1. Introduction

Optical configurations often exhibit reciprocity, or symmetry with respect to the interchange of point sources and receivers [1–3]. It is precisely this symmetry that enables adaptive optics (AO) correction of laser beam profiles delivered to targets in random media. AO correction uses the light received from a low power emitter, or beacon, on or near the target to adjust the laser beam’s spatial profile [2,4–9]. In a reciprocal configuration, every ‘ray’ in the beacon has a reciprocal partner in the beam. These rays traverse the random media along identical paths but in opposite directions. Thus by reversing the rays along the phase front, or phase conjugating, the beacon irradiance profile can be reproduced at its source. Often, however, the rays on the incoming and outgoing paths experience differing dielectric environments. The medium evolves, or as is the interest here, the power in the beam surpasses that of the beacon, leading to differences in the nonlinear refraction on the outgoing and incoming paths.

Here we examine the nonlinear breakdown of reciprocity occurring when a low power beacon informs the phase correction of a high peak power laser beam. In doing so, we provide the first analysis of AO applied to nonlinearly propagating beams. We introduce a metric, an overlap of the beacon and the beam fields, that quantifies the breakdown, and provides a necessary and sufficient condition for reciprocity. The metric, henceforth referred to as the reciprocality, is applied to the specific case of field conjugated high power beams propagating through Kerr-nonlinear turbulent atmosphere. The reciprocality is found to drop rapidly at powers approaching the critical power for self-focusing. In the strong turbulence regime, the reciprocality increases due to spatial incoherence and turbulence inhibiting nonlinear propagation. A rough scaling, explaining this behavior, is derived. Finally, we find that the drop in reciprocality is dominated by phase differences between the beacon and beam, suggesting AO correction can be effective when the on-target irradiance is important, but not the phase.

2. Reciprocity and adaptive optics

While there are several types of beacons and variations on AO implementations [2,4–9], we consider a simple optical configuration that illustrates the salient physical phenomena. The configuration is displayed in Fig. 1. A static random medium separates the target plane on the right from the receiver plane on the left. The beacon resides in the target plane, and the receiver plane coincides with the laser beam transmitter plane. The beacon light propagates through the random media and is collected in the receiver plane where its phase and amplitude are measured. The conjugate phase and amplitude are then applied to a laser beam, which thereby inherits any spatial incoherence acquired by the beacon. The laser beam then propagates back to the target through the same random media. In Fig. 1 the different colors of the beacon and laser beam are for illustrative purposes only; their wavelengths, in actuality, would be quite similar to limit chromatic effects.

 

Fig. 1 A beacon located on a target embedded in a random medium informs the phase and amplitude of a laser beam incident on the target.

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To model the light propagation, we use the scalar paraxial wave equation. We note, however, that the conceptual discussion of reciprocity and its breakdown applies to other wave equations as well, including the vector and scalar Helmholtz equations. The transverse electric field, $E_{\perp }$, consists of a carrier wave modulated by a slowly varying envelope, $E$: $E_{\perp }(x,t)=\frac{1}{2}E(x)\exp [i(kz-\omega t)]+\text{c}\text{.c}.$ where $\omega$ is the carrier frequency,$k=\omega n_0/c$, and $n_0$ is a reference refractive index. The envelope evolves according to

In the following, we make use of the Green functions for Eq. (1). The Green function is the linear solution to Eq. (1) driven by an impulse and is used to propagate the field between any two planes. In particular, we define

