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Abstract
We present theoretical and laboratory experimental results on a robust interferometric device based on pupil inversion, or 180° rotational shearing interferometry. The image of an astronomical object degraded by the atmosphere turbulence can be restored (ideally up to the diffraction limit) by a numerical post-processing of the interferogram. Unlike previous Michelson configurations that return half of the light to the sky, the Mach-Zehnder interferometer has no fundamental losses when both outputs are used. The interferogram is formed by two overlapped images of the telescope pupil, but one of them is spatially inverted, and out of phase by π/2 only in its half. This optical operation is achieved in a robust way by inserting a refractive optical image inverter and a binary phase plate in one of the arms of the interferometer. In this way, the system has no polarization dependence or moving parts since the plate allows the object to be retrieved numerically from just one interferogram (single exposition) or a few independent interferograms. For that, several algorithms are proposed. Likewise, we include a laboratory proof-of-concept in which a diffraction-limited image is obtained in spite of presence of aberrations and photon noise.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Optical resolution of ground-based telescopes is severely degraded by atmospheric turbulence, being a more important limitation than diffraction for telescope diameters ($D$) greater than Fried’s parameter also called atmospheric coherence diameter: $r_0\sim$ 200 mm. Several methods have been developed to circumvent the effects of turbulence [1–8]. The aim of this work is to show that an improved inversion pupil interferometer can help in this difficult task in a simple, economic and scalable way.
The first method to deal with the turbulence was the Fizeau/Michelson stellar interferometer [9,10] which selects two separate portions of the wavefront coming from a star (each subaperture being smaller than $r_0$). A low resolution image that is modulated by a fringe pattern is obtained. Although the turbulence randomly moves the pattern, its fringes can be detected if short enough exposure times ($\sim$20 ms) are used. The visibility mainly depends on the distance between the subapertures (baseline) and the object shape; in fact, the baseline at which the fringes are lost provided the first angular size of a star.
More generally, the Van Cittert-Zernike theorem states that the complex coherence function (which includes modulus and phase of the visibility) of an unperturbed wavefront coming from a monochromatic incoherent far object is the Fourier Transform of its intensity. The atmosphere alters the phase of this function but it does not its modulus, provided the angular size of the object is smaller than the so-called isoplanatic angle.
Aperture synthesis and aperture masking interferometry [11] can be seen as evolved versions of the Fizeau/Michelson method. These use the van Cittert-Zernike theorem to reconstruct the image of arbitrary objects from many coherence measurements for a narrow spectral band. In particular, aperture masking method incorporates three or more subapertures that are carefully located in order to the fringe pattern from each subaperture pair have an unique set of period and orientation. So each pair gives information of one point in the Fourier plane ($u-v$ plane). Moreover, the sum of the phases of each triplet of subapertures (phase closure) is independent of the turbulence or the telescope aberration, allowing the phase of visibility to be accurately measured. On the contrary, the modulus of the visibility is reduced in presence of turbulence since different wavefront tilts in each subaperture result in poorly overlapped images. This introduces the need of a turbulence calibration and imposes an upper bound to the signal-to-noise ratio (SNR) of the modulus of the visibility [12]. Other drawbacks are that the mask typically wastes more than 90 % of the light captured by the telescope and that a series of measurements must be made to fill the $u-v$ plane. Even so, these points are usually scarce and their location is not evenly distributed in the $u-v$ plane. The lack of points introduces artifacts in the image that must be mitigated by algorithms which could introduce extra assumptions [13].
Speckle interferometry [14] method uses the full pupil, so it access to the entire Fourier plane simultaneously. However, we can consider that many pairs of fictitious sub-apertures contribute to a given spatial frequency in the image. Since each pair is perturbed by its own atmospheric phases, the fringe visibility is low and its phase is lost, resulting in a speckled pattern. The remaining information provides the autocorrelation of the intensity of the object, but the SNR per frame is lower than one, even for brilliant objects, so a great amount of exposures are mandatory. Moreover, frequent calibrations of the atmosphere turbulence are necessary, which requires a star acting as a point source. It has been widely used to resolve binary stars.
