1. Introduction

In astronomy, long–baseline stellar interferometry allows the synthesis of practically impossible large–aperture optical telescopes using separate telescopes of smaller diameter. By coherently combining the light from small apertures, the coherence function of the light fields reaching the telescopes can be measured. Therefore, the brightness distribution of an object in the sky can be reconstructed through the Van Cittert-Zernike theorem which relates the coherence function to the brightness distribution through a Fourier transform.

Long-baseline optical interferometry could one day spatially resolve exoplanets orbiting nearby stars, using 10 to 100 km baselines. This would be best achieved with a space-based interferometer operating in the mid infrared (7 to 20 $\mu$m) where the contrast between the planet and the star is $10^6$ compared to $10^9$ in the visible. Also, in space, the light wavefront does not experience distortions from atmospheric turbulence. An integrated-optics beam combiner would drastically reduce the size and weight requirements, increasing the stability with regards to thermal and vibration changes, and reducing the optical components for a space mission.

The first coherent combination of the light from two separate telescopes was obtained in 1975 by Labeyrie [1]. Since then, several long–baseline interferometers were built in the visible [2–4], in the near infrared [5,6] and in the mid infrared [7,8]. The current generation of interferometers are capable of resolving angular diameters as small as 500 $\mu as$ [9]. These can combine up to four telescopes and reach limiting magnitudes of the order of 17 in the H-band ($1.5 \mu m$) [10], although these capabilities are not all achieved at the same interferometric facility. Photonics in the form of fibre optics and planar integrated optics has largely benefited stellar interferometry.

To our knowledge, the first application of planar integrated optics to astronomy was achieved in stellar interferometry by testing a 3-telescope beam combiner at Infrared Optical Telescope Array (IOTA) [11,12]. Later, this technology was deployed at the Very Large Telescope Interferometer (VLTI) with the PIONIER [13] and GRAVITY [14] beam combiners.

While the majority of integrated-optics devices are planar and offer a large variety of designs and applications, they have limitations especially in the context of beam combiners. When more than two input beams are used, waveguide crossovers emerge as part of the beam combiner design. This can be avoided in devices written using ultra-fast laser inscription (ULI) due to their 3D capabilities [15]. Several achievements have been made possible with the the help of ULI: photonic components such as waveguides, splitters and couplers have been manufactured [16–19]; on-sky tests using photonic reformatters for performing pupil remapping have been demonstrated [20]; combiners beyond mid-IR wavelengths have been built [21,22]; scalability to combine more telescopes has been demonstrated [23,24]; and on-sky verification of a device for nulling interferometry has been achieved [25].

The main limitation of ULI is the rather modest change of refractive index that can be achieved with the laser-writing process. However, studies have shown where depending on the substrate and laser writing parameters, high contrast waveguides have also been achieved [26–28]. The low contrast waveguides renders the bends needed in couplers and splitters lossy. The bend-induced losses can be avoided using a beam-combiner design based on a photonic lattice, the discrete beam combiner. This structure overcomes the problem of bending and transition losses by using parallel waveguides organised in a square lattice that exploits evanescent coupling for beam combination [29–31]. The success of the combiner in retrieving the complex visibility lies in the existence of next-nearest coupling between the waveguides (WGs) [32].

Saviauk et al., 2013 [33] have studied the bandwidth operation of square-lattice DBC and retrieved the complex visibilities experimentally with a visible source at 660 nm and 17 nm bandwidth. However, the theoretical treatment of transfer matrix of the DBC was missing under the influence of broadband light. In this paper, we extend the research by characterizing the DBC with larger bandwidth light. Here, we define a quasi-monochromatic transfer matrix of the DBC to retrieve the complex visibilities and show the simulation and experimental results obtained for a 4-telescope zig-zag based DBC. However, the theoretical treatment is valid for N telescopes and also independent of the geometry and wavelength operation of the DBC. The DBC was manufactured in Gallium Lanthanum Sulphide (GLS) substrate using ULI, and was operated in the astronomical L-band at a bandwidth of 69 nm. According to the theoretical models and analysis of broad-band data, visibilities were constructed both from simulation and experiment. We present the theoretical background of DBC in section 2, simulation and experimental results in section 3 and 4 respectively, and conclusion in section 5. Finally, there is an additional appendix section to discuss the extra findings of the experiment.