We continue by describing an idealized AO system that illustrates the importance of reciprocity [2]. We denote the beacon and laser beam electric field envelopes as $E_B(r,z)$ and $E_L(r,z)$ respectively. The receiver/transmitter resides at $z=0$ and the target at $z=z_T$. For this example, we take $\delta n_i=0$; when $\delta n_i(x)=\delta n_i(z)$, the amplitudes at each plane can be adjusted retroactively by the appropriate exponential factor, $\exp [ik\int \delta n_i(z)dz]$. The Green function $G^{-}(r,0;r^{\prime },z_T)$ propagates the beacon field from the target to the receiver: $E_B(r,0)=i\int G^{-}(r,0;r^{\prime },z_T)E_B(r^{\prime },z_T)d{r}^{\prime }$. At the receiver, the amplitude and phase of the beacon field are measured, and the AO system applies the conjugate profile to the outgoing beam, $E_L(r,0)=-i\int G^{-*}(r,0;r^{\prime },z_T)E_B^{*}(r^{\prime },0)d{r}^{\prime }$. The Green function $G^{+}(r,z_T;r^{\prime },0)$ propagates the beam from the transmitter to the target: $E_L(r,z_T)=i\int G^{+}(r,z_T;r^{\prime },0)E_L(r^{\prime },0)d{r}^{\prime }$, which upon substitution of the outgoing beam field provides $E_L(r,z_T)=\int \int G^{+}(r,z_T;r^{\prime },0)G^{-*}(r^{\prime },0;r^{″},z_T)E_B^{*}(r^{″},z_T)d{r}^{″}d{r}^{\prime }$. If the channel is reciprocal, the on-target beam field reduces to $E_L(r,z_T)=E_B^{*}(r,z_T)$. The AO system has exploited reciprocity to illuminate the target with the conjugate field of the beacon.

The nonlinear refractive index, $\delta n_{NL}$, was excluded in this example, and neither the beacon nor the beam propagated nonlinearly. Moreover, the Green function, a linear construct, was used to define the conditions of reciprocity and reversibility. To demonstrate nonlinear reciprocity and reversibility, we divide propagation over a total distance $L$ into $N$ steps of size $\Delta z=L/N$. Forward and backward propagation over a single step are expressed as

This nonlinear reciprocity can be applied to our AO example when the beacon and beam experience identical optical configurations. From a practical standpoint, however, the propagation of the beacon light from the target to the receiver, and the propagation of the beam from the transmitter to the target can occur under different conditions. The random media may change in time or, as is the interest here, the power of the beacon and beam may differ. This results in an effective breakdown of reciprocity. The symmetry breaking can be expressed symbolically by parameterizing $H^{\pm }$ with the beacon and beam powers, $H^{+}(r,z;r^{\prime },z-\Delta z;P_L)\ne H^{-}(r^{\prime },z-\Delta z;r,z;P_B)$ where $P_j=\int I_{j}dr$. Conceptually, the beacon and beam undergo nonlinear refraction to a different extent, and, as a result, their ‘rays’ take different paths through the random media.

In order to quantify the breakdown of reciprocity along the propagation path, we define the following metric:

A few examples aid in the interpretation of $R$. First consider the idealized AO system discussed above. When the beacon and beam have identical powers, we showed that $E_L(r,z_T)=E_B^{*}(r,z_T)$, which can be straightforwardly generalized to $E_L(r,z)=E_B^{*}(r,z)$. Inserting this field into Eq. (4), we find $R(z)=1$ for all $z$. Suppose $E_L$ and $E_B$ have identical amplitudes but are everywhere phase shifted by $\pi /2_{}(\pi )$, then $R=i_{}(-1)$. If $E_L$ and $E_B$ are spatially disjoint, implying that their ‘rays’ propagate through wholly different regions of the random medium, then $R=0$. A value of $\left|R\right|<1$ does not, however, indicate a unique spatial phase difference or irradiance disjointedness.

3. Kerr-nonlinear turbulent atmosphere

To demonstrate application of this metric, we consider the optical configuration illustrated in Fig. 1 for the case in which the random media is dissipationless, Kerr-nonlinear, turbulent atmosphere. The Kerr nonlinearity, $\delta n_{NL}(I)=n_{2}I$ where $n_2$ is second order nonlinear refractive index, permits the well-known phenomenon of self-focusing [11–14]. The nonlinear refractive index reduces the phase velocity in regions of high intensity. This acts as a self-lens that limits diffraction in these regions. The ratio of the total beam power to the critical power, $P_{cr}~{\lambda }^2/2\pi n_0{n}_2$, parameterizes the effect. For a simple derivation of $P_{cr}$, one can balance the diffractive operator and the refractive index term in Eq. (1).