Two simple methods to process short-exposure speckled images are the shift-and-add and lucky imaging ones. In the former, the captured images are shifted in order to align their respective brightest speckle; the images are then added to obtain a notably improved image [15]. However an halo with the size of a long exposure image remains around each punctual source. The lucky imaging method takes advantage of the fact that turbulence is random, so each captured image has a different quality; in particular, diffraction-limited images are possible although its probability decreases exponentially with the pupil surface [16]. By applying the shift-and-add method to only the 1–10% best images (according their Strehl ratio), the halo intensity is reduced and the diffraction limit can be reached for medium-sized telescopes ($D \le 7r_0$) [7,15].
On the other hand, the adaptive optics method is based on measuring turbulence and compensating for it by deforming a mirror in real time [8]. This measurement requires a (natural or artificial) bright guide star less than a few arcseconds away from the region of interest. Although it is a very successful technique, it is also complex and very demanding in real time data processing, mechanical accuracy and stability. Moreover, the larger the telescope, the more actuators are needed in the deformable mirror and the longer its stroke must be. Therefore, scaling adaptive optics for future large telescopes is an increasingly difficult task [17].
In rotational shearing interferometry (RSI), an interferogram is formed by two overlapped images of the telescope pupil, rotated to each other [18,19]. When the rotation angle $\alpha$ is 180$^\circ$, each point of the exit pupil interferes with its diametrically opposed point. In this way, each point of the interferogram only contains information of a point of the $u-v$ plane. Moreover, all spatial frequencies of the object captured by the telescope take part in the interferogram, although this does not hold for other rotation angles. To obtain the visibility function, it has been proposed to take two consecutive interferograms, changing the phase of one of the beams in $\pi /2$ in the second one [20]. Both interferograms must be taken before the turbulence changes. From them, the modulus of the visibility is obtained fully independent of the turbulence (amplitude closure) with a theoretical SNR only limited by the photonic noise. In this point RSI is clearly superior to both speckle interferometry (at least for brilliant objects) and aperture masking [12]. On the contrary, the phases retrieved by shearing in its simplest version depend on the atmospheric turbulence, although its effects can be easily reduced in several ways as we will show. An RSI-based procedure featuring phase closure has also been demonstrated [21,22], but the optical system is more complex. On the other hand, the object field in RSI is determined by the spatial sampling frequency used to capture the interferogram. The sampling distance has to be less than about $r_0$ in the pupil, that is, the image of an atmospheric coherence cell on the camera that takes the interferogram must be larger than a pixel. If this image is much larger, sampling is denser, thinner fringes can be detected, and the object field increases; but the amount of light per pixel and the SNR are reduced. In practice, this limits the maximum object field of the method, especially with faint sources and inefficient or noisy detectors.
All proposed optical setups to implement RSI have been based on a Michelson interferometer with prisms in one or both arms, but most of them present several problems: half of the light returns to the sky; due to unbalanced reflections, there is a polarization issue that results in low contrast; and the need for two interferograms requires a piezoelectrically driven mirror synchronized with a detector whose exposure time is cut in half. We must bear in mind that at the time the RSI was proposed, there were no digital cameras and the computers were still in the beginning. Interferograms were detected with single pixel photomultipliers or photographic films, both methods being slow and impractical. The former needed a scan and the latter had low sensitivity and required film development and subsequent digitization. In addition, adaptive optics became very attractive at the nineties and many efforts were devoted to it. Since then, only a few advances in RSI can be found un the literature and for other purposes [23,24]. However, the technological advances of the last decades make inversion pupil interferometry much more feasible: present digital cameras take fast images with very high quantum efficiency and low noise (in fact, they are photon noise limited); the interferogram can be numerically processed with modern computers and algorithms; the size of the object field is less important as the resolution of the telescopes improves, since the objects to be observed are also smaller; and it is very easily scalable. So it is worth exploring the usefulness of this technique, especially in combination with any of the other methods.
