnq}^\text{c}=\frac{c}{2\bar{n}l}=\frac{1}{T}\]
Here we assume that the background optical property of the medium is not very dispersive so that the background refractive index \(\bar{n}\) can be considered a constant independent of optical frequency in the narrow range between neighboring modes of interest.
Using (11-13) and (11-40), the finesse of the lossy cavity is
\[\tag{11-44}F=\frac{\pi{R}_1^{1/4}R_2^{1/4}\text{e}^{-\bar{\alpha}_{mn}l/2}}{1-R_1^{1/2}R_2^{1/2}\text{e}^{-\bar{\alpha}_{mn}l}}\]
which is mode dependent due to the mode-dependent loss \(\bar{\alpha}_{mn}\). The longitudinal mode width, \(\Delta\nu_\text{c}=\Delta\nu_\text{L}/F\), is also mode dependent for the same reason.
For a cavity with a negligible loss, we can take \(\bar{\alpha}_{mn}=0\); then, the expression in (11-44) reduces to the familiar formula for the finesse of a lossless Fabry-Perot interferometer:
\[\tag{11-45}F=\frac{\pi{R}_1^{1/4}R_2^{1/4}}{1-R_1^{1/2}R_2^{1/2}}\]
Therefore, for a nondispersive, lossless Fabry-Perot cavity, \(\Delta\nu_\text{L}\), \(F\), and \(\Delta\nu_\text{c}\) are also independent of longitudinal and transverse mode indices though the mode frequency \(\nu_{mnq}\) is a function of all three mode indices.
Using (11-25) and (11-40), the mode-dependent photon lifetime of the Fabry-Perot cavity can be expressed as
\[\tag{11-46}\tau_{mnq}^\text{c}=\frac{\bar{n}l}{c(\bar{\alpha}_{mn}l-\ln\sqrt{R_1R_2})}\]
and the mode-dependent cavity decay rate as
\[\tag{11-47}\gamma_{mnq}^\text{c}=\frac{c}{\bar{n}}\left(\bar{\alpha}_{mn}-\frac{1}{l}\ln\sqrt{R_1R_2}\right)\]
Clearly, both \(\tau_{mnq}^\text{c}\) and \(\gamma_{mnq}^\text{c}\) are also mode dependent due to the mode-dependent distributed loss \(\bar{\alpha}_{mn}\). However, they are independent of the longitudinal mode index \(q\) under the assumption that the background refractive index \(\bar{n}\) and the loss \(\bar{\alpha}_{mn}\), as well as the mirror reflectivities \(R_1\) and \(R_2\), are not sensitive to the frequency differences among different longitudinal modes. If any of these parameters vary significantly within the range of longitudinal modes of interest, then the dependence of \(\tau_{mnq}^\text{c}\) and \(\gamma_{mnq}^\text{c}\) on the index \(q\) cannot be ignored.
The Fabry-Perot cavity for a typical laser is a high-\(Q\) cavity. Even in a high-gain laser with low mirror reflectivities, \(Q\) is still very large. For example, consider a high-gain InGaAsP/InP semiconductor laser emitting at 1.3 μm wavelength with \(n=3.5\), \(l=300\text{ μm}\), and \(R_1=R_2=0.3\). Assuming a negligibly small \(\bar{\alpha}\) for simplicity, we find that \(\tau_\text{c}=2.9\text{ ps}\), \(T=7\text{ ps}\), \(\Delta\nu_\text{L}=142.86\text{ GHz}\), \(F=2.46\), and \(\Delta\nu_\text{c}=58\text{ GHz}\). Using (11-28), we obtain \(Q=4.2\times10^3\), while the approximate relation (11-30) yields a slightly smaller value of \(Q=4.0\times10^3\). A \(Q\) value on the order of \(10^3\) is relatively low for a laser cavity. Even so, the difference between (11-30) and (11-28) is only about \(5\%\). For a low-loss cavity, \(Q\) can easily be as high as \(10^8\), and the result obtained from (11-30) is essentially the same as that from (11-28).
