Photonic solution to phase sensing and control for light-based interstellar propulsion

Chathura P. Bandutunga; Paul G. Sibley; Paul G. Sibley; Michael J. Ireland; Robert L. Ward

doi:10.1364/JOSAB.414593

1. INTRODUCTION

The Breakthrough Starshot Initiative [1] aims to send a probe to image a planet orbiting Proxima Centauri and return data, at a distance of four light-years from Earth with a transit time shorter than a human lifetime. The energy scales required for this effectively rule out chemical propulsion, and the best remaining option is the acceleration of a vehicle through radiation pressure [2–4] from a laser field. The concept is for a ground-based laser array, known as the light beamer or photon engine, to illuminate a sail-endowed spacecraft—the sailcraft [5,6]—accelerating it via radiation pressure to $0.2c$ within a few minutes [7]. To illuminate the sailcraft during acceleration with limited diffraction losses, the ground-based laser array will be need to be kilometers in scale. To limit losses due to atmospheric turbulence [8], the photon engine will require an adaptive optics system. If we assume a system with piston-phase actuation and a Fried scale of roughly 10 cm, we conclude that of order ${10^8}$ piston-phase actuators and relevant sensing are required per square kilometer of array size. In addition, the photon engine will need to support the ability to provide beam shaping to enable the sailcraft to stably ride the beam [9–11].

Here, we present a modularized, internally sensed fiber optical phased array (OPA) architecture for the photon engine of the Breakthrough Starshot program, with a sensing technique scalable to ${10^8}$ emitters, and a method to sense atmospheric phase fluctuations. The architectural choices are predicated on the limits of currently available technologies with only modest extrapolations. We also propose and simulate the performance of an interferometric satellite laser guide-star technique for measuring and compensating for atmospheric and external path length changes.

Coherent beam combination at optical wavelengths by phase matching of multiple paths has been done through gradient descent algorithms [12] or multiplexed phase measurements from individual emitters [13–15]. Using the latter approach, we consider an internally sensed OPA based on an architecture initially demonstrated in [16,17]. Being internally sensed, this system requires no external beam sampling optics that might prove problematic for kilometer-scale arrays. The system employs digitally enhanced heterodyne interferometry (DEHeI) [18], using pseudo-random binary codes to uniquely identify and demodulate the interference signal from multiple lasers. This capability to uniquely identify optical paths permits the measurement of the optical phase acquired along the non-common light paths to each emitter in the OPA. Our architecture proposes a hybrid method that combines DEHeI multiplexed phase sensing with wavelength division multiplexing (WDM) to enable a multilayer hierarchical control scheme capable of individual phase control of an arbitrary number of optical emitters. This hybrid multiplexing platform is able to surpass the multiplexed sensing limits of both DEHeI and WDM, making the scale required for the photon engine attainable. Furthermore, the phase control scheme can additionally be used for high speed (megahertz bandwidth) beam forming and atmospheric pre-correction via piston/phase actuation on each emitter with no mechanical actuation required.

Compensating for atmospheric fluctuations is critical for efficient optical power delivery to the sailcraft. Previous demonstrations of such wavefront correction using OPAs have used methods that sense reflected power from the optical receiver, in this instance, the sailcraft. These methods [19,20] use optical power measurements of back-scattered light that returns to the emitter plane and utilize gradient descent algorithms to maximize the aforementioned power. For Starshot, however, any method that relies on back-reflected light from the sailcraft will have to deal with long delays due light travel time at interplanetary distances.

To eliminate the long, and changing, delays from using light reflected from the sailcraft, we obtain the external phase measurements required for atmospheric pre-correction with a satellite-based laser guide-star named the “Beacon.” The use of Earth-based laser guide-stars for adaptive optics is commonplace in astronomy [21,22], and satellite-borne laser guide-stars have also been proposed [23,24]. These techniques may be directly employed on small-scale OPA systems. However, once scaled beyond the Fried parameter, these methods require linking of independent atmospheric measurements at each individual sub-aperture of the array. Methods to synthesize an atmospheric phase image through multiple apertures have been investigated, with the primary limit being the computational overhead as the aperture count increases [25].

Our proposal differs from previous adaptive optics solutions where atmospheric perturbations are typically measured as phase slopes via a Shack–Hartmann wavefront sensor, and suppression of these phase slopes in a control loop corrects the piston-free phase over a broad wavelength range. Instead, we directly measure the wavefront phase at each emitter aperture. The required pre-correction is then fed-forward to the outgoing emitter lasers. For this, the algorithm we propose is able to take the inter-emitter phase difference with a form of two-color interferometry using light from the Beacon-based guide-stars and project to the phase difference at the emitter wavelength. As the phase difference (similar to a gradient) between neighbor emitters or each array aperture can be directly computed, given a single reference point, for example, an aperture near the center of the array, this would allow for a phase correction for the entire array to be computed using nearest neighbor measurements.

The bandwidth of useful phase corrections is thus limited by the light-transit time of the lower atmosphere; as this time-scale (1 ms) is much shorter than the time-scale of turbulent perturbations at good sites (10–50 ms), it does not pose any practical limitations. Furthermore, due to the wavelength separation between the Beacon light and the light emitted from the array, this wavefront sensing architecture is resistant to phase errors due to internal back-scattering events from the photon engine emitters and able to resolve the phase ambiguity from off-plane inter-emitter positioning errors up to 1 m in magnitude.