More generally, the beam consists of pockets of high intensity whose individual powers can be compared to the critical power [13]. When $P<P_{cr}$ a pocket of elevated intensity undergoes slowed diffraction. If $P~P_{cr}$, the pocket’s diffraction is nearly eliminated, and barring any phase front distortion or depletion, the propagation is collimated. When $P>P_{cr}$ a pocket can collapse into a smaller and smaller area until some other process, such as ionization or harmonic generation [15–17], causes refraction or depletion. For a beam with an initial Gaussian profile of spot size $w_0$, a formula for the collapse distance in uniform media was developed by Marburger: $z_c=0.18k{w}_0^2/{\{{[{(P/P_{cr})}^{1/2}-0.85]}^2-.022\}}^{1/2}$ for $P>P_{cr}$ [11]. Here we limit the propagation to distances well less than $z_c$.

Turbulent fluctuations in the linear refractive index of atmosphere, $\delta n_f$, continually distort the phase of a laser beam, causing a cumulative loss of spatial coherence. In particular, the transverse coherence length, ${\rho }_0=1.67{(k^2{C}_n^{2}z)}^{-3/5}$ where $C_n^2$ is the refractive index structure constant, decreases with propagation distance [18]. During the early stage of propagation, the beam acquires ‘whole-beam’ phase aberrations from large-scale fluctuations, $(w/L_f)<<1$ where $w$ is the spot size and $L_f$ the fluctuation scale-size. These large-scale aberrations result in centroid wander, and are the predominant contributor to the reduction in coherence length until ${\rho }_0/w~1$ [19]. As the beam continues to propagate, ‘sub-beam’ pockets of coherence develop, ${\rho }_0/w<1$, due to small-scale fluctuations, $(w/L_f)>>1$. Through diffraction, these pockets develop into intensity fluctuations, scintillating the beam. The development of small-scale phase and intensity structures, in turn, accelerate beam spreading.

The accelerated beam spreading can be illustrated using the long-term spot size equation for a Gaussian beam provided in [19]: $w^2(z)=w_0^2+(1+2w_0^2/{\rho }_0^2){(z/Z_R)}^2$, where $w_0$ is the initial spot size and $Z_R=\frac{1}{2}k{w}_0^2$ the Rayleigh length, the characteristic distance for vacuum diffraction. The equation shows that the beam undergoes spreading at a rate determined predominately by its smallest spatial structure, $w_0$ or ${\rho }_0$. Furthermore, as ${\rho }_0$ decreases, the rate of spreading increases.

The continual decrease in the transverse coherence length inhibits nonlinear focusing in turbulence. The interplay of nonlinear and turbulent refraction depends on the particular realization of turbulence, precluding a deterministic prediction akin to the Marburger formula. Nevertheless, the long-term spot size equation above can be generalized, using the moment approach [20], to include nonlinear refraction. The result, $w^2(z)=w_0^2+(1-P/P_{cr}+2w_0^2/{\rho }_0^2){(z/Z_R)}^2$, provides a scaling for the effective self-focusing critical power in turbulence, $P_{cr,eff}=(1+2w_0^2/{\rho }_0^2)P_{cr}$, and illustrates the condition that a pocket of coherence must possess a critical power, $(P/P_{cr})({\rho }_0^2/w_0^2)~1$. A similar condition can be derived for an initially, partially incoherent beam, $(P/P_{cr})(w_s^2/w_0^2)~1$, where $w_s$ is the speckle radius [13]. The primary difference being, of course, that the initial incoherence inhibits nonlinear propagation as opposed to the incoherence developed during propagation.

In the situation considered here, the initially coherent, low power beacon propagates linearly, $P_B<<P_{cr}$, through the turbulence, acquiring spatial incoherence along the way. The outgoing laser beam inherits the intensity profile and the conjugate phase of the received beacon, and thus begins partially incoherent. At low powers, $P_L<<P_{cr}$, the laser beam would recover spatial coherence as it traversed the reverse path of the beacon, reforming the coherent beacon profile at the target—a consequence of reciprocity. At high powers, $P_L~P_{cr}$, nonlinear refraction will inhibit the beams ability to recover spatial coherence—that is, reciprocity is broken. Phase conjugation makes this situation distinct from the scenarios of coherent, nonlinear beam propagation through turbulence, or partially coherent, nonlinear beam propagation. However, both the initial partial incoherence and that developed during propagation can potentially inhibit nonlinear propagation of the high power beam. To examine this situation in more detail, we turn to beam propagation simulations.

