## Abstract

This paper conducts experiments that demonstrate the utility of a general scaling law (GSL) for far-field propagation. In practice, the GSL accurately predicts the diffraction-limited peak irradiance in a far-field plane, regardless of the beam shape in a near-field plane. Within the experimental setup, we use a reflective, phase-only spatial light modulator to generate various beam shapes from expanded and collimated laser-source illumination, including both flattop and Gaussian beams with obscurations, in addition to phased arrays with these beam shapes. We then focus the resulting near-field source plane to a far-field target plane and measure the peak target irradiance to compare to the associated GSL prediction. Overall, the results show excellent agreement with less than 1% error for all test cases. Such experiments present a convenient and relatively inexpensive approach to demonstrating laser-system architectures (of varying complexity) that involve far-field propagation.

## 1. INTRODUCTION

Laser systems are multifaceted systems that involve many sub-system components including the laser-source gain medium, laser-source cavity, beam-clean-up optics, beam-train optics, beam-control optics, and beam-director optics [1–4]. Modeling such systems from first principles [5–9] is a cumbersome and time-consuming process. Thus, to alleviate the mathematical burden, one can develop simple formulas or “scaling laws” as a first-order model of the system.

Traditional scaling-law formulations start with a Gaussian fit to the Airy disk [4,10]. From there, one can account for the far-field peak irradiance and beam spread by adding the variances of independent physical processes. Such a formulation is convenient from the standpoint that one can use an error-budget tree to account for all of the variance calculations [2,4]. In general, error budgets help to simplify the process of designing multifaceted laser systems so that system engineers can keep track of the requirements for the various sub-system components.

By starting with a Gaussian fit to the Airy disk, traditional scaling-law formulations assume the use of a uniform, flattop beam shape. However, if the beam has a different shape, then one must resort to approximations, often of uncertain accuracy, to estimate the diffraction-limited peak irradiance and beam spread [11–14]. One such example is that of a centrally obscured and clipped Gaussian beam, which has no closed-form solution for far-field propagation.

The aforementioned approximations recently motivated the development of a general scaling law (GSL) [15]. In practice, this scaling-law formulation accurately predicts the diffraction-limited peak irradiance and small angle beam spread in a far-field plane, regardless of the beam shape in a near-field plane. The GSL is simple in its formulation and accurately predicts, for example, the diffraction-limited peak irradiance and small angle beam spread associated with the far-field propagation of a centrally obscured and clipped Gaussian beam, in addition to phased arrays with these beam shapes [3,4]. In turn, the GSL is of utility to system engineers because it allows for more accurate error budgets when designing multifaceted laser systems.

With the benefits of the GSL in mind, there are limitations to its
usefulness in terms of predicting the non-diffraction-limited, far-field
peak irradiance and beam spread. Analogous to traditional scaling-law
formulations, the GSL, as formulated in Ref. [15], relies on the extended Maréchal approximation and a
total-variance Gaussian fit to account for the non-diffraction-limited,
far-field peak irradiance and beam spread, respectively. Such a
formulation unfortunately gives inconsistent results, particularly when
dealing with non-independent physical processes. For instance, the
preliminary analysis of Bingham *et al.* led
to inconsistent results when comparing the predictions of the GSL to those
obtained with wave-optics simulations for turbulent propagation [16]. These shortcomings, however, are
more so associated with the limitations of the extended Maréchal
approximation [17–19] and
the total-variance Gaussian fit [2,4,10], as opposed to the exact, diffraction-limited
predictions of the GSL.

The aforementioned limitations recently motivated the development of an extension to the GSL [20,21]. This extension started with the GSL, as formulated in Ref. [15], and derives a formula for the far-field irradiance pattern associated with partially coherent beam shapes of varying complexity. In addition to diffraction effects, the formula accounts for multiple random effects including turbulence and jitter using multiplexed modulation transfer functions. As such, one can readily account for additional random effects (e.g., beam quality [22,23]). The extended GSL is simple in its formulation and accurately predicts, for example, the turbulence-limited peak irradiance and beam spread associated with the far-field propagation of a centrally obscured and clipped Gaussian beam, in addition to phased and unphased arrays with these beam shapes [24,25]. What is more, the analysis shows that a partially coherent beam propagated under the influence of multiple random effects is the convolution of the irradiance associated with a partially coherent beam propagated in vacuum with the total-system’s point spread function representing the random effects. Thus, the scaling-law formulation is extensible to non-independent physical processes like thermal blooming [26–29].