The paper hereafter is divided into three parts. In Section 2, we describe our solution for scaling OPA sensing and control to the required ${10^8}$ emitters using a hierarchical extension to existing methods; in Section 3, we discuss a method to sense the phase perturbations that result from atmospheric turbulence, fully compatible with the architecture from Section 2; in Section 4, we present a set of simulation results that demonstrates that the techniques presented in Sections 2 and 3 can meet the requirements of Breakthrough Starshot.

2. INTERNAL PHASE SENSING

A. Emitter Laser Phase Stabilization

The fundamental unit of the OPA is an emitter. For the OPA output to achieve global coherence, the relative phase between each emitter of the array must be sensed and stabilized. This requires each individual emitter to have a set of phase measurements that sense dynamic optical path length fluctuations from all contributing optical paths. We then require phase actuation back onto the array at specific points to ensure we can independently control and stabilize each emitter.

Based on the frequency-offset phase-locked OPA architecture demonstrated by Roberts et al. [17], we proceed with emitters that are individually phase locked to a common reference field [local oscillator (LO)]. The sensing of each emitter can be achieved using two detection points, and subsequently controlled via two actuation points, shown in Fig. 1. The first of these, the forward lock [Fig. 1(a)], senses the phase accrued due to dynamic path length fluctuations in the emitter output path relative to the OPA arm. This measurement is made with respect to an RF frequency shifted LO field (emitter LO). The emitter forward phase ${\phi _{e,f}}$, the relative phase between the emitter, and the emitter LO can then be written as per Eq. (1):

(1)$${\phi _{e,f}} = n({\lambda _e})\left[{\frac{{{L_c}[e]}}{{{\lambda _e}}} - \left({\frac{{{L_a}}}{{{\lambda _e}}} + \frac{{{L_b}[e]}}{{{\lambda _e}}}} \right)} \right],$$

where ${L_a}$, ${L_b}[e]$, and ${L_c}[e]$ are physical path lengths through the emitter arm of the OPA, and $n(\lambda)$ is the refractive index evaluated at ${\lambda _e}$, the emitter wavelength. The forward lock uses this phase measurement directly as a feedback signal, actuating on the emitter laser frequency. When stabilized with feedback, at Fourier frequencies below the locking bandwidth of the system, the emitter forward phase is driven to zero.

Fig. 1. (a) The forward lock uses the measurement made at the forward path detector to stabilize the phase difference between the main laser and its corresponding local oscillator. This stabilizes the phase accrued from the main laser output to the corresponding OPA arm. (b) The return lock uses the differential measurement between the two OPA arms, each measured with respect to their local oscillator. The difference is used as a control signal to actuate on the emitter OPA arm. This is done through an electro-optic modulator and ensures the OPA arms are phase locked.

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The forward locking loop is independent for each emitter and reference laser, as shown in Fig. 1. In the case of the reference laser, such as the module reference laser, this measurement is made using a separate RF shifted LO field (module LO). This allows for the module reference and LO wavelength to be well separated from the emitter wavelength, easing requirements on WDM hardware. The LO fields are offset phase locked to each other using a wavelength reference, such as an optical frequency comb (OFC).

Although the emitter and module reference lasers can be phase locked to their respective OPA arms using the forward path lock, the path lengths of these arms may drift. The return path lock, shown in Fig. 1(b), is then used to phase lock OPA arms from each emitter to a common reference path, in this case, the module reference. For this, measurements for each emitter and the reference path must be made at a common detection point. When the forward path is locked, the return path detector can be used to obtain phase measurements for both the emitter and module reference OPA arms. This detector uses the small amount of retro-reflected light from the OPA output, which in the case of the module reference is replaced by a mirror. The measurement made at the emitter wavelength is shown in Eq. (2) with an analogous expression for the measurement at the module reference wavelength:

(2)$${\phi _{e,r}} = n({\lambda _e})\left[{\frac{{2{L_d}[e]}}{{{\lambda _e}}} + \frac{{2{L_b}[e]}}{{{\lambda _e}}} + \frac{{{L_i}}}{{{\lambda _e}}} - \frac{{{L_j}}}{{{\lambda _e}}}} \right],$$

where ${L_b}[e]$ and ${L_d}[e]$ are the double passed paths seen by the reflected emitter field. The return path control loops use the differential measurement between the emitter and module reference arm, shown in Eq. (3):

(3)$$\begin{split}{\phi _{m,r}} - {\phi _{e,r}} &= n({\lambda _m})\left[{\frac{{2{L_d}[m]}}{{{\lambda _m}}} + \frac{{2{L_b}[m]}}{{{\lambda _m}}} + \frac{{{L_i}}}{{{\lambda _m}}} - \frac{{{L_j}}}{{{\lambda _m}}}} \right] \\&\quad- n({\lambda _e})\left[{\frac{{2{L_d}[e]}}{{{\lambda _e}}} + \frac{{2{L_b}[e]}}{{{\lambda _e}}} + \frac{{{L_i}}}{{{\lambda _e}}} - \frac{{{L_j}}}{{{\lambda _e}}}} \right].\end{split}$$

Lengths ${L_i}$ and ${L_j}$ are paths common to both the module and emitter measurements, but due to the wavelength separation between the emitter and module reference, this results in a static phase difference in the differential. This is actuated onto the electro-optic modulators (EOMs) located in each emitter arm, as shown in Fig. 1(b).