2. Theory

The DBC consists of evanescently coupled waveguides (WGs) where power observed at one of the output WGs can be written as a linear superposition of input electric-fields. The coefficients of the superposition describes the coupling strength between the WGs. Hence, in a multiple-field interferometric beam combiner there exists a complete transfer matrix $\{U\}$ relating the N input fields to M output fields. Therefore, the power $P_{m}$ at the m$^{th}$ output port from the coupled mode theory can be written as [34]:

The self- and mutual-coherence functions are indicated with $\Gamma _{\textrm {ii}}$ and $\Gamma _{\textrm {ij}}$. The relationship in equation (Eq. (1)) can be written in matrix form given by Eq. (2).

The $\overrightarrow {J}$ in Eq. (2) are the complex coherence terms and are represented in Eq. (3). The $\overrightarrow {P}$ (power output vector) and $\overrightarrow {J}$ are linked through the $M\times N^{2}$ Visibility-to-Pixel Matrix (V2PM in other words, transfer matrix $\{U\}$ [35]) whose elements are the modulus and the real and the imaginary part of the complex transfer function of the device (see [29,30] for details).

Equation (2) is over-determined (that is the device has more outputs $M$ than the square of the number of inputs $N$) and the V2PM possesses a Moore-Penrose pseudo-inverse matrix (Pixel-2-Visibity matrix or P2VM) which can be applied to the output vector $\vec {P}$ to extract the coherence functions. Since it is a pseudo-inverse problem, the sensitivity of extracting the coherence functions depends on the conditioning of the V2PM (see section 3 for details). The coupling strength and interaction length between the WGs are carefully chosen to get a low-conditioned V2PM.

The V2PM approach described above is strictly valid for monochromatic light. We will now consider the operation of V2PM for a light source with bandwidth. The coherence terms are defined as:

Where $\Gamma _{ii}(\tau )$ is the self-coherence which describes the correlation of the electric field at the $m^{th}$ WG (in Eq. (1)) from the $i^{th}$ input WG and $\Gamma _{ij}(\tau )$ is the mutual-coherence of $m^{th}$ WG (in Eq. (1)), where $\tau$ is the difference in the time delay between the $j^{th}$ WG and the $i^{th}$ WG. The monochromatic frequency of the light is denoted by $\nu$. Eq. (4) and Eq. (5) are substituted in Eq. (1) and integrated over the frequency range of the light:

Where $\phi _{mij} = \tan ^{-1}\Big [\frac {\Im (U_{mi}U_{mj}^{*})}{\Re (U_{mi}U_{mj}^{*})}\Big ]$ represents the phase of the transfer function matrix. The transfer matrix {U} contains frequency-dependent modal amplitudes of the electric field and solving the integral in Eq. (6) is cumbersome. However, the operation bandwidth of our light source is $< \pm 50 nm$ at centre wavelength (see description in section 3 and 4). Within this bandwidth the change in the mode-field radius describing the modal fields in the transfer matrix {U} changes by $< \pm 5 \%$ of the centre wavelength. Hence, the frequency dependency of $U_{n}(\nu )$ can be neglected within the narrow bandwidth and the integration of Eq. (6) is performed to get:

Equation (7) is valid for any generalized multi-field integrated optics beam combiner and it will be shown that our simulations and experiment support the above consideration. It describes the temporal evolution of fringes at the $m^{th}$ output WG where, in reality any light source doesn’t contain a single frequency but has a finite width. Equation (7) also tells us that the visibility retrieval using P2VM requires a certain degree of achromaticity within which the frequency dependence of {U} can be neglected. Now, we will describe the calibration procedure of the V2PM under polychromatic illumination used in the paper for analyzing the experimental data.

Under monochromatic illumination, the calibration of V2PM matrix described by Eq. (2) is carried out by filling the first N columns with the square modulus of the field transfer function and the remaining N(N-1) columns with the real and imaginary parts of the product of the field transfer function. In other words, the first N columns are filled with the normalized photometry of the output channels by exciting one input at a time. The remaining N(N-1) columns are filled with the temporal fringes by exciting two inputs at a time and adding a known phase delay between them. Generally, we follow a photometric correction procedure (see [33] for details), where we get rid of the photometric transmission terms ($\Gamma _{ii}$ and $\sqrt {\Gamma _{ii}\Gamma _{jj}}$) in Eq. (7) for $m^{th}$ WG to extract the amplitude and phase that will go in filling the elements of the V2PM. Hence, Eq. (7) is photometrically corrected to get:

In Eq. (8), we used the notation $\delta = c\tau$ and converted the frequency bandwidth to wavelength bandwidth using $\Delta \nu = c\frac {\Delta \lambda }{\lambda _{0}^{2}}$. A non-linear least square fitting is performed to the photometrically corrected interferogram given by Eq. (8). The amplitude $=|U_{mi}U_{mj}^{*}|$ and phase $= \phi _{mij}$ of the complex transfer function is extracted for each of the M output WGs, hence filling the N(N-1) columns of the V2PM. Thus, the above method for the V2PM calibration leads to a more robust result in extracting the visibilities using narrow bandwidth light sources. In this context, we define such a V2PM as quasi-monochromatic V2PM. For our analysis, we use the Michelson fringe visibility, which is routinely used in astro-interferometry [33]:

3. Simulation results

The commercially available software BeamPROP from Rsoft was used to calculate the field transfer function of the beam combiner at 51 wavelengths distributed over a band of 300 nm around the carrier at 3.5 $\mu m$. The bandwidth up to 300 nm was used to verify the theory in section 2 and to investigate the response of the $\{U\}$ as bandwidth increases. Gaussian launch fields were used for our simulation which was mode matched with the input WG of the DBC. The interferograms and photometry were evaluated at each output of the device according to Eq. (7). As was pointed out in section 2, the approximation is valid for narrow-bandwidth light and it is similar to coherently combining the power of all WGs at each spectral channel. Next, a nonlinear least square fitting was used to extract the amplitude and phase terms for the V2PM elements from the photometrically corrected interferograms (Eq. (8)) described in section 2.

The simulated DBC consists of 4 single WG inputs followed by a zig-zag array of 23 evanescently coupled WGs as shown in Fig. 1. The number of output WGs has been determined in previous work, where 23 output WGs have been found to provide optimum SNR as well as low conditioning of the V2PM for a 4-input DBC [34]. The results reported in the (Diener et al., 2017) paper were for the optimized DBC configuration at L- band. Hence, we used the same experimental parameters of the DBC ($P_{x},P_{y},W,H$) in this paper for our simulation, including the length of input WGs and the length of DBC interaction region. Furthermore, our experimental results in section 4 are for the same device that was used in Diener et al., 2017 [34].

 

Fig. 1. Representation of the Discrete Beam Combiner (DBC). Interested readers may look at the end facets of a similar 4-input and 23-output DBC device which shows the ULI inscribed waveguides. [36, Fig. 5] (a) 3D figure where 4 inputs are shown in blue and 23 outputs are shown in red color. The outputs are arranged in a zig-zag pattern, the upper waveguides displaced by half a period with respect to the lower waveguides. (b) 2D figure showing the numbering of the waveguides and parameters ($P_{x}, P_{y}, W, H$) that have to be determined in order to get a stable transfer matrix of the DBC. The input positions are marked at the waveguide locations 5, 10, 14 and 19.

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3.1 Visibility retrieval

The condition number of any coefficient matrix estimates the propagation of error while inverting it to find the unknowns. Our case is a over-determined problem $(M>N^2)$ to find the $N^2$ complex coherence variables of the transfer matrix {U}, where M WGs serve as input parameters to solve Eq. (2). Hence, one of the criteria is that the quasi-monochromatic V2PM should be low-conditioned for a stable pseudo-inverse (i.e. P2VM) to exist. Figure 2(a) shows the dependence of the V2PM’s condition number (CN) as a function of the wavelength values that are used as discrete inputs for our simulations. The mean (= 16.9) and standard deviation of the CN (= 5.3), across the wavelength range that is used for our simulation. The simulation assumes that each wavelength is represented by a delta function, which is an ideal representation of monochromatic light having zero bandwidth. We want to investigate the influence of the V2PM when the light is no longer ideal but has a finite bandwidth. Figure 2(b) shows the CN as a function of bandwidth centred at $\lambda = 3.5 \mu m$. The CN increases monotonically as the bandwidth increases and fitting shows that the dependence is polynomial. At increasing bandwidths, the modal fields describing the {U} become chromatic, which start to affect the CN and thus, validating the assumption in performing the integral of Eq. (6). We know that the CN should be a global minimum for any variables in the parameter space of the DBC [32]. But, we don’t have an estimate to quantify the high CN of the V2PM. The CN itself, doesn’t provide a conclusive statement of accuracy of the V2PM in retrieving the coherence functions [33]. Hence, we will apply the P2VM to the output power of the DBC and quantify the error in extracting the visibilities with respect to the results from the background theory in section 2.

