1. Introduction

A desire for options beyond the Gaussian beam profiles produced by spherical-mirror laser resonators has spurred researchers to investigate a variety of other cavity designs in order to produce custom beam shapes [1]. They proceed as follows. First, the designer specifies a desired intensity profile at the front mirror in terms of its amplitude f that may be taken as pure real in the usual case of a flat output mirror. Next, the designer calculates the field as propagated a distance l to the back mirror, extracts the wavefront shape Φ (here expressed as a phase), and fabricates a back mirror with a matching profile. Under the assumption that both mirrors are infinite in extent, placing such a mirror a distance l behind the front mirror results in a cavity with a mode having the specified field f at the output of the front mirror. Furthermore, this procedure works well with finite mirrors provided that the intensity of the desired mode is negligible at the mirror edge.

Of course, this only specifies one mode of the cavity. All other cavity modes must simply be accepted as side effects of the design procedure. The designer does not know beforehand either their spatial distributions or their losses. The latter is particularly important insofar as the designer probably only wants the designed mode to oscillate; consequently its low loss should contrast with the high losses of all the others. Indeed, the implicit assumption of infinite mirrors yields modes with no diffraction loss and the design procedure remains ignorant of the loss of even the designed mode. The lack of a model for the relative mode losses makes it difficult to design a selective cavity by anything but trial and error. While efforts [2] have been made in this regard, the results have been heuristic at best.

In [3], Paré et al. showed that for cavities short compared to the Rayleigh range, called here “near-field” cavities, the resonator eigenvalue problem (in the absence of mirror edges) could be approximated by a time-independent Schrödinger equation with a potential term determined by the desired fundamental mode. Higher order modes appeared as additional solutions to this equation. This suggested describing the laser modes either as corresponding to bound states that rapidly decayed away from the mirror center or as corresponding to unbound states that retained significant amplitude away from the mirror center. Presumably, with a sufficiently large finite mirror the former could be made low loss while the latter would inevitably remain lossy, a suspicion confirmed by the authors’ numeric cavity loss calculations. Nevertheless, the analytic results remain restricted to the assumption of infinite mirrors and do not yield mode losses.

Similarly, in [4], Kuznetsov et al. argued via an analogy between the cavity modes, quantum mechanics, and fiber waveguide modes that there should be a categorical difference between bound and unbound modes. This led to a classification of modes into bound and unbound based on the mirror sag and the frequency offset of a given mode essentially identical to [3]. Experimentally they showed the feasibility of building microcavities with a single bound mode.

The present work applies the methods of Weinstein [5, 6] to calculate the losses of custom laser cavities for both bound and unbound modes. Previously used for Fabry-Perot [5] and spherical resonators, both stable [7] and unstable [8, 9], this method tips the cavity on its side, treats it as a waveguide, and calculates the power radiated out the ends of the waveguide (ie., diffracted past the mirror edges). This method proves eminently suitable for the cavity geometries of present interest and quantitatively justifies previous intuition about the qualitatively different behaviors of bound and unbound modes. We find that the bound modes have losses that decline exponentially with mirror width and have frequencies that remain well separated. In contrast, we find that the unbound modes have losses with only inverse power law dependence on mirror size and tend to become frequency degenerate.

2. Basics of the waveguide approach

Consider a flat front mirror a distance l from a back mirror with sag Φ/k and width 2a. To simplify the presentation, we take this to be one-dimensional. Call the field at the flat front mirror f. To describe the field at the rear mirror, unfold the cavity into a lens train as in Fig. 1, split the effect of the mirror into two identical exp(-i Φ) phase screens, and call the field between the screens g. Note that when f is a mode of the cavity and is phase conjugated by the back mirror, the wavefront of g will be flat. Assume a suppressed time dependence exp(-iωt), where ω may be complex to allow for the decay of the modal field, let k = ω/c, introduce a dimensionless coordinate ξ = x√k/(2l) related to the square root of the Fresnel zone number as seen across the cavity and let M = √2ka ²/l define the spatial extent of the back mirror. It proves convenient to write the round trip phase as 2kl = 2π(q + p), where the longitudinal mode number q is an integer and |p| < 1. Define χ = 2πp, whose real and imaginary parts will relate to the cavity frequency shift and loss respectively. Then using Fresnel propagation and enforcing consistency across one period of the lens train yields

 

Fig. 1. Laser cavity unfolded into a lens train.