B. Hierarchical Locking Scheme

We begin describing the hierarchy with the smallest OPA structure of the array, shown in Fig. 2. This is the module, with each module consisting of 100 emitters. LO fields are injected in from the module input. Using the emitter LO field, each emitter individually makes a forward path phase measurement, shown in Fig. 2(a), which is used to offset phase lock the emitter laser to the emitter LO. This is analogous to the single emitter scenario presented in Fig. 1.

Fig. 2. Schematic overview of a single module, the primary sub-array unit for the proposed array. The phases of individual emitter lasers are measured at the (a) forward path, and (b) return path detectors with respect to the emitter LO field. The module reference, highlighted in green, consists of a (c) reference arm, the phase of which is measured at its (d) forward and (e) return path detection points with respect to the module LO. Each emitter within the module is tagged with a unique DEHeI code to facilitate individual demodulation at the emitter RPD. The module reference is also tagged with a DEHeI code that is used for hierarchical control. (f) The module reference and emitters are wavelength separated to enable WDM demultiplexing prior to the return detection point, which also enables the integration of dedicated photodetectors for atmospheric correction using wavelength separated Beacon satellites.

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The return path phase measurement, shown in Fig. 2(b), however, is now multiplexed for all emitters within a module. This multiplexing capability is achieved using a unique DEHeI modulation for each emitter, applied by the EOM within each emitter arm.

Using a common, wavelength separated module reference, shown in Fig. 2(c), all emitters within the module can be phase locked to each other. The module reference phase measurements, with respect to the module LO field, are made at forward and return path reference detection points, shown in Figs. 2(d) and 2(e), respectively. From this measurement and those of individual emitters, we can generate error signals, described previously to phase lock each emitter to the module reference. Once locked, this results in a phase coherent output from all emitters within the module. Furthermore, the relative phase between each emitter and the module reference can be individually offset and modulated. This enables the module to generate arbitrary output wavefronts for beam steering and atmospheric pre-correction.

To link together two or more modules, we require only that the module references from each be phase locked. We define a new, wavelength separated “sector” reference to which all individual module references are compared. This next level of hierarchy (the sector, composed of 1000 modules) is then similar to the module, with the module reference arms taking the place of the individual emitters at the module level. In a fashion analogous to an individual module, the sector control layer can be locked using a set of measurements made at equivalent forward and return detection points. The multiplexing of the these measurements is again achieved using DEHeI; however, as this requires unique codes only for the module and sector references, the length of the codes required for non-degenerate demodulation is significantly reduced from the total emitter count. The process can be repeated for larger structures, creating a nested OPA topology. To achieve the ${10^8}$ total emitters requisite for the Starshot photon engine while keeping the signal multiplexing requirements below 1000 channels on a single detector [26], we require only one further layer: the array, which is composed of 1000 sectors. The array is shown in Fig. 3.

Fig. 3. Overview of the hierarchical structure for the proposed OPA consisting of 1000 sectors of 1000 modules each with 100 emitters. At the array level, LO lasers generate fields at the (a) emitter, (b) module reference, (c) sector reference, and (d), (e) atmospheric sensing wavelengths, separated by ${\sim}100\;{\rm GHz} $. The LO fields for each control layer are phase locked to an optical frequency comb (OFC) and are disseminated forward through the array and boosted by in-line LO amplifiers located at the sector level to reach the outer extremities of the array. Each control layer consists of a forward path detector (FPD) and return path detector (RPD). The phases of all emitters within a module are measured with respect to the emitter LO at the emitter FPD and emitter RPD. These are locked to the phase difference between the module reference and the module LO at their respective detection points. At the sector level, this process is repeated for all module references by using a measurement at the module lead path detector (LPD) enabling them to be locked to the sector reference. At the array level, the sector references are locked to a single sector reference, which can be arbitrarily chosen.

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At the top control layer, all LO fields used in the array are referenced to an OFC. This ensures the relative wavelength separation between control layers is kept constant, which is required for the atmospheric phase sensing and correction scheme.

The primary technical challenge with the hierarchical structure is the introduction of phase offsets due to the wavelength separation between the hierarchical layers. While these are common and cancel between emitters within a module, at higher hierarchical layers, they depend on the absolute optical path length to the module and sector references. Once active phase control is engaged, the continuous phase tracking combined with DEHeI compensates for dynamic changes but does not measure the static offsets. An initial calibration prior to active phase control would enable compensation for these offsets. This pre-compensation approach has already been implemented for chip-scale OPAs [27,28], and could be adapted to calibrate chip-based sub-assemblies for the Starshot photon engine.

Fig. 4. Conceptual overview of the “Beacon” sensing approach. A two wavelength optical source within the isoplanatic patch of the OPA is used to illuminate all emitter telescopes. Light from the Beacon is coupled into the OPA arms, combined, and then interfered with a local oscillator field (LO injection). Using the multiplexing technique digitally enhanced heterodyne interferometry, the Beacon wavefront, as seen by individual emitters, can be separately measured. Using a combination of emitter measurements at the two Beacon wavelengths, we reconstruct the phase correction required for the OPA wavelength enabling dynamic correction of atmospheric wavefront distortions as well as static phase offsets between emitters due to inter-emitter positioning errors.

