1. Introduction

The emergence of computer-controlled spatial light modulators (SLMs) enabled the dynamical control of complex optical wavefronts. This made possible the generation of beams in which their amplitude, phase, and polarization could be arbitrarily shaped on demand [1]. With feedback-based methods, such beams could be reliably created even despite strong scattering and aberrations [2–4]. Furthermore, a linear model of a complex system can be acquired in the form of an optical transmission matrix [5]. The employment of these wavefront shaping techniques enables the control of a complex wavefront also at the distal end of a multimode fibre [6–10], making it possible to utilize a fibre for optical tweezing [11] or to project complex wavefronts [12]. Through projection of orthogonal wavefronts, multimode fibres can be used as hair-thin endoscopes with single pixel imaging via point-scanning [13–16] or compressive sensing [17,18]. Digital micromirror devices (DMDs), as an alternative to liquid-crystal SLMs (LC-SLMs), are gaining in popularity in the field of complex photonics, particularly when high modulation rates are sought [19–22]. On the other hand, DMDs are limited to binary amplitude-only holograms and present low diffraction efficiencies [23].

Stable wavefront shaping quality following the calibration of the optical systems is essential for emerging applications such as in-vivo imaging through holographic endoscopes based on single multimode fibres [15,24]. As opposed to LC-SLMs, which have been specially designed and improved for scientific purposes, DMDs have been developed mainly for fast projection of incoherent light, and in that sense are still used “off-label” by wavefront shaping setups.

In previous experiments employing DMDs as SLMs for high-speed imaging through complex media, we noticed that the imaging quality decreased quickly over time, as illustrated in Fig. 1(a), most prominently at the highest modulation rates. Through external stabilization of the DMD temperature with a thermoelectric cooler (TEC) element, such rapid drop in imaging quality could be fully inhibited, as shown in Fig. 1(b). The comparison of both imaging sessions, with and without thermal stabilization of the DMD modulator, is shown also in Visualization 1. These observations urged an investigation of the temperature-dependent change of the wavefront quantitatively in order to understand, and improve, the cause of unstable imaging quality.

 

Fig. 1. Imaging stability of holographic endoscopy via a single multimode optical fibre. Evolution of imaging quality (a) without and (b) with active temperature stabilization of the DMD modulator operating at its maximum frame rate (22.7 kHz). Video recordings available as Visualization 1.

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In this work, we study the temperature-induced wavefront deformations of the DMD modulator, in order to understand the origin of degradation in wavefront shaping performance. Further, we assess their impact on two important case study systems of complex photonics: focussing coherent light behind a ground-glass diffuser – as an example of highly-scattering medium – and across a multimode optical fibre. A simple solution consisting on active temperature stabilization of the DMD modulator is shown to overcome this issue, extending the stability of wavefront shaping to several hours without the need to recalibrate the optical system.

2. Experimental methods

The optical setup was built in a modular fashion to perform all experiments under identical conditions. Its three different configurations, illustrated in Fig. 2, were used to study the effect of temperature on the synthesized wavefronts at the DMD (configuration 1), on the foci generated across a diffuser (configuration 2), and on the foci created through a multimode optical fibre (configuration 3). The particular methods used in each experiment are detailed below.

 

Fig. 2. Schematic of the optical setup used in the experiments, comprised of 3 configurations: (1) wavefront correction of the DMD modulator, (2) focussing behind a ground-glass diffuser, and (3) focussing through a multimode optical fibre (MMF). Inset: schematic of the assembled DMD module allowing active temperature control. Laser: Coherent Sapphire SF NX 488 100 mW @10 mW; CMOS: Basler acA640-750um; DMD: ViALUX V-7001; TEC: Thorlabs TECF2S; L1: Thorlabs AC254-200-A-ML; L2: Thorlabs AC254-30-A-ML; L3: Thorlabs AC254-80-A-ML; L4: Thorlabs AC254-150-A-ML; MO1: Olympus PLN 20X; MO2: Olympus UPLFLN 20X; MMF: Thorlabs FG050UGA; D: Thorlabs DG10-220-MD; APT: Thorlabs SM1D12; BS: Thorlabs BS010; QWP: Thorlabs WPQ05M-488 LP: Thorlabs LPVISE100-A.