With all the benefits of the GSL and extended GSL in mind, the work reported here conducts experiments that demonstrate the utility of the GSL for far-field propagation. For this purpose, we experimentally generate various beam shapes from expanded and collimated laser-source illumination using a reflective, phase-only spatial light modulator (SLM) within the experimental setup. These beam shapes include both flattop and Gaussian beams with obscurations, in addition to phased arrays with these beam shapes. We then focus the resulting near-field source plane to a far-field target plane and measure the peak target irradiance to compare to the associated GSL prediction. The GSL, at its core, accurately predicts the diffraction-limited peak irradiance in a far-field plane, regardless of the beam shape in a near-field plane. Accordingly, comparisons of the GSL predictions to the experimental results ultimately show agreement with less than 1% error for all test cases. Such experiments, in turn, present a convenient and relatively inexpensive approach to demonstrating multifaceted laser-system architectures that involve far-field propagation.

In what follows, Section 2 formulates the GSL referred to throughout this paper. Section 3 then discusses the details of the experimental setup used to demonstrate the utility of the GSL, and Section 4 presents the experimental results. A conclusion to this paper follows in Section 5.

## 2. BACKGROUND THEORY

In this section, we follow the approach taken in Ref. [15] and formulate the GSL referred to throughout this paper. For this purpose, we consider a generic laser system comprised of laser-source illumination with an arbitrary beam shape being transmitted through an optical system (e.g., a beam-expanding telescope with an obscuration or a phased array of such telescopes) and thereafter focused at a distant target. As shown in Fig. 1, we represent the limiting aperture of this optical system using a pupil function, $p({{x_1},{y_1}})$, where ${x_1}$ and ${y_1}$ represent the transverse coordinates at the exit pupil plane (also referred to as the near-field plane). Thus, $p({{x_1},{y_1}})$ is analogous to an aperture transmittance function in Fourier optics [5–9].

If the target range satisfies the conditions for far-field propagation [5–9], then the results discussed herein should be equally applicable. However, in what follows, we appropriately assume that the quasi-monochromatic light leaving the exit pupil plane focuses at a distant target with the use of a thin-lens transmittance function. The Fresnel diffraction integral, as a result, reduces to the following Fraunhofer diffraction integral:

With Fig. 1 in mind, it is convenient to derive precise formulas for the far-field metrics of interest. One such metric of particular interest is the diffraction-limited peak (on-axis) irradiance, ${I_Z}$. Thus, setting ${x_2} = 0$ and ${y_2} = 0$, ${I_Z}$ follows from Eq. (1) as

To accommodate different laser-system architectures, we define another near-field source parameter referred to as the field-effective area, ${A_{{\rm{eff}}}}$. In units of area,

We also define the unitless pupil-field factor, ${F_f}$, such that

It is important to note that the definitions for the field-effective area, ${A_{{\rm{eff}}}}$, and pupil-field factor, ${F_f}$, involve the complex-optical field. These formulations are in contrast to traditional scaling-law formulations, which typically involve the irradiance [11–14]. Nonetheless, both ${A_{{\rm{eff}}}}$ and ${F_f}$ are straightforward to derive for common beam shapes. We summarize these derivations in Table 1 and refer the reader to Ref. [15] for further insights.

With Table 1 in mind, what is particularly intriguing about the GSL formulated in Eq. (5) is that regardless of the beam shape, the diffraction-limited peak irradiance, ${I_Z}$, is the same given equivalent near-field source parameters.