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This equation proves especially convenient in the near-field case where the fundamental mode is wide and smooth enough to acquire minimal wavefront curvature at the back mirror In the opposite far-field case, when the fundamental mode is narrow enough to have an essentially spherical wavefront with radius l at the back mirror, it proves convenient to pull out this dominant effect by defining Φ(ξ) = ξ ² - π/4 - $\tilde{\Phi }$(ξ) and obtain

Note that even simple ray tracing of the hemispheric baseline cavity leads to the conclusion that g(ξ) should depend on g(-ξ) after one round trip. A symmetric mirror Φ(ξ) = Φ(-ξ) makes the cavity modes either even or odd and permits the assumption g(-ξ) = ±g(-ξ). Thus let χ = (m - 1)π - $\tilde{\chi }$, m = 1,2,3,‧ and write

Here m serves as a mode number, (m - 1)π gives the frequency shift (half the longitudinal mode spacing) of the m ^th transverse mode of a simple hemispheric $\tilde{\Phi }$ = 0 cavity and $\tilde{\chi }$ represents the additional frequency shift and loss of such a cavity made aspheric and truncated. While the content of (2) is exactly equivalent to (1), Sec. 3 shows that they yield approximations valid in physically distinct scenarios.

To begin solving (1), first consider the case of an infinite plane-parallel (Fabry-Perot) resonator, Φ = 0 and M = ∞. Fourier transforming the integral equation gives g̃(s) = exp(iχ - is ²/2)g̃(s), implying that g̃(s) = δ(s - S_j), where

and j is an integer. The modes g(χ) = exp(±is_jξ) then correspond to waveguide modes with propagation parameter s_j along the ξ-direction. What follows builds the laser cavity modes out of these waveguide modes. Of course, the same technique solves Eq. (2) with Φ = 0. While a hemispheric cavity may not seem like a waveguide, the similarity of (1) and (2) means that viewing such a cavity as an equivalent waveguide allows effortless adaptation of near-field results to the far-field case.

2.1. Finite plane-parallel and hemispheric cavities

Next, consider a finite plane-parallel (Φ = 0) or hemispheric (Φ̃ = 0) resonator. Due to the rapid oscillation of the Fresnel kernel in (1) and (2), the field g near ξ = -M/2 after one round trip will be insensitive to the exact nature of the field for ξ ≫ 0 and, for sufficiently wide cavities (conservatively, M > 4√π [10]), the upper limit of integration ξ = M/2 in (1) or (2) may be replaced by ξ = ∞. This motivates the study of the half-infinite plane-parallel waveguide with modes described by

The Weiner-Hopf method [6], which we briefly review in the appendix, yields solutions near the left end of the guide as a sum of a waveguide mode exp(-is_jξ) incident on the open end and a series of reflected waveguide modes R_jk exp (is_kξ). Shifting this solution along with the waveguide opening to ξ = M/2, using similar reasoning near the right edge ξ = M/2, and enforcing consistency in the central region where these descriptions overlap results in a large system of equations. However, for flat cavities, p ≪ 1, allowing the approximation [6]

that further implies |R ₀₀| ≫ |R_jk| for j,k ≠ 0. This yields a simple model for the resonator modes, namely a single waveguide mode e ^±is₀ξ reflected back and forth by R ₀ = - exp[i(1 + i)βs ₀] at each end and a simple self-consistency condition, exp(i2s ₀ M)R ² ₀ = 1, so that

for positive integers m, as obtained in [5].

2.2. General finite cavities

More generally, suppose the back mirror consists of a possibly complicated central region around ξ ≈ 0 and sections with simpler functional forms to the left (ξ ≫ 0) and right (ξ ≪ 0). Suppose that, as illustrated in Fig. 2, far to the left, the field can be approximated by leftward and rightward traveling waveguide modes Bu _-(ξ) and Au ₊(ξ) and that, far to the right, the field can be approximated by leftward and rightward traveling waveguide modes Dv _- (ξ) and Dv ₊ (ξ). Here A,B,C, and D are constants related by a scattering matrix S:

 

Fig. 2. Reflected and transmitted modes in a waveguide.