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3. SENSING OF ATMOSPHERIC AND EXTERNAL PATHS

A. Phase Sensing Architecture for a Laser Guide-Star Beacon Satellite

To achieve phase compensation at each emitter aperture of the photon engine, we require a coherent, phase-stable optical reference point. A solution to meet these requirements is a satellite laser guide-star or “Beacon.” The concept, as illustrated in Fig. 4, is to illuminate the emitter apertures with a laser source affixed to an accompanying mothership satellite from within the isoplanatic angle of the sailcraft. While this is highly dependent on atmospheric seeing conditions, we can approximate it to be between 5–8 arcseconds at a 1 µm wavelength at a good site [29]. So long as the array is in the same inertial frame as the Beacon, it will remain in the isoplanatic patch during the launch. This can be partly achieved by ensuring that the orbital velocity matches the velocity of the array about the Earth’s axis ($460\;{{\rm ms}^{- 1}}$ on the equator), which is possible in highly elliptical orbits of period at least ${\sim}4$ sidereal days. For non-equatorial sailcraft positions, the length of the launch window for a non-accelerating Beacon is then limited by the rotational acceleration of the array about the Earth’s axis to ${\sim}20$ min.

When operating, the Beacon is directed towards emitter apertures such that the sensing beam covers the entire array, which is on the order of a square kilometer with 100 million emitters. Given these nominal operating parameters and an optimal scenario of even illumination, the amount of Beacon light received by each emitter is $1 \times {10^{- 8}}$ the total captured light, prior to losses within the receiving optics.

As the optical power from each emitter of the array is on the order of kilowatts, any reflection from the sailcraft back to the array at the emitter wavelength is expected to be overwhelmed by atmospheric back-scatter near the Earth’s surface. To distinguish the Beacon from back-scattered emitter light, we require the Beacon to be wavelength separated from the emitter wavelength. As the rate of phase accumulation scales with the wavelength, using separate wavelengths therefore requires a projection from the Beacon phase measurement back to the emitter wavelength. As phase measurements are modulo $2\pi$, ambiguity arises from integer wavelength path length differences. This is addressed using a second Beacon situated on the same spacecraft but at another, third wavelength. By having two Beacon wavelengths of a known frequency difference probing the same optical path length, we can account for the integer path length difference, enabling the correction at the emitter wavelength. This is akin to two-color interferometry [30], the key difference, however, being that the projection is made with no phase information at the target, emitter wavelength.

We consider an architecture for receiving and measuring the Beacon wavefront that is integrated with the photon engine OPA, sensing common optical paths. In Fig. 5, we show an optical schematic for a 100 emitter OPA, which can be considered a standalone unit, or a sub-array of a larger OPA structure. The key requirement for this atmospheric sensing technique is the stabilization of internal paths within this array. While several architectures are able to achieve this, we consider a nominal design based on an internally sensed, offset-locked, DEHeI phase measurement and control system. An overview and demonstration of a DEHeI internally sensed OPA configuration has been exposited by Roberts [16], and has also been experimentally demonstrated in alternative configurations [14,15].

Fig. 5. Beacon wavefront sensing architecture for an internally sensed OPA of 100 emitters. All emitters are internally phase locked to the OPA reference using DEHeI phase sensing and control, which ensures the internal path lengths are stabilized using methods described in [16]. (a) For atmospheric correction, once fiber coupled from the emitter telescopes, (b) the Beacon fields are passed through the their respective OPA arms where they are tagged with DEHeI modulation to enable identification in demodulation. (c) Once combined into a single optical fiber, (d) the received fields are interfered against their respective LO fields. Finally, the Beacon fields are split from the OPA internal control fields and each other using standard dense wavelength division multiplexing (DWDM) methods and detected on individual detectors (Beacons 1 and 2 RPD).

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Once light is coupled through the emitter apertures, both Beacon wavelengths propagate back through the OPA, sharing the same optical path as the emitter internal control fields. They are then combined into a single optical fiber through a $1 \times 100$ splitter/combiner, as shown in Fig. 5. Prior to combination, each field is phase modulated with DEHeI phase modulation, a pseudo-random bit sequence. This effectively imparts an emitter dependent identifier to each Beacon field. After combination, the received Beacon fields are interfered against a frequency offset LO, resulting in a heterodyne beatnote at the difference frequency, nominally on the order of 10 MHz. For multiple Beacons, we use individual LOs, which allows for flexibility in demodulation and further isolation of individual Beacon wavelengths by choice of heterodyne frequency. The interfered fields are then split onto separate photodetectors using standard dense WDM (DWDM) techniques. We therefore impose a requirement of at minimum 100 GHz frequency spacing between each Beacon and emitter fields to coincide with current multiplexing methods [31].

The second requirement we impose addresses optical losses within the OPA. If we consider a 100 emitter OPA, shown in Fig. 5, the $1 \times 100$ splitter in conjunction with the asymmetric couplers immediately results in 40 dB of optical loss. If we consider this as a sub-array of the Starshot photon engine, factoring in nominal insertion losses for other optical components, this translates to an optical loss from the emitter output to the photodetector of $- 53.9\;{\rm dB}$. While weak light phase tracking down to 100 fW using digital phase-locked loop architectures has been demonstrated by Francis et al. [32], we propose an additional condition of a minimum of 1 pW of detected optical power per Beacon wavelength, per emitter to ensure reliable continuous phase tracking over a nominal sensing Nyquist bandwidth of 10 kHz. For a final design, this bandwidth requirement may be tuned to ensure that the phase noise dynamics of the system are resolved.