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The same DMD module (Vialux V-7001, based on the Texas Instruments DLP-7000 chip) is used in all experiments as the wavefront shaping element, and is overfilled by a collimated laser beam (488 nm wavelength) under an incidence angle of ≈24°. The active region of the DMD chip is limited to its central $768\times 768$ pixels, and we use the Lee hologram approach – whereby a linear grating is superimposed to the binary-amplitude DMD holograms – to modulate the phase in the off-axis diffraction orders [25]. To control the temperature of the DMD chip, a thermoelectric cooler (TEC) is placed between the DMD module and a heat sink, and a temperature sensor (thermistor) is fixed close to the contact surface between the DMD and the TEC (inset of Fig. 2). For cooling, the TEC transports heat from the DMD to the heat sink and for heating vice versa. This allows a precise and fast stabilization of the DMD temperature, not limited by the time constant of natural thermal equalization. It is important to note that, in the cooling state, the heat sink should be able to dissipate sufficient heat to the surrounding air for the required temperature set. Otherwise, the temperature difference between DMD and heat sink rises to the point where the power of the TEC is not sufficient for stabilization. Due to a frame rate dependent variable thermal load of the DMD module, its temperature changes during the display operation. In our experimental settings we could find an increase of 16.5 °C above room temperature at the maximum display frame rate (22.7 kHz). Such a temperature deviation could cause small deformations on the DMD chip which could in turn distort the projected wavefronts, as investigated in the first experiment below.

2.1 Wavefront correction of a focussed beam

In the simplest arrangement (configuration 1 in Fig. 2), the beam reflected off the DMD is focussed through lens L1 onto the sensor array of a CMOS camera. The DMD displays a binary grating with a 4-pixel period along both orthogonal directions, and aperture APT placed next to the Fourier plane of lens L1 isolates the resulting first diffraction order, while filtering out the remaining. The focus recorded by the CMOS camera is affected by aberrations of lens L1, as well as from the curvature of the DMD chip itself. To measure the wavefront deformation, the transmission matrix of this simple system is acquired using the sub-domain based method [23], whereby the active area of the modulator is segmented into a $48\times 48$ regions (sub-domains) with size $16\times 16$ pixels. Each of such sub-domains sequentially displays the 4-pixel period grating with four phase steps of $ {\pi }/{2}$, and light reflected off each one is interfered at the CMOS camera sensor with the signal originating from another sub-domain – displaying a static grating throughout the calibration procedure – serving the purpose of an internal phase Ref. [26]. The recorded interference patterns on the CMOS camera allows building a transmission matrix of the optical system, which encodes the phase map necessary to apply on the DMD in order to correct for the wavefront deformation. An example of such phase maps with $48\times 48$ pixels can be seen in Fig. 3(a). The transmission matrix is acquired for temperatures from 25 °C to 40 °C in steps of 3 °C. To ensure that the wavefront deformation is caused by temperature, rather than by misalignment introduced by mechanical drift in the optical mounts, the temperature is decreased back to 25 °C in the same manner.

2.2 Focussing behind a ground-glass diffuser

To evaluate the impact of temperature-induced wavefront deformations of the DMD modulator in the highly-scattering regime, a wavefront correction generating diffraction-limited foci behind a ground-glass diffuser (D) is performed at a reference temperature. The experimental setup is depicted in configuration 2 of Fig. 2. The beam reflected off the DMD is spatially filtered by aperture APT placed in the Fourier plane of lens L1 to isolate the first diffraction order, and lens L2 images the DMD onto ground-glass diffuser D. Transmitted, forward-scattered coherent light forms speckle behind the diffuser, captured by a CMOS camera placed at a distance such that the speckle grain size spans over several camera pixels. Linear polarizer LP2, aligned with linear polarizer LP1, placed between the diffuser and the camera ensures that only one polarization state is observed.