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Further suppose that the reflections at the left and right edges are given by coefficients R_u and R_v so that

and the resonator eigenvalue equation becomes

Experience with spherical [7, 8, 9] and tilted [11] cavities, as well as more general theoretical considerations [12], suggests that the reflections at the ends of the guide are essentially identical to the simpler plane-parallel case. Shifting the reference points from the mirror edges gives

where

is the propagation parameter of the mode at the left edge and similarly for s_v at the right edge. For the u’s and v’s we typically take asymptotic forms valid as x → ±∞. This proves reasonable provided the mirror is not truncated too close to its central feature. In the case of most interest presently, where the mirrors are asymptotically flat, we take u ₊ (ξ) = v ₊(ξ) = exp(isξ) and u _-(ξ) = v _-(ξ) = exp(-isξ), with common propagation parameter s. With the additional assumption that the back mirror has left-right symmetry, S and R may be written as

where r and t are the reflection and transmission coefficients of the central feature and I is the identity matrix. Furthermore, the eigenvalues λ _±=r±t of S correspond to even and odd modes and reduce the resonator eigenvalue equation (7) to [1 - λ ₊ exp(isM)R ₀][1 - λ - exp(isM)R ₀] = 0.

3. Perturbation calculation of the mirror phase, waveguide modes, and resonator eigenvalues

There remains the problem of calculating S. We begin by finding infinite mirror modes, M → ∞. In the case of asymptotically flat cavities with wide fundamental modes we can proceed as in [3]. This section reviews that method and shows how it may be applied to the case of asymptotically hemispheric cavities with narrow fundamental modes. To cast the resonator integral Eqs. (1) and (2) as a perturbation problem, introduce a length scale 1/ε and take g in the form g(ξ) = ψ(εξ). Introducing $\overline{\xi }$ = εξ and expanding the Fresnel integral via steepest descent yields, in the near-field case,

where

and similarly in the far-field case (let χ → $\tilde{\chi }$, Φ → $\tilde{\Phi }$ and i → -i).

Finding the mirror profile requires separate consideration of the near and far-field cases. In the near-field case, take the desired fundamental mode f(ξ) on the flat mirror, propagate to the back mirror, and extract the wavefront shape:

where Ψ($\tilde{\xi }$) = f($\overline{\xi }$/ε). For this expansion to be useful requires the operator (ε/2)∂/∂ $\overline{\xi }$ = (1/2)∂/∂ξ to be small when applied to the Ψ of interest. Note that {¼∫|∂f/∂ξ|²dξ}^1/2 = σ _f̃/2 where σ² _f̃ is the second moment of the normalized angular power spectrum (in units of k/2l). More convenient may be the form M ² _Ψ/4σ_f where σ_f ² is the second intensity moment of f (in units of 2l/k) and M _Ψ ² is the beam quality factor. Hence the above approximation requires a wide mode with a simple beam structure.

In the far-field case, with f(ξ) still the desired fundamental mode at the front mirror, write

Recognizing the Fourier integral form of the above, letting f̃ be the Fourier transform of f, using the convolution theorem, and letting $\tilde{\Psi }$($\overline{\xi }$) = f̃(2ε $\overline{\xi }$) yields

We again require the operator (ε/2)∂/∂ $\overline{\xi }$ = (1/2)∂/∂ξ to be small when applied to the $\tilde{\Psi }$ of interest. In this case applying it to a particular fundamental mode f yields validity in the limit σ_f/√2π → 0. Note that now only the rms spot size is relevant.

As the order-ε ² terms of the expansions (10) and (11) are of special interest, define

in the near-field case and

in the far-field case. For convenience, when considering asymptotically flat or hemispheric mirrors, take V($\overline{\xi }$) → 0 as $\overline{\xi }$ → ± ∞, making the constant P the first order shift of the resonant frequency of the fundamental mode with respect to that of a plane-parallel or hemispheric cavity.