We illustrate this last point in Fig. 2 for two different beam shapes in the near-field source plane [20]. In particular, one beam shape is a uniform, flattop beam with a central obscuration, and the second is a Gaussian beam with a central obscuration. They both have the same wavelength, $\lambda$, of 1 µm. The flattop beam has a diameter of 15 cm, and the Gaussian beam has a (1/e) field diameter of 10.6 cm. Furthermore, the obscuration radius is 3 cm for the flattop beam and 2.17 cm for the Gaussian beam. For both beam shapes, the laser-source illumination has a total power, ${P_L}$, of 1.0 W, and the field-effective area, ${A_{{\rm{eff}}}}$, and pupil-field factor, ${F_f}$, are $0.018\;{\rm m}^2$ and 0.837, respectively. These near-field source parameters result in an identical diffraction-limited peak irradiance, ${I_Z}$, of $123.8\;{\rm W} / {\rm m}^2$ for both cases when focused at a target range, $Z$, of 10 km [cf. Eq. (5)]. Note that we calculated the diffraction-limited irradiance profiles shown in Fig. 2 using a wave-optics approach that efficiently solves the Fresnel diffraction integral using matrices [30]. Also note that even though both profiles have the same ${I_Z}$, the impact of the differing beam shapes showed up in the sidelobes away from the on-axis peaks at ${x_2} = 0$ and ${y_2} = 0$.

## 3. EXPERIMENTAL SETUP

In this section, we describe the experimental setup needed to demonstrate the utility of the GSL formulated in Section 2. As shown in Fig. 3, we used a diode-pumped, solid-state laser (Oxxius LBX-532S-300) operating in the green at 532 nm. We expanded the specified beam waist of 0.65 mm (${{1}}/{{\rm{e}}^2}$) ${{40}} \times$ by a combination of ${{2}} \times$ and ${{20}} \times$ beam expanders (Thorlabs BE2M-A and BE20M-A). The collimated output greatly overfilled the active region of a reflective, phase-only SLM and essentially provided uniform illumination.

The SLM (Holoeye Photonics AG Pluto-VIS) had ${{1920}} \,\times {{1080}}$ pixels on an 8 µm pitch with 92% fill factor. We interfaced the SLM to a desktop computer using the DVI port on a standard graphics card. After calibration, the SLM provided programmable ${{0 {-} 2}}\pi$ phase shifts on a pixel-by-pixel basis. Thus, we had complete control over the global and local aperture transmittance functions commanded to the SLM. We also had complete control over the total-multiplexed transmittance function and ensured far-field propagation by commanding a global thin-lens transmittance function. This function focused the expanded and collimated laser-source illumination at the target, which we located 9.4 m away from the SLM.

The target was simply the annular region of a commercially available compact disc (CD) with a diffuse-white optically rough surface. During experimentation, we spun the CD with a DC motor to provide frame-to-frame speckle averaging relative to the camera-integration time. We then used a universal serial bus (USB) camera (Point Grey Research CMLN-13S2M-CS monochrome) and adjustable zoom lens (Sigma APO150-500 DG OS HSM F) to image the target irradiance pattern. The camera provided ${{1280}} \times {{960}}$ pixel resolution with 12-bit digitization. A USB connection allowed the computer to grab and store the imaged camera frames. Through this connection, we also had complete control the camera-integration time and camera-signal gain.

It is important to note that the relatively low fill factor of the SLM pixels (92%) created a phase-grating effect that led to a large number of reflected diffraction orders. To place the target irradiance pattern away from these orders, we commanded a global tilt-phase transmittance function to the SLM, leading to the target irradiance pattern falling between diffraction orders in a quiet zone. Given the SLM-pixel pitch (8 µm) and, hence, the diffraction-order spacing, the required global tilt-phase transmittance function led to the wrapped phase being commanded to the SLM, as illustrated in Fig. 4.

With Fig. 4 in mind, it was desirable to have both flattop and Gaussian beam shapes as part of the experimental setup, despite only having uniform illumination incident on the active region of the reflective, phase-only SLM. In practice, one can achieve Gaussian beam shapes by modulating the heights of the phase grating formed by the SLM pixels. The grating efficiency, $\eta ({x,y})$, follows as

where ${\rm{sinc}}(\circ) = {{\sin ({\pi \circ})} / ({\pi \circ})}$ and $\varphi ({x,y})$ is the wrapped phase commanded to the SLM. Thus, we set the grating efficiency equal to the desired Gaussian beam shape [i.e., $\eta ({x,y}) = G({x,y})$] and solved for $\varphi ({x,y})$. We then commanded this wrapped phase to the SLM.As shown in Fig. 5, by modulating the heights of the phase grating formed by the SLM pixels [cf. Eq. (6)], we achieved the desired near-field Gaussian beam shape. Specifically, in Fig. 5(a), we show an irradiance pattern, along with Gaussian curve fits. Furthermore, in Fig. 5(b), we show horizontal and vertical irradiance profiles (through the near-field peak), along with Gaussian line fits. From these fits, we determined the horizontal and vertical (${{1/}}{{\rm{e}}^2}$) beam widths as 63 and 62 camera pixels, respectively. These results indicate that we effectively produced Gaussian beam shapes, despite only having uniform illumination incident on the active region of the reflective, phase-only SLM.

