1. INTRODUCTION

The power levels of diode-pumped solid-state and fiber lasers have steadily increased over the past two decades [1], which has enhanced the technical viability of high-power applications such as lidar, free-space communications, astronomy, and medical imaging [2,3]. However, high-power beams tend to accumulate wavefront distortion from turbulence and variations in the propagation medium, reducing the efficacy of power delivery.

There are a number of methods to measure the local aberrations in a wavefront, such as Shack–Hartmann sensors [4], shearing interferometers [5], and holography [6,7], some of which can be used in array/subaperture forms [8]. The Shack–Hartmann is the canonical solution for wavefront measurement, using an imaging sensor and a lenslet array to measure localized tilts in the beam and using these to infer the shape of the wavefront; however, it suffers from a weakness in measuring strong wavefront distortions across small apertures. Several alternative sensor designs, including pyramid and curvature wavefront sensors, share this inability to handle the many waves of distortion and apparent null-intensity discontinuities—“branch points”—commonly caused by horizontal atmospheric propagation in long-distance laser applications. Methods that can handle these, such as quality metric estimation, plenoptic sensing, or interferometry, tend to add complex processing algorithms and/or a great deal of large, sensitive hardware [9].

There exist relatively few technologies that can measure natural atmospheric turbulence in real time. Scintillometry is a classic technique based on the turbulence-induced intensity variations in a distant light source, but the requirements of a carefully aligned yet far-separated source and detector can be onerous: past measurements for high-altitude turbulence staged the experiment’s halves in separate aircraft flying in formation [10,11]. “Hot wire” measurements, which measure temperature variations in microthermal probes, also require the physical presence of instrumentation in the measured medium [12]. Some remote-sensing alternatives have been demonstrated, such as using real-time camera images to observe fluctuations in the edges of distant objects [13]; while these techniques are promising, they require either significant hardware or software processing or both. There is space in this technical area for single-point measurement techniques that could directly indicate certain features of an aberrated wavefront and the turbulence that caused it.

In this paper, it is shown that an aberrated wavefront coupled to an optical cavity will excite a series of cavity spatial modes. These spatial mode profiles vary from aberration to aberration, and the spectral width of the spatial mode profile tends to increase as the number of waves of aberration increases. Note that in this work, the cavities are perfectly passive, and no new frequencies are created that did not exist in the original light beam. Since high-power lasers tend to have large spectral bandwidths—many fiber lasers even intentionally broaden their bandwidths, on the order of $10\mathrm{GHz}{\mathrm{kW}}^{-1}$, to control stimulated Brillouin scattering—the light source can be safely assumed to support a large number of spatial modes [1,14].

2. RELATED WORKS

Although to our knowledge this simple technique has never been described in literature, its component parts are certainly well traveled by workers past and present. The decomposition of fields into Gaussian mode bases is an established subfield often referred to as Gaussian beam mode analysis (GBMA). Its applications are often focused towards efficient ways of performing diffraction calculations [15,16], notably including the propagation of a beam with imposed Seidel aberrations by Trappe et al. [17].

Recent use of extremely long optical cavities in gravitational wave observatories has yielded much attention to the Laguerre–Gaussian modes within resonators; for instance, Bond et al. (2011) have simulated the ability of mirror aberrations, expressed using Zernike polynomials, to redirect power among Laguerre–Gaussian modes, while Gatto et al. (2014) experimentally demonstrated these effects for several aberrations [18]. Mode expansions are used in quite a few other measurement systems, such as the laser beam quality and alignment system of Kwee et al. [19].

In shorter resonators, Liu and Talghader examined the effects of imperfections in tunable micromirror cavities on Gaussian beams; however, only a relatively limited selection of aberrations could be easily treated [20]. Perhaps the most superficially similar work we are aware of belongs to Takeno et al., who also adopted the Fabry–Perot cavity’s mode discrimination for aberration sensing. However, their technique measures the intensity of the higher order spatial modes reflected away from the cavity, and does not rely on spectral information [21].