Furthermore, expanding an arbitrary mode ψ and eigenvalue exp(iχ) as

and similarly for $\tilde{\chi }$ and using (9) and (12) in (1) and (2) yields χ ₀ = χ̃ ₀ = 0 and the first order solution

where E = 2χ ₁ in the near-field case and E = 2$\tilde{\chi }$ ₁ in the far-field case.

Note that this Schrödinger equation has solutions of the form ψ ₀ ($\overline{\xi }$) ~ exp(is̅ $\overline{\xi }$) as $\overline{\xi }$ → ±∞ where s̅ ² = E and take these as approximations to the waveguide modes used in Sec. 2.2. Also, χ = ε ² E/2 + 0(ε ⁶) implies that, in the small ε limit, the frequency shifts are small compared to the longitudinal mode spacing, as needed to justify the approximation of Eq. (5) and the assumption that the cavity mode is dominated by a single waveguide mode.

4. Bound and unbound modes

For the moment, continue ignoring the edges of the mirrors and hence any loss. With s̅ pure real, the asymptotic solutions exp(±is$\overline{\xi }$) have the form of propagating waveguide modes as $\overline{\xi }$ → ±∞. These unbound solutions correspond to E > 0 and hence frequencies higher than the unperturbed plane-parallel case or lower than the unperturbed hemispheric case (see also [4]). With s̅ pure imaginary, most solutions blow up either to the left or right, except those at the poles of the scattering matrix S where there exist bound solutions that decay exponentially as $\overline{\xi }$ → ±∞. These correspond to E < 0 and hence frequencies lower than the unperturbed plane-parallel case or higher than the unperturbed hemispheric case.

Truncating the cavity to finite width confines the unbound modes by partial reflection but concurrently introduces radiation loss. Consider a reflectionless guide (as illustrated in Sec. 5) so that r = 0. Writing δ = -ilogt and using (8), the resonance condition of (7) becomes

where m is an integer. Note the similarity to Eq. (6); the new term δ simply reflects the additional phase incurred as the mode’s propagation parameter changes with the variable cross section of the central region. Because δ depends on s, one must generally solve this iteratively starting at s = 0 (and using the correct branch of the logarithm). Unfortunately, in many cases of interest, it is difficult to obviate this need through approximation. Still, the case M → ∞ may be relevant to the microcavities considered in [4]. In that case, if δ = δ ₀ + a s̅ + O(s̅ ²), the round trip power loss of an unbound mode becomes

and the round trip phase shift of an unbound mode becomes

First, note that the loss decreases like M ^-3 as the mirror size M → ∞. Second, note that the frequency spacing decreases like M ^-2 as M → ∞. If, instead of Eq. (16), the loss becomes limited to some fixed amount due to mirror reflectivity or misalignment, then for some sufficiently large M the unbound modes will lose all resonant character, as appears to have been the case in the experiments described in [4].

Since the geometry of the central section of the guide already confines the bound modes, we expect truncation of the guide far from the interesting central features to produce considerably less radiation loss. In showing this, it is convenient to use decay parameters, so define s̅ = i$\overline{\sigma }$. Recall that the bound modes of the infinite system occur at the poles of the scattering matrix S. For concreteness, consider an even bound mode corresponding to a simple pole at $\overline{\sigma }$_p and expand the transmission coefficient near $\overline{\sigma }$_p as

Recalling that λ _- = r - t, avoidance of an odd mode at the same frequency requires canceling the pole in t so

For the odd modes let

A peculiarity of bound modes makes it preferable to use the above inference rather than trying to find r directly. For example, for the unbound modes, u($\overline{\xi }$) ~ exp(is̅$\overline{\xi }$) as ξ → ±∞ implies r = 0, but in the case of the bound modes u($\overline{\xi }$) ~ exp(-$\overline{\sigma }$$\overline{\xi }$) as $\overline{\xi }$ → ±∞ tells us nothing as u(ξ)+ Aexp($\overline{\sigma }$$\overline{\xi }$) ~ exp(-$\overline{\sigma }$$\overline{\xi }$) as $\overline{\xi }$ → -∞ as well.