Example 11-1
A Nd : YAG microchip laser, shown in Figure 11-6, is made of a Nd : YAG crystal of the same properties as that of the Nd : YAG laser amplifier described in Example (10-9) [refer to the laser amplifiers tutorial], except that it is thinner and its surfaces are coated differently to form a laser cavity. It consists of parallel Nd : YAG plates of 500 μm thickness. The surfaces of the plate are coated for \(R_1=100\%\) and \(R_2=99.7\%\) at the 1.064 μm laser wavelength to form the laser cavity but for \(R_1=R_2=0\) at the 808 nm pump wavelength to allow only a single pass of the pump beam. The refractive index of Nd : YAG is \(n=1.82\). The distributed loss of the laser cavity is found to be \(\bar{\alpha}=0.5\text{ m}^{-1}\).
(a) Find the round-trip optical path length, the round-trip time, and the longitudinal mode spacing of this cavity.
(b) Find the finesse of this cavity.
(c) What are the cavity decay rate and the photon lifetime?
(d) What are the longitudinal mode width and the \(Q\) factor of the cold cavity?
(a)
This is a Fabry-Perot laser cavity with a filling factor of \(\Gamma=1\). Therefore, \(\bar{n}=n=1.82\). Then the round-trip optical path length is
\[l_\text{RT}=2nl=2\times1.82\times500\text{ μm}=1.82\text{ mm}\]
The round-trip time is
\[T=\frac{l_\text{RT}}{c}=\frac{1.82\times10^{-3}}{3\times10^8}\text{ s}=6.07\text{ ps}\]
The longitudinal mode spacing is
\[\Delta\nu_\text{L}=\frac{1}{T}=\frac{1}{6.07\times10^{-12}}\text{ Hz}=164.8\text{ GHz}\]
(b)
The finesse of this cavity has to be found by using (11-44), not (11-45), because there is a distributed loss of \(\bar{\alpha}=0.5\text{ m}^{-1}\). For \(l=500\text{ μm}\), \(\bar{\alpha}l=2.5\times10^{-4}\). With \(R_1=100\%\) and \(R_2=99.7\%\), we find that the finesse is
\[F=\frac{\pi{R_1}^{1/4}R_2^{1/4}\text{e}^{-\bar{\alpha}l/2}}{1-R_1^{1/2}R_2^{1/2}\text{e}^{-\bar{\alpha}l}}=\frac{\pi\times(0.997)^{1/4}\times\exp(-2.5\times10^{-4}/2)}{1-(0.997)^{1/2}\times\exp(-2.5\times10^{-4})}=1793\]
(c)
The cavity decay rate can be calculated using (11-47):
\[\gamma_\text{c}=\frac{c}{\bar{n}}\left(\bar{\alpha}-\frac{1}{l}\ln\sqrt{R_1R_2}\right)=\frac{3\times10^8}{1.82}\times\left(0.5-\frac{\ln\sqrt{0.997}}{500\times10^{-6}}\right)\text{ s}^{-1}=5.78\times10^8\text{ s}^{-1}\]
Then, the photon lifetime is
\[\tau_\text{c}=\frac{1}{\gamma_\text{c}}=\frac{1}{5.78\times10^8}\text{ s}=1.73\text{ ns}\]
(d)
Using the definition of the finesse in (11-17), or (11-19), we find the following longitudinal mode width for the cold cavity:
\[\Delta\nu_\text{c}=\frac{\Delta\nu_\text{L}}{F}=\frac{164.8\times10^9}{1793}\text{ Hz}=91.91\text{ MHz}\]
Using the approximate relation give in (11-29), we find that \(\Delta\nu_\text{c}\approx\gamma_\text{c}/2\pi=91.99\text{ MHz}\), which is almost the same as the accurate value obtained above. Using the definition given in (11-28), the \(Q\) factor of the cold cavity is found to be
\[Q=\frac{\omega}{\gamma_\text{c}}=\frac{2\pi{c}}{\lambda\gamma_\text{c}}=\frac{2\pi\times3\times10^8}{1.064\times10^{-6}\times5.78\times10^8}=3.065\times10^6\]
Though the cavity is very short, it is a high-\(Q\) cavity because of the very high reflectivities of both of its end mirrors. If we use the approximate relation in (11-30), we find \(Q=3.068\times10^6\), which is only slightly different from that found by using (11-28). For a high-\(Q\) cavity, there is little difference between (11-30) and (11-28) for the \(Q\) value and between (11-29) and (11-17) for the mode width.
The next tutorial covers laser oscillation.