3. CAVITY MODE DECOMPOSITION AND TRANSMISSION SPECTRA

The eigenmodes of a stable optical resonator are a set of Gaussian beams: the Laguerre–Gaussians in cylindrically symmetric systems and the Hermite–Gaussians in Cartesian coordinates [22]. For the former, the field for integer transverse mode indices $n$ ($\ge 0$) and $\alpha$ is given by

Each of the Hermite–Gaussian and Laguerre–Gaussian families form a mutually orthogonal basis set, which may be used to express any arbitrary optical beam meeting the paraxial conditions [23,25]. The electric field amplitude of each Gaussian mode $E_{n,\alpha }(r,\varphi ,z)$ excited by an input field $E_{\mathrm{in}}(r,\varphi ,z)$ can be computed by the overlap integral [26],

If a laser beam of finite spectral bandwidth, with an imposed wavefront aberration, is incident upon a lossless resonator, it will couple some of its energy into the eigenmodes of the cavity. Aberrations or other phase front deformations to the incident wave will change the spatial distribution of energy, and thus the amount of coupling—where supported by the source spectrum—into the higher order transverse modes of the cavity. The excitation of these modes can be easily seen in the transmission of light at the resonant frequencies.

The process suggests a reversible correlation. Since each cavity mode corresponds (usually degenerately) to a transmitted resonant frequency, the spectral content of the transmitted output of the cavity will correlate to the strength and nature of any aberrations imposed on the input beam. Given spectral transmittance information from such a cavity, it will be possible to deduce the presence of wavefront aberrations and limited information about their number and magnitude.

4. CONCEPT DEMONSTRATION—SINGLE ABERRATIONS

We can illustrate this concept by numerical application of the above equations. The choice of the optical cavity that we might use to analyze a wavefront is certainly not unique; one must consider cavity issues of the longitudinal mode spacing, spatial mode spacing, spectral width of each resonance, and other practical factors such as mirror shape and cavity materials. The laser wavelength and line shape are also important to the design and analysis.

Let us consider an aberrated wavefront originally emitted from a high-power laser at wavelength 1.064 μm with a spectral linewidth of about 100 GHz. Since we will be analyzing the wavefront using the transverse modes of an optical cavity, the cavity length (and material) are chosen such that the longitudinal mode spacing is somewhat greater than the spectral linewidth of the laser, and excitation is thus limited to a single longitudinal mode number. A fused silica (refractive index $n_{\mathrm{refrac}}\approx 1.45$) blank of thickness 1 mm would lead to a longitudinal mode spacing of about 103 GHz. We wish to have a large number of spatial modes in the system, so erring on the side of too many, we can choose one spatial mode every 1 GHz. We also wish to allot a large-enough spectral width to each cavity mode so that we will have reasonable light throughput for each mode, but not so much that they overlap and are difficult to distinguish. With this in mind, a spectral width of approximately 300 MHz would be useful. Using $\mathrm{\Delta }\nu =c/(2\pi n_{\mathrm{refrac}}{z}_0)$ for the separation between transverse modes for near-planar mirrors and $\mathrm{\Delta }{\nu }_{d/2}=(c/(2\pi n_{\mathrm{refrac}}l))·((1-R_{\mathrm{mirror}})/\sqrt{R_{\mathrm{mirror}}})$ for the full width at half-maximum (FWHM) [23], these parameters lead to a cavity with mirror reflectivity of $R_{\mathrm{mirror}}=99.1\%$, fed by a beam with waist size $w_0\approx 100\mathrm{\mu m}$.

For calculation of cavity transmission spectra, we presume that the laser supplies, with equal intensity, every wavelength necessary to observe the transmission spectrum from the range of modes that we sample. A more realistic light source would typically have variations in intensity with frequency, but these deviations from uniformity will vary from source to source and can be easily handled mathematically after a spectral power measurement of the laser beam. We will assume that the beam may be sampled to measure the total beam power entering the cavity. Figure 1 conceives a measurement setup that might enable our simulated scenarios.

 

Fig. 1. Conceptual diagram of the measurement setup emulated by the simulations below. A laser provides a source beam that has some phase aberrations imposed upon it. The light is fed—optionally with the help of collection optics—into an optical cavity, and the transmitted output is spectrally analyzed using some kind of selectable spectral filter and a detector. A small portion of the light may be sampled to measure the total beam power.

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To simplify our initial examples, we will assume that the source beam has exactly the correct $w_0$ and $R$ to transmit all of its power into the cavity’s fundamental mode. Without aberration or perturbation, we confirm from evaluation of the overlap integral Eq. (3) that only the fundamental mode $(n,\alpha )=(0,0)$ is excited, as we would expect.

We may now add a basic wavefront modification and observe its effects. The Seidel aberrations, the five simple “third-order” transverse ray aberrations almost omnipresent to some degree in every optical imaging system, seem a natural choice. The Seidels are listed, in terms of the exit pupil spatial coordinates $(x,y)$ and image plane coordinate $x_0$ appropriate for an imaging system, in Table 1 [27].