## 4. EXPERIMENTAL RESULTS

In this section, we conduct experiments that demonstrate the utility of the GSL formulated in Section 2. With Eq. (5) in mind, we provide results for a wide range of beam shapes with equivalent and varying near-field source parameters. Overall, the results show excellent agreement with less than 1% error.

#### A. Experimental Results with Equivalent Near-Field Source Parameters

The purpose of the following experiment was to show that regardless of beam shape in the near-field source plane, the diffraction-limited peak irradiance in the far-field target plane was the same given equivalent near-field source parameters. For this purpose, we conducted an experiment with five different aperture transmittance functions, all with an equivalent pupil-field factor, ${F_f}$ [cf. Eq. (4) and Table 1]. As shown in Fig. 6, we used a circular aperture with both on-axis and off-axis circular and square obscurations. The circular aperture had a radius of 128 SLM pixels, and the circular obscurations had radii of 64 SLM pixels. We then ensured that the circular and square obscurations had the same area (to within 0.23% error).

In accordance with the experimental setup described in Section 3, we commanded the aperture transmittance functions shown in Fig. 6 to the SLM, and we used the camera to image the target irradiance pattern. The camera-integration time and camera-signal gain remained constant throughout the experiment. For each case, we collected 10 camera images for speckle averaging.

In Fig. 7, we provide target irradiance patterns for two different aperture transmittance functions [cf. Figs. 6(b) and 6(e)]. The leftmost column shows the corresponding aperture transmittance functions, whereas the middle and rightmost columns show the target irradiance patterns from the experiment and a wave-optics calculation, respectively. Analogous to Fig. 2, the wave-optics calculation used an efficient-matrix approach [30] to solve the Fresnel diffraction integral for the same near-field source parameters as the experiment, but with observation directly in the far-field target plane. Overall, the experimental results agree well with the wave-optics calculations; thus, we concluded that 10 camera images were enough for speckle averaging.

With Fig. 7 in mind, we obtained target irradiance patterns for each aperture transmittance function shown in Fig. 6. Thereafter, we determined the horizontal and vertical target irradiance profiles (through the far-field peaks), as shown in Fig. 8. It is important to note that, in Fig. 8, we present the results in terms of the raw camera signal; thus, we did not scale the peaks. As predicted by the GSL formulated in Section 2 [cf. Eq. (5)], the peaks are invariant to the position and shape of the obscurations. To quantify this last statement, we calculated the percent error as $\varepsilon = 100 \times {{| {N_{{\rm{Cam}}}^{{\max}} - N_{{\rm{Cam}}}^{{\min}}} | / N_{{\rm{Cam}}}^{{\max}}}}$, where $N_{{\rm{Cam}}}^{{\max}}$ and $N_{{\rm{Cam}}}^{{\min}}$, respectively, denote the maximum and minimum peaks obtained from the raw camera signals. Overall, we obtained less than 1% error for all test cases presented in Figs. 8(a) and 8(b).

#### B. Experimental Results with Varying Near-Field Source Parameters

In the next set of experiments, we varied the pupil-field factor, ${F_f}$ [cf. Eq. (4) and Table 1]. Again, the purpose was to show that regardless of beam shape in a near-field source plane, the GSL formulated in Section 2 accurately predicted the diffraction-limited peak irradiance in the far-field target plane. For this purpose, we increased the complexity of the laser-system architecture of interest and used hexagonal close-packed (HCP) subapertures [3,4] with varying on-axis circular obscurations for the aperture transmittance functions. We also used both Gaussian and flattop beamlets for the laser-source illumination (cf. Fig. 5). In turn, we created phased arrays with seven-HCP subapertures and Gaussian beamlets, in addition to phased arrays with 61-HCP subapertures and flattop beamlets.