 

Fig. 2. (a) Variation of the condition number of the V2PM across the wavelength spectrum. (b) Variation of the condition number of the V2PM across a bandwidth centred at 3.5 $\mu m$.

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We know that our source is a point source in the spatial domain and contains a narrow spectrum of light in the temporal domain. Hence, we define the complex visibility model of our source from Eq. (8):

$SS_{res}$ is defined as the residual sum of squares resulting from the difference between the model given by Eq. (10) and simulated visibilities. $SS_{tot}$ is the total sum of squares estimated from the simulated visibilities by applying the P2VM to the simulated data. Figure 3 shows the dependence of $R^2$ over bandwidth for all pair-wise combinations of input WGs. The pairwise combinations of input WGs, also known as pairwise visibility, is defined according to Eq. (9). It can be seen that $R^{2}$ is greater than 0.95 up to bandwidth of 100 nm for all visibility amplitudes suggesting that our model is able to account for more than $95 \%$ of the visibilities. This is an indication that our simulation visibilities is well-replicated by our ideal model given by Eq. (10). The retrieval of the visibility phase also shows a similar trend as it decreases with increasing bandwidth but the goodness of the model in describing the phase relationship is bad as $R^2 < 0.9$ for all pair-wise combinations. We later show the oscillations around the visibility curves (Fig. 4), which are due to phase mismatch errors arising from the simulation. Taking a modulus of our theoretical model (Eq. (10)) shows that it is not adapted to predict the oscillations, thus giving rapid reduction in $R^2$ for phases. We can easily calculate the predicted phase from Eq. (10) since, we have define a known $\delta$ ( = optical path difference) used in our simulation. But, we found out that the retrieved phases from the simulation, don’t start from the same position as that of predicted phases, giving reduced $R^2$. However, we can define an another metric adjusted-$R^2$ [37], which can be increased by adding a phase term in the exponential part of Eq. (10) and accounting for the phase mismatch between our model and simulated phases along the bandwidth of the source. Both curves from Fig. 3 follow the same trend of decreasing $R^2$ as bandwidth increases indicating that the stability of V2PM decreases when bandwidth of the light source increases. As a compromise between both the Figs. 3(a) and 3(b), we conclude that the DBC should be operated with input light which has a bandwidth below 100 nm, as larger bandwidths lead to rapidly accumulating errors when constructing the coherence functions as well as substantial deviations from the predictions made by the model in Eq. (10).

 

Fig. 3. Coefficient of determination $(R^2)$ calculated for the retrieved simulated visibilities. The visibilities are defined according to Eq. (9). $R^2$ decreases as the bandwidth increases indicating that the operation of the DBC is bandwidth-limited. (a) Visibility Amplitude (b) Visibility Phase.

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Fig. 4. Simulated retrieved visibility amplitudes for our 4-input DBC for all 6 combinations of the input beams. The modulus of the complex visibilities is shown here using a light source with a bandwidth of 72 nm. Blue is the retrieved value from simulation and the red curve is the model given by Eq. (10). Inset is a zoom view of 20 $\mu m$ around the central optical path difference showing periodic oscillations around the model curve which arise due to decorrelation of phases with increasing bandwidth. The fringe visibility amplitude $V_{ij}$ is defined according Eq. (9).

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The retrieved amplitudes and phases are shown in Fig. 4 and Fig. 5, respectively for a bandwidth of 72 nm centred at $3.5 \mu m$. The results are shown for 72 nm to compare with the experimental results as we were limited by a filter bandwidth of 69 nm. From Fig. 3(a) it can be seen that $R^{2}$ is small especially for $V_{12}$ and $V_{34}$ and the correlation is seen in Fig. 4. It can also be seen that there are periodic oscillations around the model curve given by Eq. (10), which is shown in the inset of Fig. 4. The frequency of these oscillations are approx. $\lambda /2 = 1.75 \mu m$ indicating a phase difference of $\pi$. It seems there are residual phase errors of $\pi$ between the real and imaginary parts of the complex visibilities, which accumulate with departure from zero optical path difference (OPD) and with increasing bandwidth. The retrieved phases shown in Fig. 5 were examined to verify our findings. For the sake of clarity, only a difference of $20 \mu m$ is shown in Fig. 5. At zero OPD a perfect correlation between the expected and retrieved phases exists. However, moving away from the zero OPD, the correlation between the phases of individual wavelengths decreases with increasing bandwidth. Thus, the phases accumulates a value of $\pi$ and doesn’t give the exact phase difference of $\pi /2$ between the real and imaginary parts of the complex visibilities leading to periodic oscillations around the model curve. The oscillations increase with increasing bandwidth and the frequency dependence of $U_{n}(\nu )$ in Eq. (6) cannot be neglected, thus limiting the bandwidth operation of the DBC.