Table 1. Five Seidel Aberrations and Their Functional Forms, in the Context of a Lens or Imaging System^a

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The Seidels are most often seen in the context of the lenses or imaging systems they plague, but free space does not possess the same refocusing or imaging properties. Consequently, we use the Seidels here only to describe the surface of a phase front modification, multiplicatively applied to the propagated field of our light source. Examining the Seidel functional forms while disregarding the image plane spatial coordinate $x_0$, we notice that there are two basic components: the directionally sensitive term $r\cos \varphi$, seen isolated as distortion, and the circularly symmetric $r^2$ term, at the core of field curvature. The other three Seidel phase fronts are able to be expressed as products of these two terms, making distortion and field curvature the fundamental examples. Each aberration shape is scaled by a coefficient $W$ and multiplied as a phase delay onto our cavity’s incident field, producing

 

Fig. 2. Phase fronts and difference in resulting imaginary field components $\mathfrak{I}(E_{\mathrm{aberrated}}-E_{\mathrm{source}})$ of the Seidel aberrations of distortion (left) and field curvature (right). Each aberration is scaled such that a maximum of one wave of phase delay occurs within the area where the source field magnitude $|E_{\mathrm{source}}|>1/e^2$. Periodic folding limits the phase delay displayed to $\pm 0.5$ wave. (a) Distortion phase delay. (b) $\mathfrak{I}(E_{\mathrm{withdistortion}}-E_{\mathrm{source}})$. (c) Field curvature phase delay. (d) $\mathfrak{I}(E_{\mathrm{withfieldcurvature}}-E_{\mathrm{source}})$.

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Thus aberrated, these fields now encounter the cavity. We may view the excitation amplitudes of the cavity modes for a variety of different aberration strengths by making use of Eq. (3). The results are shown in Fig. 3.

 

Fig. 3. Transverse cavity mode power distribution caused by (a) distortion and (b) field curvature as the aberration strength increases from 0.5 to 3 waves of maximum phase delay.

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Since field curvature is circularly symmetric, with a phase delay dependent only upon $r$, we might expect its effects would eschew any modes that are asymmetric. Figure 3 confirms this suspicion. As increasing strengths of field curvature are applied, power diffuses upward along $n$ but remains only in modes where $\alpha =0$; the fundamental mode remains the single-most energetic mode even as it loses power to an increasing number of others. Distortion, on the other hand, involves both $r$ and $\varphi$ in its phase delay expression, and thus the expansion takes place in both of the respective corresponding mode indices $n$ and $\alpha$. As the aberration strength increases, power is transferred from the source beam’s fundamental mode into a “pulse” of additional modes, which widens as its center moves upward in mode index $n$.

Finally, we translate the cavity mode activity information to the experimentally observable transmission spectrum. Equation (2) gives the resonant frequency of each transverse mode (degenerately shared with others of the same $l+m$); thus, the transmission at each resonant peak is equal to the sum of the intensities of all the cavity modes that share that frequency. We will display here only the peak intensity at each resonance and not concern ourselves with the full transmission curve, which—since we have expressly designed the cavity to separate the resonant peaks—should clearly display the results without materially affecting our conclusions. Figures 4(a) and 4(b) display the resulting transmission spectra for distortion and field curvature, respectively.

 

Fig. 4. Intensity spectrum transmitted by the cavity as a result of transverse mode excitation by an input Laguerre–Gaussian, with and without varying amounts of (a) distortion or (b) field curvature. Each spatial mode’s transmission peak contribution is taken to have Lorentzian line shape, approximating the typical response of dielectric mirrors.

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The transmission spectrum from the distortion aberration is simple to predict: since there is no activity except where mode index $m=0$, each resonance frequency directly corresponds to a value of $l$, and the spectrum reads exactly as the mode activity plot viewed from one side. The “pulse” of power is again clearly visible traveling upwards and spreading in frequency. Field curvature produces a markedly different spectrum, with power diffusing upwards in frequency from the fundamental mode and skipping every other cavity resonance.

In many turbulence detection applications, the goal is to estimate a statistical characteristic such as $C_n^2$ to express the overall amount of turbulence present. For these control-oriented or real-time uses, a full spectrum may be prohibitively slow or complex to capture. We may instead define simpler quantities that still present a strong correlation to the unseen complete spectrum. For instance, we define here the spectral power distribution to be the portion of the total transmitted light intensity that passes within a certain bandwidth above the fundamental mode. This can be measured with two photodetectors, one each with and without a fixed spectral bandpass filter. The simulated results for various values of bandwidth are shown in Fig. 5.

 

Fig. 5. Spectral power distribution, or the portion of light intensity within a certain spectral bandwidth above the fundamental mode frequency, resulting from various strengths of (a) distortion and (b) field curvature.