The system is calibrated with the DMD stabilized at 25 °C in the same manner described above, using $48\times 48$ sub-domains as input fields and a square grid of $11\times 11$ focal points (located at camera pixels) as the output basis. A sequence of 121 binary-amplitude holograms for generating foci behind the diffuser at each calibrated position is then calculated from the transmission matrix and uploaded to the on-board memory of the DMD module. The temperature of the DMD is increased in steps of 3 °C, and for each the 121 foci are generated sequentially with their individual intensity profiles recorded by the CMOS camera. Since these distributions span over the 8-bit depth of the camera, high-dynamic-range (HDR) images are reconstructed from four measurements taken with different exposure times spanning four orders of magnitude. After reaching 40 °C, the measurements are repeated for decreasing temperatures back to 25 °C.

2.3 Focussing and imaging through a multimode fibre

Configuration 3 in Fig. 2 is used to assess the influence of temperature-induced wavefront deformations of the DMD modulator when focussing coherent light through a step-index multimode fibre (MMF). In this geometry, the DMD is imaged onto the back-focal plane of microscope objective MO1 by lens pair L1-L3 forming a telescope with a demagnification ensuring that the beam fits within the acceptance cone of the fibre. Quarter-wave plate QWP positioned before MO1 creates a circularly polarized beam, such that the polarization state is well preserved through the MMF [9]. The circularly polarized beam emerging from the MMF is converted back to linear polarization by a second QWP, and the distal fibre facet is imaged onto a CMOS camera by microscope objective MO2 in combination with tube lens L4. Beamsplitter cube BS merges this beam with an equally polarized external phase reference.

The calibration is conducted with the plane-wave based method [9], where each input is formed by a plane wave truncated by the active area of the DMD propagating with a distinct wavevector, forming all together a grid of foci at the input fibre facet. Each such input field is displayed sequentially by the DMD with four phase-steps of $ {\pi }/{2}$, and the resulting speckles emerging from the distal fibre facet interfere with the external phase reference at the CMOS camera. The phase drift between the two signals is tracked by displaying a reference binary grating after every four phase-step scan of each input mode [6]. A transmission matrix is built from the recorded interferograms, and a sequence of 92 binary-amplitude holograms for generating diffraction-limited foci on the distal fibre facet is calculated and uploaded to the DMD’s on-board memory. These 92 distal foci are generated sequentially and HDR images of their intensity profiles recorded in the same manner as described above in sub-section 2.2, for varying temperatures in increasing (followed by decreasing) steps of 3 °C using the same initial calibration taken at the reference temperature of 25 °C.

The experimental setup in this configuration, as well as the holographic methods described, were used also to assess the impact of the temperature-induced wavefront deformations of the DMD modulator under its own thermal load, over time. In this experiment, the wavefront optimization is carried with the DMD modulator initially at room temperature (≈22°), and the sequence of holograms generating distal foci is calculated and uploaded to the DMD memory. The DMD is set to display this sequence continuously at its maximum frame rate. Every 30 s the high-speed display is interrupted to acquire HDR images of 60 foci at the highest frame rate allowed by the CMOS camera at each exposure time used, with the high-speed display being then resumed. This experiment is repeated also using active temperature stabilization of the DMD modulator at 34.5 °C using the TEC element throughout both the transmission matrix measurement and acquisition of HDR images of generated foci on the distal fibre facet.

For the qualitative imaging demonstration shown in Fig. 1, the calibration plane is set to 20 µm in front of the distal fibre facet by offsetting microscope objective MO2 by this same amount. Once the wavefront optimization is completed, MO2 is further displaced away from the fibre facet, and a negative 1951 USAF resolution test chart is inserted at this plane. The DMD displays a sequence comprising ≈20 000 holograms, corresponding to raster-scanning the calibration plane with an equal number of sequential foci. For each focus, the imaging system (microscope objective MO2 together with tube lens L4) collects the light transmitted through the 1951 USAF test chart, which is integrated over a region of interest on the camera sensor, and the resulting intensity assigned a grayscale value on the corresponding image pixel. Because in these experimental settings the imaging speed is limited by the acquisition frame rate of the CMOS camera, we again set the DMD to continuously display the hologram sequence at its maximum frame rate, with an interruption every 30 s to acquire a full image at the maximum frame rate supported by the camera. This demonstration is performed once with, and once without active temperature stabilization of the DMD module.