 

Fig. 5. Simulated retrieved visibility phases for a 4-input DBC for all 6 combinations of the input beams. The bandwidth of the light source is 72 nm. Blue is the retrieved curve from simulation and red is the model curve. The visibility phase $\phi _{ij}$ is defined according to Eq. (9).

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4. Experimental results

Here, the experimental results of the DBC characterization are discussed and compared with the simulations. As we mentioned already in section 3, we have used the same device for our characterization that is reported in Diener et al., 2017 [34]. A dedicated interferometric setup in Cologne is used for combining up to 4 beams and it is is shown in Fig. 6. The setup can be operated at K- (2–2.5 $\mu m$) and L- (3–4 $\mu m$) astronomical bands independently. The samples were fabricated using ultra fast laser inscription (ULI) on GLS substrate ordered from ChAMP, university of Southampton due to its long transparency window in mid-IR [38]. An ultra-fast pulsed laser is focused on the substrate, where non-linear processes, such as two-photon absorption, lead to a permanent change of refractive index. The pulse energy, duration and speed are carefully monitored to control the refractive index of the core, resulting in waveguides within the substrate. The focus of the laser spot can be detuned to exhibit the 3D properties of ULI.

 

Fig. 6. Experimental optical setup that is used to characterize the 4-input DBC chip. The 4 mirrors M1, M2, M3 and M4 behave as 4 telescopes and are coupled in the input waveguide locations at 5, 10, 14 and 19 respectively as shown in Fig. 1. Filter: 3.745 $\mu m$ centre and 69 nm bandwidth; APH: Adjustable Pin Hole; BS: Beam Splitter; M: Mirror; L: Lens; DL: Delay Line; CAM: Camera.

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Our input laser consists of a super continuum source and we used a filter centred at $3.75 \mu m$ having a bandwidth of 69 nm. The centre wavelength for the experiment ($\lambda = 3.75 \mu m$) is slightly different from the wavelength used for our simulations but the throughput over the spectrum across this type of GLS written beam combiners is nearly constant over the L band [22]. From Fig. 2(a), the conditioning of V2PM shows variations across the relevant wavelength range, but the results for the simulation will still be valid at a centre wavelength of 3.75 $\mu m$. A setup of 4 beam splitters was used to split the light into equal amounts of $12.5 \%$. The setup had 3 delay lines that can independently delay 3 of the beams, keeping one of the beam fixed. This enabled using all 6 pairwise beam combinations from the 4 input beams simultaneously for fast characterization of the transfer response of the DBC to extract the fringe visibilities. The beams from the 4 mirrors were made overlapped in the camera before placing the chip in the optical path. The chip was then placed on a multi-axis stage. By carefully, translating the chip on the stage, we saw that the light from all the 4 mirrors were coupled in one of the input WGs of the DBC. Finally, we used the tip/tilt of the kinematic mirror mounts and the beams from the mirror 1, 2, 3 and 4 were coupled in the input waveguide locations at 5, 10, 14 and 19 respectively (see Fig. 1). The final output image of the DBC was magnified onto the IR Camera by choosing two lenses, L3 and L4 of appropriate focal lengths.