Given this power requirement, and following photodetection, the DEHeI demodulation process can decode and reverse the pseudo-random modulation on a per emitter basis. This process can be parallelized on digital signal processing (DSP) hardware, such as a field programmable gate array (FPGA) enabling each emitter to be simultaneously processed [15,26]. Once decoded, heterodyne phase extraction methods, such as quadrature demodulation or digital phase-locked loops may be employed to extract the emitter phase from the recovered heterodyne beat notes. Using this method, we are able to measure the phase difference between each returning Beacon field and its LO at a single return path detector. We label the Beacon 1 phase measurement of the $k$th emitter ${\Phi _{b1,RPD}}[k]$ and similarly ${\Phi _{b2,RPD}}[k]$ for the Beacon 2 measurement.

To isolate the atmospheric phase difference between adjacent emitters, we take the subtraction between phase measurements from different emitters. This yields two different measurements, ${\phi _{b1}}$ and ${\phi _{b2}}$, for the two Beacon wavelengths, respectively. We can write them in terms of the OPA optical paths as per Eqs. (4) and (5), here using emitters 1 and 2 as an example:

(4)$$\begin{split} \Delta {\phi _{b1}} & = {\Phi _{b1,RPD}}[2] - {\Phi _{b1,RPD}}[1] \\& = \phi H{3_{b1}}[2] + \phi H{2_{b1}}[2] + \phi A{1_{b1}}[2] \\&\quad- \phi H{3_{b1}}[1] - \phi H{2_{b1}}[1] - \phi A{1_{b1}}[1],\end{split}$$

(5)$$\begin{split} \Delta {\phi _{b2}} &= {\Phi _{b2,RPD}}[2] - {\Phi _{b2,RPD}}[1]\\& = \phi H{3_{b2}}[2] + \phi H{2_{b2}}[2] + \phi A{1_{b2}}[2]\\&\quad - \phi H{3_{b2}}[1] - \phi H{2_{b2}}[1] - \phi A{1_{b2}}[1].\end{split}$$

In Eqs. (4) and (5), $\phi H{3_x}[k]$ is the phase accumulated external to the emitter $k$, at wavelength ${\lambda _x}$. When internal phase control is active, the remaining terms, $\phi H{2_x}[k]$, the phase from the emitter to the 99:1 splitter, and $\phi A{1_x}[k]$, the phase accumulated through individual arms of the OPA are phase locked for all emitters and therefore cancel. This means that when internally locked, the Beacon phase measurements given by Eqs. (4) and (5) are sensitive solely to external phase disturbances.

B. Phase Mapping between Optical Wavelengths

The phase differences between emitters measured at the beacon wavelengths, as given by Eqs. (4) and (5), need to be converted to the corresponding phase difference at the emitter wavelength. This mapping depends on $\Delta L$, the optical path length difference between two emitters. This encompasses micrometer-scale changes due to atmosphere, as well as physical separations in emitter positioning, which may be several orders of magnitude greater.

Given this path length difference, Eq. (6) determines the phase difference between two emitters at the emitter wavelength ${\lambda _e}$, which we split into sub-cycle and integer components in the following expression:

(6)$$\Delta L/{\lambda _e} = \Delta {\phi _e} + {N_e},$$

where $\Delta {\phi _e}$ is the sub-cycle phase difference between two emitters, defined at the emitter wavelength, and is the quantity of interest. The integer cycle difference between the two emitters is then given by ${N_e}$. Similar to Eq. (6), analogous expressions may be written for the Beacon wavelengths ${\lambda _{b1}}$ and ${\lambda _{b2}}$. Using the expressions at the emitter wavelength, and the first Beacon, we can determine the difference between the phase measurements at the two wavelengths $\Delta {\phi _e} - \Delta {\phi _{b1}}$, written as follows:

(7)$$\Delta {\phi _e} - \Delta {\phi _{b1}} = \Delta L\left({\frac{1}{{{\lambda _e}}} - \frac{1}{{{\lambda _{b1}}}}} \right) + ({N_{b1}} - {N_e}).$$

Here, Eq. (7) shows the residual phase shift due to the difference in wavelength. To simplify this, we consider the synthetic wavelength ${\Lambda _{e,b1}}$, generated from ${\lambda _e}$ and ${\lambda _{b1}}$. This is a common concept in two-color interferometry, where the synthetic wavelength represents an effective wavelength for the two-color system [33], and is defined by ${\Lambda _{1,2}} = {\lambda _1}{\lambda _2}/({\lambda _1} - {\lambda _2})$. Using the synthetic wavelength ${\Lambda _{e,b1}}$, we can determine the residual phase shift over the path length $\Delta L$ due to the wavelength separation between ${\lambda _e}$ and ${\lambda _{b1}}$. Making this substitution and rearranging for the phase difference of interest, $\Delta {\phi _e}$, we arrive at the following expression:

(8)$$\begin{split} \Delta {\phi _e} &= \Delta {\phi _{b1}} + \underbrace {\frac{{\Delta L}}{{{\Lambda _{e,b1}}}} + ({N_{b1}} - {N_e})}_{\text{residual phase shift}}\bmod 1, \\\Delta {\phi _e} & = \Delta {\phi _{b1}} + \frac{{\Delta L}}{{{\Lambda _{e,b1}}}}\bmod 1,\end{split}$$

where the phase shift describes the mapping from ${\lambda _{b1}}$ to ${\lambda _e}$. The expression is modulo 1, as $\Delta {\phi _e}$ is defined only as a fractional phase, which by definition removes the difference in integer cycles: ${N_{b1}} - {N_e}$. Equation (8), however, still requires knowledge of the path length difference, $\Delta L$, and for this we use the phase measurement at the second Beacon wavelength, ${\lambda _{b2}}$. Taking the difference between Beacon 1 and Beacon 2 measurements, we can arrive at an expression analogous to Eq. (7), however, in terms of the two Beacons. This can be solved for the path length difference in terms of the two measured phases, $\Delta {\phi _{b1}}$ and $\Delta {\phi _{b2}}$, as shown in Eq. (9):

(9)$$\begin{split} \Delta {\phi _{b1}} - \Delta {\phi _{b2}} &= \frac{{\Delta L}}{{{\Lambda _{b1,b2}}}} - ({{N_{b1}} - {N_{b2}}} ) \to \Delta L\\& = {\Lambda _{b1,b2}}\left({\Delta {\phi _{b1}} - \Delta {\phi _{b2}} + {N_{b1}} - {N_{b2}}} \right).\end{split}$$

From Eq. (9), we can see that absolute knowledge of $\Delta L$ can be obtained in the instance where ${N_{b1}} = {N_{b2}}$. This condition is analogous to the ambiguity range of two-color interferometry techniques [33,34]; however, it is not required here, as the absolute distance $\Delta L$ is not the measurement of interest. Therefore, by substituting the expression for $\Delta L$ from Eq. (9) into the phase correction at the emitter wavelength given by Eq. (8), we can obtain an expression without explicit dependence on the path length difference:

(10)$$\begin{split}\Delta {\phi _e} &= \underbrace {\Delta {\phi _{b1}} + \frac{{{\Lambda _{b1,b2}}}}{{{\Lambda _{e,b1}}}}(\Delta {\phi _{b1}} - \Delta {\phi _{b2}})}_{\text{Measurable Correction}} \\&\quad+ \underbrace {\frac{{{\Lambda _{b1,b2}}}}{{{\Lambda _{e,b1}}}}({N_{b1}} - {N_{b2}})}_{{\rm Residual}}\quad\bmod 1.\end{split}$$

Using the phase measurements from the two Beacons, and knowledge of wavelengths ${\lambda _{b1}}$ and ${\lambda _{b2}}$, the first term of the above expression can be experimentally derived. The first term therefore is the estimate of $\Delta {\phi _e}$, the phase difference at the emitter wavelength, provided the residual term is optimized to be negligible. We label the first term as the Beacon phase correction, $\Delta {\phi _{e(c)}}$, which for completeness is written in Eq. (11):

(11)$$\Delta {\phi _{e(c)}} = \Delta {\phi _{b1}} + \frac{{{\Lambda _{b1,b2}}}}{{{\Lambda _{e,b1}}}}(\Delta {\phi _{b1}} - \Delta {\phi _{b2}})\quad\bmod \,1.$$

C. Determining the Optimal Beacon Wavelength

In determining the optimal parameters for the Beacon wavelengths, we optimize the residual error in Eq. (10), which is achieved when it is equal to an integer number of cycles. We can approximate the residual from Eq. (10) in terms of the path length difference and the Beacon synthetic wavelength as follows:

(12)$$\begin{split}\Delta {\phi _{\textit{err}}} &= \frac{{{\Lambda _{b1,b2}}}}{{{\Lambda _{e,b1}}}}\left({{N_{b1}} - {N_{b2}}} \right) \equiv \frac{{{\Lambda _{b1,b2}}}}{{{\Lambda _{e,b1}}}}\left({\left\lfloor {\frac{{\Delta L}}{{{\lambda _{b1}}}}} \right\rfloor - \left\lfloor {\frac{{\Delta L}}{{{\lambda _{b2}}}}} \right\rfloor} \right) \\&\approxeq \frac{{{\Lambda _{b1,b2}}}}{{{\Lambda _{e,b1}}}}\left\lfloor {\frac{{\Delta L}}{{{\Lambda _{b1,b2}}}}} \right\rfloor \;\;\text{(all mod 1)},\end{split}$$

where $\lfloor x \rfloor$ constitutes a floor operation, and all expressions are modulo 1 cycle. By inspection, we can see that as both ${N_{b1}}$ and ${N_{b2}}$ are integers, and the residual error term remains an integer under the condition that ${\Lambda _{b1,b2}}/{\Lambda _{e,b1}} = M$, where $M$ is also an integer. The simplest instance of this, and the instance we consider, is where $M = 1$, which yields ${\Lambda _{b1,b2}} = {\Lambda _{e,b1}}$. Physically, we can interpret this as the synthetic wavelength ${\Lambda _{b1,b2}}$ being equal to ${\Lambda _{e,b1}}$, and therefore both accumulating the same integer number of cycles over the unknown path length $\Delta L$. Under this condition, given a fixed ${\lambda _e}$ and ${\lambda _{b1}}$, we can determine an expression for an optimal ${\lambda _{b2}}$:

(13)$${\lambda _{b2}} = \frac{{{\lambda _{b1}}{\lambda _e}}}{{{\lambda _{b1}} - 2{\lambda _e}}}.$$

As a lower bound on the wavelength separations, we impose a further restriction of a minimum frequency separation of 100 GHz between any adjacent wavelengths. This is consistent with the frequency separation used for standard DWDM optical communications, with the intention of using similar wavelength separation techniques to isolate individual Beacon phase measurement. The lower bound on the frequency shift has the further effect of pushing the Beacon wavelengths outside of the Brillouin scattering window of approximately 20 GHz around the high power emitter wavelength [35].