4. Simulations

The simulation involves three steps. In the first step, the beacon field is propagated from the target to the receiver using Eq. (1) with $\delta n(x)=\delta n_f(x)+n_{2}I$. The turbulent refractive index, $\delta n_f(x)$, is included using phase screens [21,22]. In the phase screen approximation, the envelope acquires the accumulated phase distortions due to turbulence at discrete axial points along the propagation path. Specifically, after every $\Delta z_s$ the phase $\theta$, is applied to the envelope, $E\to E{e}^{i\theta (r)}$_{, where}

The second simulation step initiates the laser beam envelope with the conjugated and amplified receiver plane beacon envelope: $E_L(r\text{,0})=\eta E_B^{*}(r,0)$ with $\eta >1$. The outgoing high power laser beam thus starts with the full degree of spatial incoherence acquired by the beacon beam during propagation. In the third step, the high power beam is propagated to the target, encountering the same phase screens as the beacon at the appropriate axial positions.

The use of identical phase screens for the beacon and beam propagation creates a monostatic turbulent channel, an example of which is displayed in Fig. 1. This allows us to explore the reciprocity breaking due to nonlinear propagation without the complication of evolving turbulence. In practice, the channel can be considered monostatic if the turbulence correlation time, $\tau ~{\rho }_0/u$ where $u$ is the wind speed, exceeds the total of the beacon and beam transit times and the AO system correction time.

In all of the simulations presented, the initial beacon field had a Gaussian profile, $E_B(r,0)=E_0\exp (-r^2/w_0^2)$. The amplitude, $E_0$, was chosen such that the power, $P_B=\frac{\pi }{4}c{\epsilon }_0{n}_0{w}_0^2{E}_0^2$ was far below $P_{cr}$, ensuring linear propagation. The initial beam power, $P_L$, was varied from below $P_{cr}$ to above $P_{cr}$. Statistical quantities, such as ensemble averages, denoted by ${〈}_{}〉$, and standard deviations, were obtained by simulating the propagation through 10³ statistically independent realizations of monostatic turbulence.

We considered an atmospheric propagation regime where four parameters are required for characterization: $P_L/P_{cr}$ and $z_T/Z_R$ which have already been discussed, the Rytov variance ${\sigma }_r^2=1.23C_n^2{k}^{7/6}{z}^{11/6}$, and the inner scale length ${\ell }_0$. For simplicity, the propagation distance was limited to $z_T=0.12Z_R$, such that in the absence of index fluctuations the beacon would be collimated. The Rytov variance describes the normalized intensity variance of a plane wave, and provides a convenient metric for the optical turbulence strength [18]. In particular ${\sigma }_r^2>1$ provides a rough condition for strong optical turbulence. We note that by using the Rytov variance and the Rayleigh length, the transverse coherence length becomes a redundant parameter: ${\rho }_0/w_0=1.3{\sigma }_r^{-6/5}{(z/Z_R)}^{1/2}$. The ratio of the inner scale to the laser spot size determines the relative importance of beam spreading and wander, with wander dominating when ${\ell }_0/w_0>>1$. In these simulations, the inner scale length was fixed at ${\ell }_0=w_0/8$. The use of normalized parameters in presentation of the simulations allows mapping of the results to a wide range of parameters values.