In section 2, we present the theory of an improved pupil inversion interferometer to address the above problems and take advantage of the technological advances: a Mach-Zehnder configuration that has a refractive inversion system, a phase plate and no moving parts. Moreover three different possibilities to process the interferograms are presented. In section 3, the experimental laboratory setup is described in detail including an artificial object and a simulated turbulence. In section 4, we explain a slight modification of the processing algorithms which is intended to address residual aberrations of the interferometer. These algorithms were checked with experimental interferograms from faint sources and a bias from the photon noise was corrected. Two possible hybridizations of pupil inversion interferometry with other methods are also outlined. Finally, the conclusions are presented in section 5.
2. Pupil inversion Mach-Zehnder interferometry with a phase plate
Let us consider a spatially incoherent light stellar object. For sake of expository clarity we describe the mentioned object by a large but discrete number of light points although a continuous object could also be used. The light coming from these light points reaches the entrance pupil of a telescope which forms an image near or inside a Mach-Zehnder interferometer; this only matters for vignetting purposes (see Fig. 1). In each point on the plane $\boldsymbol {\rho }_{o}$ of the entrance pupil there is a complex optical field $E_1(\boldsymbol {\rho }_{o},t) =\sum _j A_{j}e^{i\phi _{j}(t)}\, e^{i\boldsymbol {k}_{j}\boldsymbol {\rho }_{o}}\,e^{i\alpha (\boldsymbol {\rho }_{o},t)}\,e^{-i\omega _{c}t}$, where $\phi _{j}(t)$ are very fast random phases of each light point emitter ($j$) of the object due to the optical incoherence; $\boldsymbol {k}_j$ is the wavenumber projection of each emitter on the pupil plane; and $\alpha (\boldsymbol {\rho }_{o},t)$ is the phase fluctuation due to the atmosphere, common to all sources (isoplanatism). Moreover, $\omega _{c}$ is the central frequency of a chromatic filter which is placed at the input of the interferometer; this justifies the use of the temporal phase $e^{-i\omega _{c}t}$. The filtered optical field propagates through both arms of the interferometer. In one of them, a spatial inversion $\boldsymbol {\rho }$ $\rightarrow$ $-\boldsymbol {\rho }$ is implemented by a refractive image inverter. In the other arm, only compensating elements are introduced. The inverter system can be considered as a Kepler telescope with a particular field lens [25]. A sketch of it is shown in Fig. 2 and a more detailed explanation is given in appendix A. It is designed to obtain an image of the pupil with magnification $M=-1$ and located just at its own plane. So, the pupil and its inverted image fully overlap when viewed from either output port of the interferometer. A lens after the last beam splitter forms the final direct image and the inverted one on a camera, where the interference is registered. Note that the refractive system does not alter the polarization, in contrast with the inverters based on prisms. The use of both outputs of the interferometer would allow for collecting all the light entering the telescope; unlike Michelson-type interferometers which resend half of the light to the sky. Since both interferograms are complementary, their difference gives the fluctuations with an improved SNR.
It has been shown [20] that the random atmospheric phases can be removed from the modulus of the complex visibility function. This is achieved by using only two consecutive interferograms which differ in an extra phase $\pi /2$. However, as will shown further, each interferogram almost has inversion symmetry (except for a constant phase) so the information that provides is duplicated. We can take advantage of this property if we introduce the extra phase in one half of the pupil. Thus we have all the information needed to recover the visibility modulus in just one interferogram. Fortunately, there is a conjugated plane of the pupil inside one arm of the Mach-Zehnder, between the second and third lens of the optical image inverter. Here, we place a so-called biphase plate which introduces this phase retardation $\pi /2$ between its two halves: $\delta _p(\boldsymbol {\rho })=\theta (y)\pi /2$, being $\theta$ the Heaviside function.