Following this restriction, we can reparameterize the system in terms of the optical frequency difference, $\Delta {\nu _{b1}}$ between ${\lambda _e}$ and ${\lambda _{b1}}$. Similarly, the optical frequency difference between ${\lambda _e}$ and ${\lambda _{b2}}$ can be written as $\Delta {\nu _{b2}}$. Following this substitution, we can simplify Eq. (13) to the following relation:

(14)$$\Delta {\nu _{b2}} = 2\Delta {\nu _{b1}}.$$

We can set $\Delta {\nu _{b1}}$ to be the minimum frequency shift of 100 GHz from the emitter wavelength of 1064 nm, which yields an optimal Beacon 2 frequency shift of 200 GHz, maintaining the 100 GHz separation between all wavelengths. This is satisfied only for $M = 1$. While these are the optimal set-points for the Beacon wavelengths, the accuracy at which they can be tuned relative to the emitter determine the residual error. In practice, this residual due to limits on the wavelength tuning accuracy of the Beacons means a small fractional phase error will remain, and increase with the path length difference.

4. SIMULATED PERFORMANCE

A. Inter-emitter Beacon Phase Correction and Displacement Tolerance

To evaluate the robustness of the atmospheric phase correction architecture, we implement a time domain simulation of a two-emitter system, evaluating the correction listed in Eq. (11) and the residual error from Eq. (12). Into this simulation, we use an off-plane inter-emitter path length comprising an off-plane distance offset, $\Delta {L_o}$, and time varying atmospheric turbulence, modeled as a Brownian noise process on the optical path length, $\Delta {L_a}(t)$. We also include the uncorrelated shot noise contribution of each Beacon detector.

Given a minimum power requirement of 1 pW detected per Beacon wavelength, we arrive at a shot noise limited phase sensitivity of $4.3 \times {10^{- 4}} \;{\rm rad}/\sqrt {{\rm Hz}}$. To determine the coupling requirements for the array, we use the optical loss model for an array of 100 emitters to obtain a 250 nW minimum requirement on the received Beacon optical power for each emitter.

Alongside the Beacon phase measurement, atmospheric correction requires knowledge of the absolute wavelength of both Beacons and the emitter lasers. One method to achieve this is through measuring each of the optical wavelengths with respect to an OFC. For the Breakthrough Starshot phased array, an OFC already underpins the internal phase control scheme; therefore, knowledge of the relative shift between the emitter and Beacon wavelengths is readily obtainable. We consider multiple tolerance levels’ accuracy of the Beacon and emitter wavelength set-points.

The metric of interest for atmospheric correction is the fractional phase residual between the two emitters, and is defined as the difference between the fractional phase at the emitter wavelength, $\Delta {\phi _e}$, and its projected value, $\Delta {\phi _{e(c)}}$, given by Eq. (11). This is computed by taking the difference and subtracting the nearest integer number of cycles, returning a fraction phase residual within the bounds: ${\pm}0.5$ cycles. In Fig. 6, we simulate the fractional phase residual as a function of the optical path length offset between the two emitters. We see that as the Beacon 2 wavelength approaches the optimal optical frequency set-point, as defined by Eq. (14), the maximum path length able to be tracked with shot noise limited performance increases. To maintain shot noise limited operation, the inter-emitter length difference requires path length matching, which can be done internally by tuning the fiber length between the asymmetric output coupler and telescope assembly for each emitter shown in Fig. 5. With the appropriate path length matching, down to within 1 m and a frequency accuracy down to 1 MHz, a fractional phase error over a 10 kHz Nyquist bandwidth of $1.5 \times {10^{- 2}}$ cycles or an RMS stability of ${\sim}\lambda /64$ between two emitters can therefore be achieved.

Fig. 6. Simulation of the fractional phase error between two emitters in the wavefront measurement for different tolerance levels on Beacon 2 wavelength accuracy. This is plotted alongside the residual phase error (dashed lines) calculated from Eq. (12) and the simulated shot noise level for 1 pW of Beacon power detected per emitter. With the Beacon 2 wavelength tuned within 10 kHz of the optimal value given by Eq. (13), path length differences up to a dynamic range of 100 m can be tracked with shot noise limited performance.

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B. Array Phase Stability Model

We proceed to model the phase sensing precision for the Starshot photon engine using this hierarchical control method in the case of shot noise limited operation. The 100 million emitter lasers are each specified to output 1 kW of optical power at 1064 nm, delivering a total of 100 GW at their combined output. All reference and LO lasers are limited at 10 mW output to minimize the challenges of high power handling within the bulk of the array. We model the path length losses over the array using single mode polarization maintaining fiber at 1064 nm, with an attenuation coefficient of 2 dB/km. For wavelength separation, we consider 30 dB of wavelength isolation, which is required to supplement DEHeI crosstalk suppression [26].

To determine the incident optical power at all detection points throughout the array, we develop a loss model for the internal phase control optics. This model considers nominal optical losses given currently available technologies. A list of the optical components, their losses, and other key parameters for the simulations are given in Table 1.