5. Results

Figure 2(a) displays the ensemble averaged $R(z_T)$ as a function of $P_L/P_{cr}$ for a turbulence strength of ${\sigma }_r^2=6.8$. The dots, squares, and triangles represent the means of $|R(z_T)|$, $\mathrm{Re}[R(z_T)]$, and $\mathrm{Im}[R(z_T)]$ respectively. The swath boundaries illustrate +/− the standard deviation of $|R(z_T)|$. While not plotted, the standard error in the mean is less than 2% for all simulations. The real (imaginary) component of $R(z_T)$ decreases (increases) with increasing beam power consistent with modified propagation of the beam due to nonlinear focusing. We return to the apparent scalings, $\mathrm{Re}〈R(z_T)-1〉\propto -{(P_L/P_{cr})}^2$ and $\mathrm{Im}〈R(z_T)〉\propto (P_L/P_{cr})$ for $P_L/P_{cr}<1.0$, below. The standard deviation of $|R(z_T)|$ increases with the beam power, demonstrating that, even when phase-corrected, high power beam propagation is sensitive to the specific realization of turbulence. As an example, Figs. 2(c) and (d) show two instances of on-target intensity profiles for a beam with $P_L/P_{cr}=1.5$. Figure 2(b) displays the initial beacon intensity profile for comparison. In Fig. 2(d) $\left|R\right|=0.96$, which, by visual inspection, reproduces the beacon profile more closely than Fig. 2(c) where $\left|R\right|=0.28$. However, as we will see below, the degree of reciprocity cannot be judged solely by similarity of the intensity profiles.

 

Fig. 2 (a) Ensemble average of $R(z_T)$ as a function of $P_L/P_{cr}$ for ${\sigma }_r^2=6.8$. The dots, squares, and triangles show the means of the magnitude, and real and imaginary components respectively, and the swathes +/- the standard deviation. (b) the initial on-target beacon intensity profile. (c) and (d) examples of low, $\left|R\right|=0.28$, and high, $\left|R\right|=0.96$, reciprocality, at $P_L={1.5}_{}{P}_{cr}$. The reciprocality drops with increasing power due to nonlinear refraction of the beam.

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In Fig. 3 the quantities ${(P_L/P_{cr})}^{-2}\mathrm{Re}〈R(z_T)-1〉$and ${(P_L/P_{cr})}^{-1}\mathrm{Im}〈R(z_T)〉$ are plotted as a function of ${\sigma }_r^2$ for three different powers. The curves nearly overlap illustrating the $P_L/P_{cr}$ scaling. The real component firsts drops with turbulence strength; then, counterintuitively, increases. The imaginary component shows the opposite trend. A rough scaling can be derived to explain this behavior. The total nonlinear phase acquired by the beam can be approximated as $|k\int n_2{I}_L(x)dz|~2(P_L/P_{cr})(z/Z_R)\equiv 2\alpha$. If this phase is small, we can condense it into a single screen applied to the beam at the transmitter. Using this approximation and reciprocity, one can show

 

Fig. 3 Ensemble average of ${(P_L/P_{cr})}^{-2}\mathrm{Re}〈R(z_T)-1〉$ and ${(P_L/P_{cr})}^{-1}\mathrm{Im}〈R(z_T)〉$ as a function of ${\sigma }_r^2$ for $P_L={0.25}_{}{P}_{cr}$ red triangles, $P_L={0.5}_{}{P}_{cr}$ green squares, and $P_L={1.0}_{}{P}_{cr}$ blue dot. The initial drop and subsequent rise with turbulence strength is associated with the same behavior in the scintillation index.

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The above scaling suggests that the dip and rise in $〈R〉$ with ${\sigma }_r^2$ result from the same behavior observed in the scintillation index [18,24]. In the weak turbulence regime, the spatial phase distortions increase with turbulence strength. This, in turn, enhances the irradiance fluctuations causing ${\sigma }_I^2$ to grow. The initial drop in $〈R〉$ can thus be interpreted as follows. At low powers, every beam ‘ray’ has a reciprocal beacon ‘ray’. This reciprocal ray pair propagates through the turbulence along the same path, but in opposite directions. At high powers, the beam ray undergoes nonlinear refraction, continually deviating it from the path of its reciprocal counterpart. The random refraction experienced along the deviated path increases with turbulence strength, leading to greater, on average, path differences between the rays. This leads to spatial phase difference and irradiance profile disjointedness at the target.