3. Results

3.1 Wavefront correction of the DMD modulator

The experimental setup in its simplest arrangement (configuration 1 in Fig. 2) gives rise to an aberrated focus in the Fourier plane of lens L1 – recorded by the CMOS camera – when no spatial light modulation is enforced by the DMD. Here, the sources of aberrations include the spherical surfaces of the lens, small misalignments of the optical components, and curvature of the DMD chip itself. Measuring the transmission matrix of the system in a representation of sub-domains on the DMD modulator yields information on which complex amplitudes to apply to each sub-domain in order to obtain a diffraction-limited focus at each calibrated CMOS camera pixel. With the DMD placed in the far-field plane of the foci, the phase-only information of such complex amplitudes is therefore equivalent to the discretized phase aberration map in the pupil plane. Examples of such phase maps are shown in Fig. 3(a) for wavefront optimizations (i.e. measurements of the transmission matrix) generating a focus on the same location (i.e. camera pixel), performed at distinct temperatures. The temperature-induced deformation of the DMD modulator can thus be assessed by computing the (unwrapped) phase map differences $\phi \left (x,y\right )$, where $\left (x,y\right )$ denotes the sub-domain position, to the phase map of a reference temperature (chosen to be the starting temperature of 25 °C), as illustrated in Fig. 3(b).

 

Fig. 3. Temperature-induced wavefront deformation of the DMD modulator. (a) Examples of phase maps producing a diffraction-limited focus, measured at varying temperature. (b) Average wavefront deviation (phase difference) from the reference measurements, taken at 25 °C. (c) Reconstruction of the wavefront deviations shown in (b) using Zernike polynomials $Z^m_n$ up to order $n=2$. (d) Residuals between (c) and (b). (e) Zernike coefficients and total wavefront error as function of temperature for heating (solid lines) and cooling (dashed lines). (f) Fraction $ {\sigma _{t}}/{\sigma }$ of total wavefront error, described by Zernike polynomials $Z^m_{n}$ ($n\leq 2$), as function of temperature.

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To evaluate the type and magnitude of the aberrations involved in these phase deformations, a circular region of the phase maps (marked with dashed circles in Fig. 3(b)) was projected onto Zernike polynomials $Z^m_n\left (x,y\right )$ [27], to retrieve Zernike coefficients as $c^m_n = \sum _{x,y}{Z^m_n\left (x,y\right )\left [\phi \left (x,y\right )-\bar {\phi }\right ]}$, where $\bar {\phi }$ is the mean value of $\phi \left (x,y\right )$. In Fig. 3(e), we plot the Zernike coefficients with order $n\leq 2$ obtained in this way, as a function of temperature. Zernike coefficients of order $n>2$ were comparatively neglectable, and are thus not shown on the plot. The mean value $\bar {\phi }$ is subtracted from the aberration map $\phi \left (x,y\right )$ since it corresponds to an overall phase offset, and therefore does not account for any aberrations; as a consequence, the Zernike coefficient $c^0_0$ is identically zero and is also not plotted in Fig. 3(e).

The total wavefront error, $\sigma$, calculated as $\sigma ^2 = \sum _{x,y}{\left (\phi \left (x,y\right )-\bar {\phi }\right )^2}$, is also plotted in Fig. 3(e). As result from the expansion of $\phi \left (x,y\right )$ onto the orthonormal basis of Zernike polynomials, the total wavefront error could also be obtained from the Zernike coefficients as $\sigma ^2 = \sum _{m,n}^{\infty }{\left (c^m_n\right )^2}$. By truncating this infinite sum to terms $n\leq 2$, denoted by $\sigma _t^2 = \sum _{m}\sum _{n\leq 2}{\left (c^m_n\right )^2}$, the fraction of the total wavefront error described by aberrations of order $\leq 2$ is quantified by $ {\sigma _t}/{\sigma }$. This ratio is plotted in Fig. 3(f) as a function of temperature, showing that Zernike aberrations of order $n\leq 2$ account for more than 98% of the total wavefront error for the phase deformations measured at temperatures higher than the reference value. Figure 3(c) shows the phase deformations reconstructed from the Zernike coefficients $n\leq 2$, i.e. $\phi _r\left (x,y\right ) = \sum _{m}\sum _{n\leq 2}{c^m_n Z^m_n\left (x,y\right )}$, and in Fig. 3(d) are shown the residuals between the measured and reconstructed phase deformations, $\phi -\phi _r$, indicating an error smaller than $ {\lambda }/{20}$.