The data was recorded at 100 fps and the delay line speed was kept at 0.08 mm/s. A total of 5000 frames for each of the 6 baseline pairs as well as the photometric flux for each of the 4 inputs were recorded, with the photometric measurements requiring the remaining 3 beams to be blocked using a physical shutter. It took nearly 4 hours to complete all the measurement data for filling the elements of the V2PM. A sufficient amount of time was also spent for coupling the light into the 4 inputs of the chip by using the tip/tilt kinematic mirror mounts prior to 4 hours. For data analysis, a Fast Fourier Transform (FFT) of the interferometric signal was performed to remove the high frequency noise arising from the measurement data. Photometric correction is applied to the interferometric signal and non-linear least square fitting is done using the model from Eq. (8) to extract the amplitude and phase which are used to fill the elements of the V2PM. The experimental condition number of the V2PM was 26, a low enough value for the calculation of P2VM. The P2VM is applied to the output power to retrieve the modulus and phase of the complex visibilities which is shown in Figs. 7 and 8, respectively. A close resemblance between the retrieved amplitude and the model from Eq. (10) can be seen. By looking at Fig. 7, the point of zero OPD can be compared for all baseline pairs. In principle, the point of zero OPD is expected to be at the same x-position (i.e. absolute delay line (in $\mu m$)) for all baseline pairs. However, a different zero OPD position can be noticed for all baselines in our experimental data, which is suspected to be due to the differential tip/tilt introduced by the 4 mirrors. The tip/tilt caused by the mirror mounts to couple light into the WGs shifts the zero OPD causing offsets in the central peak for all baselines. There are similar oscillations around the model curve as expected by the simulation. However, due to the noise in the extracted results, the frequency of the oscillations cannot reliably be extracted, which the simulations predicts to be $\lambda /2$. The periodicity of the oscillations (=$\lambda /2$) in the simulation does not take into account the sources of error. Hence, we believe that these non-periodic high frequency oscillations around the visibility amplitudes are due to errors from the experiment and not from the data analysis. The sources of errors include fabrication errors from ULI, precision of delay line, coupling errors of the beam from the mirror into the WG and statistical fluctuations from the camera. If there is error in coupling the light flux from the mirror into the input WGs, there will be unwanted phase differences between the WGs giving biases in the modulus of the complex visibility (see section 6 about the repeatability of the V2PM). This can be inferred by including a phase perturbation in the cosine term of Eq. (7) and expanding it. There will be uncertain coefficients in the product of the modulus of the complex visibility $|U_{mi}U_{mj}^{*}|$ for each M output waveguides, thus biasing the retrieved coherence terms. Despite these errors, we appreciate the robustness of our beam combiners in retrieving the visibility amplitudes and phases for all 6 baselines.

 

Fig. 7. Experimentally retrieved visibility amplitudes (defined by Eq. (9)) for 4-input DBC for all 6 combination of the input beams. Blue is the retrieved curve from experiment and orange is the model curve. The x-axis is the absolute delay line position (in $\mu m$) as measured in our experiment and the y-axis is the amplitude value. Inset is a zoom view of $20 \mu m$ around the central white light fringe packet.

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Fig. 8. Experimentally retrieved visibility phases (defined by Eq. (9)) for 4-input DBC for all 6 combination of the input beams. Blue is the retrieved curve from experiment and orange is the model curve and green is the residual. The standard deviation of the phase residual are also indicated for each pairwise combination in green. The x-axis is the absolute delay line position (in $\mu m$) as measured in our experiment and the y-axis is the phase value (in rad).

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The $R^2$ value was found to vary between 0.8-0.9 indicating that 80-90% of the data can be explained by our model for the visibility amplitudes for all baselines. The standard deviation of the retrieved residual phases predicted by our model lies between 0.11-0.40 rad for the experiment corresponding to residual OPD of $\lambda /57 - \lambda /15$ at $\lambda =3.75 \mu m$. However, such residual phases obtained from our device are low enough to construct good quality astronomical images where phase perturbations of $< 1$ rad is expected under good seeing conditions [39, chap. 2].

5. Conclusions

We revisited the theory describing the polychromatic response of any multi-field interferometric integrated optics beam combiner in general and DBC in particular. A quasi-monochromatic approach of filling the V2PM elements was developed. Using the pseudo-inverse quasi-monochromatic P2VM, the complex visibilities for a narrow-band light source can be retrieved. We were able to see the retrieved white light fringe envelope for all baselines in both simulations and experiment. Simulation results show that $R^2$ decreases with increasing bandwidth thus affecting the stability of V2PM in retrieving the complex visibilities and the frequency dependent V2PM cannot be ignored. The modulus of visibilities could be retrieved up to 100 nm with an accuracy of more than 95 % of our model source. A similar $R^2$ trend for retrieved phases was seen with increasing bandwidth. Periodic oscillations were observed around the white light envelope in simulations, which are due to residual phase differences $(\neq \pi /2)$ between the real and imaginary parts of the complex coherence terms retrieved from the P2VM.