In Fig. 9, we show the relative peak target irradiance, as predicted by the GSL [cf. Eq. (5)] and the experimental results, for the phased arrays with seven-HCP subapertures (with varying on-axis circular obscurations) and Gaussian beamlets. Here, the radius of each subaperture is 170 SLM pixels, and the on-axis obscuration radius of each subaperture varies from zero to 83 pixels. For each case, we collected 10 camera images for speckle averaging, and both the camera-integration time and the camera-signal gain remained constant throughout this experiment. As predicted by the GSL formulated in Section 2, the peak target irradiance is linear with the square of the pupil-field factor, ${F_f}$, since we held all other near-field source parameters constant.

In Fig. 10, we show the relative peak target irradiance, again as predicted by the GSL [cf. Eq. (5)] and the experimental results, for phased arrays with 61-HCP subapertures (with varying on-axis circular obscurations) and flattop beamlets. Here, the radius of each subaperture is 54 SLM pixels, and the on-axis obscuration radius of each subaperture varies from zero to 24 pixels. For each case, we collected 10 camera images for speckle averaging, and both the camera-integration time and the camera-signal gain remained constant throughout this experiment. As predicted by the GSL formulated in Section 2, the peak target irradiance is linear with the square of the pupil-field factor, ${F_f}$, since we held all other near-field source parameters constant.

It is important to note that in Figs. 9 and 10, we scaled the peak target irradiance value to unity creating a relative peak target irradiance. In particular, for the GSL predictions we reduced Eq. (5) to ${I^\prime _{{\rm{GSL}}}} = F_f^2$, whereas for the experimental results we scaled each peak from the raw camera signal, ${N_{{\rm{Cam}}}}$, to the maximum peak obtained from the raw camera signals, $N_{{\rm{Cam}}}^{{\max}}$, such that ${I^\prime _{{\rm{Exp}}}} = {{{N_{{\rm{Cam}}}}}} / N_{{\rm{Cam}}}^{{\max}}$. Here the primes denote a relative peak target irradiance. To compare the predictions from the GSL to the experimental results, we then calculated the percent error as $\varepsilon = 100 \times {{| {{{I}_{{\rm{GSL}}}^\prime} - {{I}_{{\rm{Exp}}}^\prime}} | / {I}_{{\rm{GSL}}}^\prime}} $.

Again, we obtained less than 1% error for all test cases presented in Figs. 9 and 10. Since these results were for both Gaussian and flattop beamlets, in addition to a wide range of subaperture and obscuration configurations, they further demonstrate the utility of the GSL formulated in Section 2. Regardless of beam shape in the near-field source plane, Eq. (5) accurately predicts the diffraction-limited peak irradiance in the far-field target plane.

## 5. CONCLUSION

This paper conducted experiments that demonstrate the utility of a GSL for far-field propagation. As part of the experimental setup, we used a reflective, phase-only SLM to generate various beam shapes from expanded and collimated laser-source illumination, including both flattop and Gaussian beams with obscurations, in addition to phased arrays with these beam shapes. We then focused the resulting near-field source plane to a far-field target plane and measured the irradiance profile to compare to the associated GSL prediction. Overall, the GSL accurately predicted the diffraction-limited peak irradiance in the far-field target plane, regardless of the beam shape in the near-field source plane, and the experimental results showed excellent agreement with the GSL predications (with less than 1% error for all test cases).

Moving forward, we hope that system engineers make use of the exact, diffraction-limited predictions of the GSL demonstrated in this paper and appropriately use them as part of their overall scaling-law formulation. In practice, different scaling-law formulations have different limits of applicability, so system engineers should proceed with caution (as discussed above in the introduction to this paper). Nonetheless, systems engineers effectively have a new tool in their toolkit to help them keep track of the requirements for the various sub-system components that make up multifaceted laser systems.

## Acknowledgment

The authors of this paper would like to thank the Joint Directed Energy Transition Office for sponsoring this research, as well as R. A. Carreras and D. E. Thornton for many insightful discussions regarding the results presented within. Approved for public release; distribution is unlimited. Public Affairs release approval #AFMC-2018-0340.

## Disclosures

The authors declare no conflicts of interest.

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