In the strong turbulence regime, the light becomes sufficiently spatially incoherent that the irradiance fluctuations saturate. The received beacon, and hence the initial transmitted high power beam’s, irradiance profile resembles that resulting from a collection of random sources [24]. As discussed above, two effects inhibit nonlinear propagation in this regime. First, the effective critical power of an incoherent beam surpasses that of a coherent beam [13]. Second, as described in [14] by Peñano et al., strong turbulence inhibits self-focusing. A combination of these effects results in linear-like propagation, and a corresponding increase of $〈R〉$ with turbulence strength.

Departures of $\left|R\right|$ from unity can occur from both spatial phase differences and irradiance disjointedness between the beacon and beam. For many applications, such as power beaming and directed energy, the on-target quality of the irradiance profile, not the phase, is of primary interest. To examine this, we defined a modified reciprocity metric

 

Fig. 4 Ensemble averages of $|R(z_T)|$, blue dots, and $|R_I(z_T)|$, red triangles, as a function of $P_L/P_{cr}$ for ${\sigma }_r^2=4.6$, top, and ${\sigma }_r^2$ for $P_L/P_{cr}=1.0$, bottom. The swathes indicate +/- the standard deviation. At the target, the drop in reciprocity is dominated by phase differences between the beacon and beam.

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6. Discussion

The simulation results presented above were specific to a Gaussian profile beacon embedded in dissipationless, Kerr-nonlinear turbulent atmosphere. Because the reciprocality is found by integrating over the spatial structure of the on-target beacon and beam profiles, similar qualitative trends can be expected for different beacon profiles. The reciprocality would be most sensitive to the exact beacon profile in weak turbulence. In this regime, the light received from the beacon has developed only mild spatial incoherence and still resembles its initial profile; recall $z_T/Z_R=.12$ (near field). The high power beam inherits the conjugate profile, and because the nonlinear refraction and self-focusing critical power depend on that profile, so to will the power and rate at which the reciprocality drops. In strong turbulence, on the other hand, the light received from the beacon is largely incoherent, bearing little resemblance to the initial profile. As discussed in section 3, the ‘sub-beam’ pockets of incoherence will determine the nonlinear refraction and effective self-focusing critical power of the high power beam. This should make the reciprocality relatively insensitive to the exact beacon profile in this regime. These arguments are, however, qualitative; in a future study one could examine the quantitative effect of different beam profiles.

As discussed in section 2, reciprocity holds in media with either gain or loss. In the simulations above the reciprocity was effectively broken because the beacon rays propagated linearly, whereas the high power ‘rays’ underwent nonlinear refraction. If the medium had gain or loss, the change in beam power along the propagation path would directly affect the nonlinear refraction. In the case of gain, the increase in beam power along the propagation path would accelerate the nonlinear refraction and result in the reciprocality dropping at lower initial beam powers. Of more relevance to atmospheric propagation is the situation of loss. During atmospheric propagation the laser pulse power depletes due to absorption and large angle scattering off of aerosols. This loss of power would limit the nonlinear refraction and lead to the reciprocality dropping at a larger initial beam power.

Finally, the nonlinear reciprocity demonstrated in section 2 holds from any real, intensity dependent refractive index. If, in the simulations, the nonlinearity were something other than the Kerr term, the reciprocality would still drop with increasing pulse power, but the scaling parameter would no longer be the self-focusing critical power. Using the nonlinear phase screen approximation used to derive Eq. (7), one can come up with an equivalent expression valid for any dissipationless random media or nonlinearity

7. Summary and conclusions

We have examined nonlinear reciprocity breakdown when AO phase correction is applied to high power laser beams propagating in random media. A metric, the overlap of a high peak power beam field and that of a beacon, was introduced to quantify reciprocity breaking. To illustrate the breaking, an ideal phase-conjugation based AO implementation was applied to propagation through Kerr-nonlinear atmospheric turbulence. As expected, the reciprocity metric, or reciprocality, was found to drop with increasing beam power. For weak turbulence and fixed power, the reciprocality dropped with increasing turbulence strength. Surprisingly, in the strong turbulence regime, the reciprocality increased with turbulence strength due to incoherence and turbulence inhibiting nonlinear self-focusing. A simple scaling was provided to explain the behavior. The drop in reciprocality with beam power was shown to be dominated by phase differences, suggesting AO correction can be effective in applications requiring irradiance, not phase, quality.