The V2PM of the DBC in GLS substrate was characterized for a 69 nm bandwidth centred at $3.75 \mu m$. A 4-Telescope Michelson setup was used, which was able to inject 4 input beams to the DBC, simultaneously. The retrieved visibilities matched the predictions from the theoretical models with an accuracy of 80-90 $\%$. We expected the frequency of the oscillations around the retrieved visibility amplitudes in the experiment to be of the order of $\sim \lambda /2$, as observed from simulations. But due to noise in the retrieved experimental results, it was difficult to extract the oscillations. The sources of experimental noise are introduced due to the writing process of ULI, precision of delay lines, coupling errors due to mirrors and statistical fluctuations in the camera. However, the standard deviation of the residual phase errors were $< 1$ rad for all baselines which is sufficient to construct good quality astronomical images. There is still room for improvement of the beam combiners at L-band in retrieving a higher accuracy of experimental visibilities when compared with simulations. Potential improvements include the use of higher precision delay lines in the experimental setup, improving the coupling from the mirrors that are being delayed, or using alternative chalcogenide substrates with better refractive index contrast. For example, IG2 has low intrinsic losses and higher refractive index changes when the same ULI waveguide writing parameters are used [40].

A DBC can be used to study the mode field pattern at the output of a photonic lantern, which can be used in wave-front correction for application in adaptive optics [41]. The results in this paper will be set as a benchmark when the 4-inputs of the DBC are used as reformatters for doing pupil remapping [42]. In order to perform a proof-of-concept experiment of this type of beam combiner, we have already executed tests at the William Herschel telescope (WHT) using a DBC at astronomical H-band (1.55- 1.65 $\mu m$), for which the data analysis is ongoing. We used pupil remappers integrated with the DBC to collect light from 4 separate sub-apertures of the WHT. We also used a deformable mirror for static tip/tilt correction and a micro-lens array for efficient injection of light into the input WGs of the DBC. However, these results are targeted for a separate journal paper. Finally, due to the simple geometrical design of the DBC, we can explore doing interferometry in all-astronomical bands.

6. Appendix

6.1. Repeatability of V2PM

In order to test the repeatability of the V2PM calibration, measurements were performed with V2PMs from two different days. All experimental parameters were kept constant: the data was recorded with the same camera settings, the speed of the delay line was kept constant at 0.08 mm/sec and we took a total of 5000 frames. The V2PM obtained from the previous day was inverted and applied to the new data. The resulting modulus of complex visibility is shown in Fig. 9. It is noticed that the uncertainties in extracting the amplitude has increased as compared to the V2PM when applied on the data from the same day in Fig. 7. Once the beams are coupled into the DBC component, the temporal stability of the DBC’s output image relies primarily on the opto-mechanical stability of the characterization setup, which may impact the stability of the input coupling, and does not depend on the one-piece glass substrate itself. Thermal effects could also impact the stability of the setup. In the standard conditions of the lab, we were not able to detect qualitatively any instability of the component’s output flux over several hours of static operation, if not the overall variations of the background flux on the camera detector. As all experimental settings have been the same for the duration of the repeatability test, hence any changes in the measured fringes or the extracted visbilities were not expected. However, between the measurement days we had powered off the components in the experimental setup (i.e. camera, motorized delay lines and laser source). The next day we repeated the experiment by powering on; the differences observed in the visibility plots of consecutive measurement days must therefore be due to unwanted changes in the components present in the experimental setup. Figure 7 shows the initial measurement (V2PM and data from the same day), where the white light fringe center is around 400 $\mu m$. It can be seen that the absolute position of the central white light fringe packet has changed in Fig. 9 i.e. the center of the white light fringe has moved with respect to the x-position. This position change is most likely due to uncertainties in the delay line position. As the delay line is moved by a mechanical process using the software interface provided by Thorlabs, the precision of the motor itself might have been too course that might have led to systematic errors in the delay line control. Since the position of the central OPD has changed, it has also impacted the amount of light flux that is coupled from the mirror into the WGs. Due to change in coupled light flux, there will be undesirable phase differences between the output WGs of the DBC. We looked at the residuals of the V2PM elements between the new data and data obtained previously from Fig. 7 although, we used the previously obtained V2PM in Fig. 9. The standard deviation of the residuals were higher for the interferometric entries than the photometric ones, indicating phase differences between the output WGs of the DBC during the measurement days. It is also evident from Eq. (7) that if a phase perturbation inside the cosine term is different for each WG, the estimated phase by least-square fitting is changed for each WG, which alters the elements of the V2PM. Hence, the conditioning of the transfer matrix of the DBC will change, causing uncertainties in the retrieved coherence functions.