Fig. 1. Sketch of the pupil inversion Mach-Zehnder interferometer at a telescope. Light coming from a stellar object propagates through the turbulent atmosphere and is collected with the primary mirror of the telescope that acts as the entrance pupil EP (not to scale). The system also consists of: a chromatic filter CF with a narrow bandwidth $\Delta \lambda$; an optical image inverter system OII located in one arm; a coarse optical path compensator OPC and a thin prism P to fine adjust in the other arm; and a final lens to form two interfering images of EP at the exit pupil XP, where a camera (preferably CCD) registers the interferogram. The OII system is made up of three lenses and an ion-exchanged glass biphase plate which introduces phases 0 and $\pi /2$ in its upper and lower half, respectively.
Fig. 2. Sketch of the optical image inverter which is made up of three positive lenses with focal distances $f,f_0$ and $f$, respectively. They are separated a distance $f$, which is suggested by a typical optical inverter composed of two lenses with focal $f$ and separated a distance $2f$. However, we introduce a central lens whose focal distance value is $f_0=f/4$, such a way that the plane containing the inverted image of the pupil coincides with the plane of the pupil itself. This result is shown by a ray tracing. Likewise, the biphase plate is placed at the conjugated plane of the entrance pupil after the second lens (see appendix A).
Now, we can write the total optical field at the plane $\boldsymbol {\rho }=(x,y)$ of the exit pupil, except a complex constant and/or a scaling factor, as a superposition of the optical fields propagated through both arms of the Mach-Zehnder interferometer, that is,
Now, taking into account that the $\phi _j$’s are uncorrelated to each other (incoherent sources), we can write the interferogram irradiance at the exit pupil as:
A possible strategy to improve the aberrated images is to apply a numerical version of some known method to the perturbed complex visibility given by Eq. (5). For example, we can easily emulate the shift-and-add method by centering the brightest pixel in each aberrated image and averaging them. The lucky image one is very similar as it only involves an extra selection. Let us analyze the former in the $u-v$ plane. The centering step is essentially equivalent to removing the linear part of the spatial Taylor’s expansion of $\Delta (\boldsymbol {\rho },t)$. As $\Delta (\boldsymbol {\rho },t)$ is free from quadratic terms, the remaining aberration, $\Delta _3(\boldsymbol {\rho },t)$, only contains cubic and higher odd terms. Then it should keep within the range $\pm \pi$ in a circle centered in the pupil having a diameter significantly bigger than $r_0$. The image averaging can be performed in the visibilities since the Fourier transform is a linear operator. This involves the average of the phasor $\exp \left \{i\Delta _3(\boldsymbol {\rho },t)\right \}$, which will result in 1 where the fluctuations of $\Delta _3(\boldsymbol {\rho },t)$ are small. Instead, the average will be zero for large values of $\rho$ where $\Delta _3(\boldsymbol {\rho },t)$ fluctuates more than $\pm \pi$. This will lead to a kind of numerical apodization if we apply this method to too large telescopes. Even so, the lucky imaging method is expected to take advantage of larger telescopes when applied to inversion interferometry images than when applied to conventional images, since the latter are affected by symmetric terms of aberration like defocus, astigmatism and spherical aberration. In fact, the lucky imaging has achieved diffraction-limited images in a 4.2 m telescope in combination with an adaptive optics system that removes only low-order aberrations [26]. Moreover, let us consider a dark object point nearby to a bright source. The aberrated image of that source given by the Fourier transform of Eq. (5) is a real function that can take positive or negative values at the position of the geometrical image of the dark point. Instead the point spread function of a conventional detected image is always positive, which generates a bias in the averaged image resulting in the halo [27]. The proposed method is free from this artifact, which can be an advantage in detecting a faint companion of a bright source.