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As observed in Figs. 4(a) and 4(b), the concentration of power breaks quickly away from the fundamental mode under distortion, whereas with field curvature power spreads upward in frequency but remains highest at the fundamental mode. Correspondingly, Fig. 5(a) quickly develops a gap containing no power next to the vertical axis, but Fig. 5(b)’s lines adhere to the fundamental mode for a much larger degree of aberration.

5. CONCEPT DEMONSTRATION—DISTANT SOURCE

The single-mode Laguerre–Gaussian source beam assumed above has the advantage of clearly isolating the effects of aberrations, but it is not likely to be replicated outside of a laboratory setting. A sensor for atmospheric turbulence will most likely sample a small patch of a source wavefront that has propagated over a long distance, and thus would (if not for the aberrations added along the way) resemble a plane wave regardless of the original source.

To approximate this scenario, let us replace our original Laguerre–Gaussian beam with a plane wave. We assume that a diffraction-limited optical system is in use to enhance the light-collection area of the cavity, and that it has a sufficiently large exit pupil to allow frequency apodization to be neglected. The input to the cavity can then be taken as the aberrated plane wave, bounded by the exit pupil function and arbitrarily magnified. We set, for convenience, a circular pupil function and a magnification that scales the now-circular field outline to approximately the size of the Gaussian cavity mode. Since our source beam is no longer a perfect match for a cavity mode, we should expect a more complicated mode structure and spectrum even without applied aberrations, as shown in Fig. 6.

 

Fig. 6. The (a) cavity mode decomposition and (b) resulting transmission spectrum for a plane wave with a flat phase front, bounded by a circular pupil with negligible diffraction effects.

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We now apply the same Seidel aberrations and observe the cavity mode excitation, displayed in Fig. 7. Mode activity for distortion spreads quickly with strength, mostly along the angular $\alpha$ index, and remains symmetric about $\alpha =0$. Field curvature displays no effect on modes other than those with $\alpha =0$, consistent with its behavior in Fig. 3(b), which is logical given the circular symmetry of the aberration; however, the single highest power mode is no longer guaranteed to have $n=0$.

 

Fig. 7. Transverse cavity mode power distribution caused by 0.5–3 waves of maximum phase delay, shaped as (a) distortion and (b) field curvature. In the case of the largest strengths of aberration, power begins to escape the sampled range of modes.

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As before, we may translate the cavity mode decomposition into the observable spectral transmission. Despite the noted differences in mode excitations, Fig. 8(a) shows recognizable similarities with the Laguerre–Gaussian-borne spectrum: power is still concentrated into a packet of limited width, which detaches completely from the fundamental to drift upward in frequency. The field curvature’s spectrum in Fig. 8(b) also exhibits a local maximum of power traveling upwards in frequency, but it is easily discerned from distortion by the significant and evenly distributed power remaining in the modes down to the fundamental.

 

Fig. 8. Intensity spectrum transmitted by the cavity as a result of transverse mode excitation by a pupiled plane wave, with and without varying amounts of (a) distortion and (b) field curvature. It should be noted that the frequencies shown have accounted for all (degenerately) contributing modes, which is a larger range of mode indices than shown in Fig. 7.

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We again test the ability of the single metric of spectral power distribution to indicate aberration strength. In the resulting Fig. 9, the spectral power distribution of distortion now departs the vertical axis—i.e., contains no power in the lowest frequency modes—starting with much lower aberration strengths, exaggerating its differentiation from field curvature; aberration strengths greater than half a wave may be estimated by summing the power within a fixed frequency of the fundamental. Field curvature, in contrast, never reaches zero on the vertical axis. In both cases, spectral width clearly and increasingly rises with aberration strength, suggesting that this metric actually becomes more effective with many waves of aberration.

 

Fig. 9. Spectral power distribution plots introduced in Fig. 5, now with a pupiled plane wave source beam instead of a cavity-matched Laguerre–Gaussian. (a) Distortion. (b) Field curvature.

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6. CONCEPT DEMONSTRATION—MANY ABERRATIONS

The single-aberration scenarios examined above were chosen to have simple and clear geometries, but real-world systems are unlikely to produce wavefronts with such characteristics. For example, wavefront aberrations could be created with thermal distortion in a complex optical system or over long distances of horizontal propagation in atmosphere. Many aberrations could exist simultaneously, up to the essentially fractal nature portrayed in Kolmogorov turbulence theory [28].