3.2 Focussing behind a ground-glass diffuser

With the experimental setup arranged in configuration 2, coherent light transmitted through the ground-glass diffuser generally forms a speckle field behind it, as shown in Fig. 4(a). The methods described above allow computing the DMD holograms from a transmission matrix recorded at the reference temperature (25 °C) to generate diffraction-limited foci at any calibrated output location (camera pixel). An example of one such focus is shown in Fig. 4(b), both in linear (b1) and in logarithmic (b2) scales, where its peak intensity is seen to have increased from its average value prior wavefront shaping by nearly three orders of magnitude. The ratio between these two values, so-called enhancement, is frequently used to quantify the focussing quality through highly-scattering media [2]. Panel b2 also shows that an uncontrolled speckle background remains present. Moreover, the diffraction pattern of the square active area of the DMD modulator is also visible, indicating that the focus is indeed close to be limited by diffraction and not by any other aberrations in the system, which are inherently compensated by the wavefront shaping method. When changing the temperature of the DMD – while retaining the wavefront correction acquired at the reference temperature – aberrations of increasing magnitude begin to manifest, modifying the shape and size of the foci, as well as reducing its peak amplitude and therefore the enhancement (Fig. 4(c)). Figure 4(d) shows the enhancement as function of temperature measured for 121 foci generated sequentially at different locations across the focal plane. The measurements were performed with a “heating” cycle, followed by “cooling” the DMD, showing that the effect is reversible, thus excluding other sources of degradation of the wavefront correction. A not fully thermally equalized DMD at calibration could be the cause of the small deviation at 25 °C.

 

Fig. 4. Focussing coherent light through a ground-glass diffuser. (a) Speckle resulting from transmission of uncontrolled coherent light across a highly-scattering medium. (b) Example of a diffraction-limited focus generated behind the diffuser by wavefront shaping. (c) Aberrated focus resulting from an increase in temperature of the DMD modulator while applying the wavefront correction used in (b). The intensities in (a1-c1) are represented in linear scale, whereas in (a2-c2) they are displayed in logarithmic scale. (d) Enhancement averaged over 121 equally distributed foci as function of temperature, generated using a wavefront correction (transmission matrix) obtained at 25 °C. The scale bars in (a-c) correspond to 50 µm. The error bars in (d) indicate standard deviations over 121 measurements (foci) at each temperature.

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3.3 Focussing through a multimode fibre

In case of configuration 3 of the experimental setup (Fig. 2), coherent light coupled into a multimode fibre generally emerges from its opposite extremity as speckle, as seen in Fig. 5(a). As before, the transmission matrix measured at the reference temperature (25 °C) allows calculating the DMD holograms that result in diffraction-limited foci at the output calibration plane (chosen as the distal fibre facet). Figure 5(b) shows one of these foci as an example, in linear (b1) and in logarithmic (b2) scales. In the latter, concentric rings surrounding an Airy disc are clearly discernible, which correspond to the diffraction pattern of the circular entrance pupil of the optical fibre (i.e. its acceptance cone). Since in this case one has access to the total optical power exiting the fibre on its distal endface, a suitable figure-of-merit frequently adopted for quantifying the focussing performance of the wavefront-shaping system is the fraction of controlled optical power. This so-called power ratio was estimated from the HDR intensity distributions of the generated foci as the ratio between the power contained in the focus (the integrated power under a fitted Airy distribution) and the total integrated intensity emerging from the distal fibre facet. Particularly in the context of holographic endoscopy, the power ratio impacts directly the imaging quality, especially when performing raster-scan imaging [16].

 

Fig. 5. Focussing coherent light through a multimode fibre. (a) Speckle resulting from transmission of uncontrolled coherent light along a multimode fibre. (b) Example of a diffraction-limited focus generated by wavefront shaping on the distal endface of the fibre. (c) Aberrated focus resulting from an increase in temperature of the DMD modulator while applying the wavefront correction used in (b). The intensities in (a1-c1) are represented in linear scale, whereas in (a2-c2) they are displayed in logarithmic scale. (d) Average power ratio of 92 equally distributed foci as function of temperature, generated using a wavefront correction (transmission matrix) obtained at 25 °C. The scale bars in (a-c) correspond to 5 µm. The error bars in (d) indicate standard deviations over 92 measurements (foci) at each temperature.