Table 1. Simulation Parameters Used to Determine Shot Noise Limited Phase Stability of the Proposed Architecture

View Table

We start by first computing the shot noise limited performance at each detection point, for each wavelength separated detector. This is a function of the total number of multiplexed optical fields, and therefore the size of each module, sector, and the array. As the multiplexed structure increases in number of measurements per detector, the shot noise performance for a given phase measurement of an individual optical path decreases. We determine this shot noise performance using the following expression for shot noise, adapted from [26]:

(15)$${\phi _{\rm{shot}}} = \sqrt {\frac{{hc}}{\lambda}\frac{{{P_{\rm{tot}}}}}{{{P_{\rm{sig}}}{P_{\rm{LO}}}}}} ,$$

where $h$ is the Planck constant, $c$ the speed of light, and $\lambda$ the optical wavelength of 1064 nm. Here, ${P_{\rm{tot}}}$ is the total power incident on the photodetector in question, ${P_{\rm{LO}}}$ is the relevant LO power, and ${P_{\rm{sig}}}$ is the signal power of interest.

When all error signals have been locked, each phase measurement within the OPA contributes to the stability of a given emitter. The total shot noise seen by that emitter, with active internal phase control, is therefore the quadrature sum of all phase measurements within the array.

In Fig. 7, we plot the shot noise limit for the OPA’s internal phase sensing, consisting of the quadrature sum of the shot noise at all detectors for a given emitter. We also plot the contribution from the atmospheric phase sensing architecture for varying received beacon optical powers at the array output, and which combined, yields the total shot noise limited stability of the photon engine. Given a picowatt detection limit for phase sensing (250 nW/emitter), we are limited by the shot noise limited performance of the atmospheric phase sensing system, at a RMS phase stability of $\lambda /64$. This can be further increased by optimizing the amount of Beacon optical power captured by each emitter, reaching an optimal stability of $\lambda /434$, given 22.5 µW/emitter captured from the Beacon.

Fig. 7. Shot noise limited phase stability of the optical phased array including atmospheric correction for instances with varying beacon optical power captured per emitter (solid color). The internal phase sensing architecture has a minimum shot noise at ${\sim}1000$ emitters per module (black). The total shot noise is, however, limited by the atmospheric phase sensing system (dashed color) for large module sizes. To maintain a three layer control structure within the limits of DEHeI, we need at minimum 100 emitters. At minimum, we require 250 nW of power received by each emitter to ensure 1 pW per emitter at the atmospheric sensing detectors. However, to achieve an optimal shot noise limited performance, this requirement increases to 22.5 µW of Beacon power received by each emitter.

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5. CONCLUSION

We have presented a nominal design and shot noise performance model for the Breakthrough Starshot photon engine, using only photonic components that are either currently available or modest extrapolations of current technology. Using a modularized architecture based on DEHeI and WDM techniques, we are able to scale the OPA structure to the required 100 million emitters for the project.

We have further proposed an integrated architecture for the interferometric measurement of an optical wavefront using the photon engine. We simulate and show that by using this multi-wavelength Beacon architecture is able to measure and correct for emitter positioning offsets on the order of 1 m while maintaining shot noise limited performance, given knowledge of the optical frequency to within 1 MHz.

When combined with internal phase sensing architecture, this nominal design for the photon engine array is limited by the shot noise performance of the atmospheric wavefront sensing system at a piston phase error of $\lambda /64$. With internally sensed phase control, integrated beam-forming and steering capabilities, and an embedded sensing for a satellite laser guide-star, this conceptual architecture presents a holistic topology for light-based interstellar propulsion.

Funding

Breakthrough Initiatives, a division of the Breakthrough Prize Foundation.

Disclosures

The authors declare no conflicts of interest.

Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Parameter	Loss	Parameter	Value
3 dB coupler (t)	3.1 dB	Fiber attenuation	2 dB/km
3 dB coupler (r)	3.1 dB	EOM insertion loss	3 dB
10 dB coupler (t)	0.6 dB	Emitter telescope AR	0.1 %
10 dB coupler (r)	10.1 dB	WDM isolation	30 dB
99-1 coupler (t)	0.14 dB	WDM insertion loss	4 dB
99-1 coupler (r)	20.1 dB	Phase sensing Nyquist bandwidth	10 kHz
$1 \times 5$ (LO forward) splitter	7.1 dB
$1 \times 4$ (LO amp) splitter	6.1 dB
$1 \times 100$ (emitter) splitter	20.1 dB
$1 \times 1000$ (module) splitter	30.1 dB
$1 \times 1000$ (sector) splitter	30.1 dB

Photonic solution to phase sensing and control for light-based interstellar propulsion

Abstract

1. INTRODUCTION

2. INTERNAL PHASE SENSING

A. Emitter Laser Phase Stabilization

B. Hierarchical Locking Scheme

3. SENSING OF ATMOSPHERIC AND EXTERNAL PATHS

A. Phase Sensing Architecture for a Laser Guide-Star Beacon Satellite

B. Phase Mapping between Optical Wavelengths

C. Determining the Optimal Beacon Wavelength

4. SIMULATED PERFORMANCE

A. Inter-emitter Beacon Phase Correction and Displacement Tolerance

B. Array Phase Stability Model

5. CONCLUSION

Funding

Disclosures

Data Availability

REFERENCES

Data Availability

Cited By

Figures (7)

Tables (1)

Equations (15)

Journal of the Optical Society of America B