Using the pole approximation yields

a round-trip power loss

and a round-trip phase shift

The exponential dependence of loss on mirror size M contrasts sharply with the power law dependence of the unbound modes in Eq. (16). Furthermore, the mode frequencies rapidly approach well-separated limits as M becomes large and may be expected to maintain their resonant character, unlike the unbound modes.

5. Numeric examples

As a concrete case, consider, as in [3],

a well-known reflectionless potential with exactly n bound states. Exact eigenstates may be expressed in terms of hypergeometric functions (which simplify considerably for small n, less so otherwise), but the procedure adopted here only requires the transmission coefficient [9]

where $\overline{\sigma }$ m = n - m + 1 for m = 1,…,n.

Figures 3 and 4 show the round trip losses and phase shifts for the first six modes as a function of mirror size M for the cases n = 1 and n = 3, holding σ_f̃ = .628 for all. We performed these calculations for the near-field case. Given the identical form (up to conjugation) of Eqs. (1) and (2), the far-field equations give identical results. We have verified this numerically but omit the redundant far-field results. The solid line calculates the resonator eigenvalues by direct numerical methods. The dashed line uses the methods of this paper. Specifically, we compute the eigenvalues of the unbound modes by iteratively solving Eq. (15) starting from s = 0 (unfortunately (16) and (17) are not especially accurate at the values of M considered here) and compute the eigenvalues of the bound modes directly from Eqs. (19) and (20).

Notice the good agreement between the losses of the bound modes and the exponential decay of (19). Even when, for a large number of bound modes, the numeric accuracy of our model declines, it still captures the basic functional form. While Eq. (15) is not quite accurate in the range of M considered here, nevertheless, the unbound modes’ losses in Figs. 3 and 4 appear to indeed be following a power law. Finally, note that the cavity with just a single bound mode exhibits the sharpest distinction between its bound and unbound modes. Shallow bound modes that decay weakly toward the edges may not obviously manifest their distinctive character unless very large mirrors are used.

Figures 3 and 4 also show the round trip phase shift for the unbound and bound modes. First and foremost, note that the bound modes stabilize to well-separated frequencies, while the unbound modes gradually converge to the plane-parallel frequency (corresponding to zero phase shift) as the mirror size M increases. While our method appears to break down for the first unbound mode for n > 1 and large mirrors, it does capture most of the trends and adds further clarification to the distinction between bound and unbound modes.

6. Conclusions

To summarize, we have shown that in the near-field limit, treating a custom laser resonator as a peculiar sort of waveguide, using a perturbation analysis as in [3] to find the waveguide modes, and employing a waveguide end loss analysis as in [5] yields an accurate picture of the losses and frequency spectra of these resonators. Furthermore, in the far-field limit, a given cavity may be transformed to an equivalent near-field cavity and the same analysis applied. While the method is more general, the case of cavities with asymptotically flat mirrors yields particularly interesting results, as the modes divide themselves into two classes, here called bound and unbound modes. The bound modes, characterized by fields asymptotic to exp(±σξ) far from the mirror center, have low losses due to the exponential decay of the field away from the modal caustic and have widely separated frequencies. The unbound modes, characterized by fields asymptotic to exp(±isξ) far from the mirror center, have high losses due to the significant fraction of the mode that extends out to the mirror edges and have a tendency to become frequency degenerate with increasing mirror width. The general form of our predictions agrees well with direct numeric loss calculations, especially in the case of cavities with a single bound mode and a corresponding large difference in the behavior between the two cases.

 

Fig. 3. Eigenvalues of bound (b) and unbound (u) cavity modes when Ψ($\overline{\xi }$) = sech$\overline{\xi }$. Solid curve is numeric solution; dashed curve is our method. Phase results are indistinguishable for the bound mode.

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Fig. 4. Eigenvalues of bound (b) and unbound (u) cavity modes when Ψ($\overline{\xi }$) = sech³ $\overline{\xi }$. Solid curve is numeric solution; dashed curve is our method. Phase results are indistinguishable for the first bound mode.

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Fig. 5. Zeros of K̃(s) and paths of integration in the complex s-plane

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