These elaborate phase fronts can be described using the Zernike polynomials, a family of surface functions that can be superimposed to describe any smooth, well-behaved surface on the unit circle. The Zernikes are often given in modified form for various purposes; however, the changes often reduce to differences in normalization and index definition [29,30]. For our purposes, the Zernike term $Z_n^m$ is given for $m\ge 0$ by [31]

 

Fig. 10. Phase delay, defined by (a) randomly weighting the first 10 Zernike terms, (b) summing them, and (c) scaling the resulting surface to achieve a phase front as previously with the Seidels, is (d) applied (seen as $\mathfrak{I}(E_{\mathrm{aberrated}}-E_{\mathrm{source}})$) to a plane wave bounded by a circular pupil function with negligible frequency component loss from diffraction. (a) Zernike term weightings. (b) Phase surface. (c) Field with one wave of phase delay. (d) Difference in fields.

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These fields are brought to our optical cavity, as before, and the resulting mode excitations are shown in Fig. 11. Unlike our geometrically simple Seidels, these mode patterns do not display perfect symmetry along either index. Some similarity to the progression of distortion is suggested by the partial “ridge” of mode power, which seems to spread downwards in $n$ and outwards in $\alpha$.

 

Fig. 11. Transmitted cavity spectrum resulting from a pupiled plane wave with varying strengths of phase delay in the shape of 10 randomly-weighted Zernike terms.

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Finally, Fig. 12 examines the intensity spectrum transmitted. The spectrum develops as a slow, rather uneven combination of a “traveling wave packet” and an equilibrating spread toward higher order modes, which were, respectively, the qualitative patterns of distortion and field curvature. Recalling that the 10 Zernike terms of our phase delay include rather weighty distortion $Z_1^1$ and field curvature $Z_2^0$ terms, and that the remaining three Seidels can be seen as products of those two, this hybrid behavior makes intuitive sense.

 

Fig. 12. Intensity spectrum transmitted by the cavity for a plane wave with a phase front derived from 10 randomly weighted Zernike terms.

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The spectral power distribution, shown in Fig. 13, again shows a clear and accelerating increase with aberration strength. We conjecture that further interpretation is possible: at lower values of $W$, the power distributions seen are difficult to distinguish from those of a slightly higher strength of distortion; as the phase delay becomes more significant, departures such as a nonuniform slope to the spectral power line—which we previously witnessed in Fig. 9(b), as the result of field curvature—begin to appear.

 

Fig. 13. Spectral power distribution of optical cavity output when stimulated by a pupiled plane wave with varying amounts of phase delay, shaped by arbitrarily weighting 10 Zernike polynomial terms.

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7. CONCLUSION AND FUTURE WORK

The results presented above show that aberrations induced in a beam will excite high-order modes when that beam is coupled into an optical cavity. This is a perfectly natural result of how aberrations are merely spatial variations in an otherwise uniform phase (and intensity) front—the more variation there is (in the form of spatial frequency content and amplitude) across the phase front, the more spatial modes will be excited in the cavity, and the higher their amplitude coefficients will be—but it also performs the valuable and somewhat surprising task of converting the spatial variations across the phase/intensity front into a distribution of spectral frequencies. Let us be very clear that no new frequencies are created that were not in the initial bandwidth of the light source; typical high-power lasers have bandwidths of many tens of GHz, so it is easy to design a cavity that has many spatial modes within this bandwidth yet avoids overlapping into the next longitudinal mode.

The simulations presented here on such a cavity suggest that input aberration strength can be well estimated from simple spectral measurements of its transmitted light. In addition, it may be possible to identify or distinguish between basic aberration types. The use of optical cavity spectral sensing may enable construction of a wavefront sensor array with fewer elements, faster readout, easier basic analysis, and the ability to measure more highly aberrated phase fronts than current technologies allow. Variations on this concept, such as the use of cylindrically asymmetric cavity mirrors, could yield further information such as wavefront orientation.

In our efforts to introduce the concept of aberration sensing with optical cavity modes, we have assumed that an imaging optical system is used to transfer the incident wavefront onto the front cavity mirror. This will work well, but it requires imaging optics, which are at least modestly complex. The simplest aberration sensors based on cavity spatial mode spectra may have optical systems limited to a single focusing lens. The cavity would then analyze the Fourier transform of the incident coherent field, which would convert the base plane wave to a delta function, and the added aberrations would have somewhat different spectral features. The mathematics of this process are very similar to what we have performed here, except for the Fourier transform.

The next steps in this work are to connect the theory introduced in this work to the practical aberrations seen in real systems. The daunting part of this effort is the sheer number of types of systems, and therefore immediate efforts will concentrate on mathematically well-characterized sets of aberration-inducing phenomena, such as Kolmogorov turbulence.