 

Fig. 9. Experimentally retrieved visibility amplitudes for 4-input DBC for all 6 combinations of the input beams. Here, the previously obtained P2VM (from Fig. 7) is applied to a new data which was taken a day later. Blue represents the retrieved visibility from experimental data, orange is the predicted curve from theoretical model. The x-axis is the absolute delay line position (in $\mu m$) as measured in our experiment and the y-axis is the amplitude value. Inset is a zoom view of $20 \mu m$ around the centre of white light fringe envelope. The error in retrieving the experimental visibilities have increased when compared with Fig. 7, indicating undesirable changes in the experimental setup between the consecutive measurement days.

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6.2. Visibility retrieval using 4 beams

It is desirable to extract the experimental visibilities when 4 input beams are injected simultaneously into the chip, firstly to study the effect of transfer matrix (V2PM) in extracting the visibilities, and secondly to simulate interferometric arrays of telescopes. Figure 6 shows the optical setup, where the delayed Mirror M3 is coupled to WG numbered 10 as shown in Fig. 1(b) while keeping the remaining 3 mirrors fixed. The output WGs of the DBC were recorded taking a total of 5000 data frames. From section 2, the column wise filling of the V2PM depends on the 4 photometric intensities and 6 pairwise combinations of the input beams by adding a known phase delay in one of the input beams. Hence, the calibrated V2PM that was taken using 2 beams at a time will still be valid for estimating the visibilities when a new data is taken with all the 4 input beams coupled into the DBC. Thus, the V2PM obtained from Fig. 7 was inverted and applied to the new data. The retrieved experimental amplitudes are shown in Fig. 10. Since the mirror M3 is modulated while the remaining mirrors are kept fixed, we should see the modulation in visibility (i.e. white light envelope) for $V_{13}, V_{23}, V_{34}$ and a constant visibility of 1 for $V_{12},V_{14}, V_{24}$, which is also seen in Fig. 10. But, the uncertainties in estimating the visibility amplitude is much larger in this case as compared to when 2 beams are injected. The errors in estimating the visibility are already huge in section 6.1, when the measurement is repeated next day with the same experimental settings. Now, due to increased number of beams, the visibility errors have increased further. The differential tip/tilt introduced by the mirror causes undesirable phase differences at the output of each WG and is highly dependent on the numbers of beams used. The unwanted phase errors are due to combined effect from the undesirable changes in the experimental setup and also, from the numbers of beams used in the experiment. We believe that the results could be improved by using more precise delay lines and reducing differential tip/tilt from the mirrors. However, we appreciate that we are able to retrieve the visibilities as expected using 4 beams at a time, but subjected to experimental errors.

 

Fig. 10. Experimentally retrieved visibility amplitudes for 4-input DBC for all 6 combinations of the input beams. Here, the previously obtained calibrated P2VM using 2 beams at a time (from Fig. 7) is applied to the new data. The new data is taken, when all the 4 beams are coupled in and beam corresponding to mirror M3 is delayed as shown in Fig. 6. Blue represents the retrieved visibility from experimental data. The x-axis is the absolute delay line position (in $\mu m$) as measured in our experiment and the y-axis is the amplitude value. The retrieved visibilities are as expected, where we see a modulation in the visibility (i.e. white light envelope) for $V_{13}, V_{23}, V_{34}$ and a constant visibility of 1 for $V_{12},V_{14}, V_{24}$. However, the errors have increased when compared with Fig. 7, indicating a combined effect of undesirable changes and also, numbers of beams used in the experimental setup.

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Funding

Deutsche Forschungsgemeinschaft (326946494); Bundesministerium für Bildung und Forschung (03Z22AN11).

Acknowledgements

We would like to acknowledge Dr. Romina Deiner for fabricating the DBC device in GLS. We would like to thank Dr. Jan Tepper for setting up the optical table for characterizing the device. We would like to heartily acknowledge Dr. Stefano Minardi for his concept on DBC. Also, it will be incomplete if we don’t mention Dr. Aline Dinkelaker and Dr. John Davenport for their proof reading of the paper.

Disclosures

The authors declare no conflicts of interest.

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