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As before, modifying the temperature of the DMD while applying the wavefront correction measured at the reference temperature leads to increasing aberrations in the spatial light modulations coupled into the fibre, resulting in a visible degradation of the focussing quality. A higher fraction of optical power is found on the uncontrolled speckled background, and the power ratio of the distal foci decreases accordingly, as seen in Fig. 5(c). Figure 5(d) shows the power ratio of 92 foci generated consecutively as function of the DMD temperature. As in the previous experiment, the measurements were performed in a cyclic fashion under increasing followed by decreasing temperatures. The fact that the two curves are congruent guarantees that focussing degradation was not caused by drift of opto-mechanical components, and that the time interval between measurements allowed the system to reach thermal equilibrium.

3.4 Stabilized wavefront shaping through a multimode fibre

The quality of wavefront shaping decreases with departing temperature of the DMD modulator from that at which the system is calibrated. This was quantified above by a diminishing power ratio of foci generated through a multimode fibre (Fig. 5), and before by a decreasing enhancement of foci created behind a highly-scattering sample (Fig. 4). In those cases, the effect was studied by setting the temperature of the DMD in a controlled manner using a TEC element. However, a freely running DMD has its own thermal load, which results in an increase of its temperature. When displaying holograms at its maximum refresh rate of 22.7 kHz, the temperature of the DMD rises quickly from the ambient temperature up to ≈35 °C over the course of ≈1 h, as shown in Fig. 6(a). This results in a fast degradation of the focussing quality, as demonstrated by the decreasing power ratio of foci generated at the distal endface of a multimode fibre. As a consequence, images reconstructed by raster-scanning an object with excitation foci quickly become impaired, as illustrated earlier in Fig. 1(a). By actively stabilizing the temperature of the DMD modulator using the TEC element, this rapid loss of focussing quality can be prevented, as demonstrated by the plot in Fig. 6(b) which shows no signs of degradation over 1 h. In the context of holographic endoscopy, this allows greatly extending the imaging stability over a much longer time span, as shown before in Fig. 1(b).

 

Fig. 6. Focussing stability through a multimode fibre. Power ratio of generated foci and temperature of the DMD modulator as function of time with active temperature stabilization (a) disabled (b) and enabled, while the DMD modulator operates at its maximum display rate of 22.7 kHz. The shadowed areas indicate standard deviations around the average values.

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4. Conclusion

Motivated by a rapid quality degradation observed in wavefront shaping systems employing DMDs as spatial light modulators, we found that very-high-speed DMD modulators, operating at their fastest display rates, experience a temperature-induced deterioration of the synthesized wavefronts arising from the thermal load of the devices.

Within the range of temperatures that the DMD can normally reach due to its own thermal load (i.e. up to ≈40 °C), we have shown that increasing temperature causes a phase deviation that can be well described by low-order aberrations (namely defocus, tip, tilt, and astigmatism), represented by Zernike polynomials up to order two. Although the magnitude of such wavefront deformations might seem small – less than one wavelength for the highest temperature tested – we further demonstrated their impact in two important case-study systems of complex photonics. When focussing light behind a highly-scattering sample, the peak intensity in the optimized foci (accounted by the enhancement factor) gradually decreased with temperature by more than three fold. Similarly, when generating foci across a multimode optical fibre, the fraction of optical power contained in the foci (described by the power ratio) decreased steadily with temperature also to approximately one third of its initial value. Using this model system, we also compared the focussing quality over time while the DMD modulator operated continuously at its maximum display rate with and without active temperature stabilization of the DMD chip, showing that our proposed solution prevents the quick degradation in wavefront shaping quality. Our studies of wavefront impairment are readily applicable to estimations of thermal instability influence in applications beyond these demonstrated here, for example structured-illumination imaging approaches.

The reliable performance achieved here allows greatly increasing the stability of DMD-based wavefront shaping systems from minutes to hours, thus lifting an important constraint in the design of experiments, especially when recalibrating the system is not possible. Particularly in the context of holographic endoscopy, as shown in Fig. 1, such long-term stability can crucially extend imaging sessions to tens of hours using a single calibration of the system.