A different processing possibility arises when the object contains a point source away from the rest, since a numerical version of the adaptive optics method can be applied. For that, we take Fourier transforms in Eq. (5) and we numerically select the region of the aberrated image of this point source (guide star) by setting the rest to zero; then we take the inverse Fourier transform to obtain $\Delta (\boldsymbol {\rho },t)$ which is finally introduced again in Eq. (5) to recover the Fourier transform of the object.
A third approach is to take the square modulus in the Eq. (5) in order to remove the phase of the turbulence $\Delta (\boldsymbol {\rho },t)$ from it:
3. Experimental setup and operation
We have implemented the proposed interferometer in the laboratory (Fig. 3) as a proof of concept of our configuration. In addition to the elements shown in Fig. 1, it consists of: a sodium spectral lamp (SSL) as an incoherent monochromatic light source that also simulates the presence of a bandpass interference filter; an artificial object (AOB) close to it; and a phase distortion plate (PDP) at L$\simeq$1.75 m from the object. The object (Fig. 4(a)) is an aluminized glass plate with several small holes made by etching. The radius of the largest hole is 62 $\mu$m, being smaller than the resolution of the subsequent optical system whose Airy’s disk radius is 92 $\mu$m when projected on the object plane. So, all holes act as point sources, appearing to be brighter the larger. Similarly, there are a couple of points which are too close (72 $\mu$m) to be clearly resolved. The PDP was fabricated by spreading and curing a few drops of optical adhesive over a 75$\times$50 mm$^2$ glass plate to obtain a wavy and smooth surface. It was qualitatively characterized in an separate Mach-Zenhder interferometer working at 633 nm. The interferograms showed about 10 to 20 fringes/cm, corresponding to a simulated Fried parameter of a few tenths of a milimeter. However, the phase distortion seems to be dominated by a tilt component. When it is removed, between 2 to 4 fringes remain, as we can see in Fig. 4(b). Our aim is not to accurately simulate the atmospheric turbulence but to qualitatively illustrate its effect and check the system behavior. Just after the PDP, an Schneider Componon-S 5.6/150 objective lens plays the role of a primary telescope lens and the entrance pupil (EP) with a focal distance $f_p = 150$ mm, and working at an aperture of f/11. The lenses of the optical image inverter (OII) have the following focal lengths: $f$ = 40.1 mm (achromatic doublet lens with central thickness 2.5 mm+10 mm) and $f_0$ = 10.1 mm (Steinheil triplet lens with thickness 1.0 mm+3.7 mm+1.0 mm). Due to the high coherence length of the source a partial optical path compensation is enough in this case, which was done with two glass plates (10+10 mm) and a prism (3 mm) inserted in the other arm. The biphase plate is located between the second and third lens of the OII, in a conjugate plane of the entrance pupil, as said above. We have fabricated it in our laboratory by using an accurate ion-exchange technique in glass [31,32], as explained in the appendix B. Finally, a CCD camera (Clara model from Andor Technology) working in photocounting mode and equipped with an objective Canon Macro Lens 100 mm, 1:2.8 USM takes the image of the exit pupil XP.
Fig. 4. (a) Microscope image of the artificial object fabricated by etching an Al layer over a glass plate; the image shown is 4.8 mm wide. (b) 10 mm diameter interferogram of the PDP, taken at 633 nm after reducing the contribution of tip&tilt for clarity. (c) and (d) Images taken through the interferometer when the object is imaged without and with the PDP, respectively. (e) and (f) Images of the pupil without and with the PDP, respectively.
The final objective can also focus the image plane of the first lens, as shown in Fig. 4(c), where the PDP is absent. Here, we see both the direct image (on the right side) and the inverted one of the object shown in Fig. 4(a). These images have a limited resolution that barely makes it possible to distinguish the weak source close to the brightest one (286 $\mu$m apart). Moreover, the direct image is weaker than the inverted one due to the lack of anti-reflective coatings on the surfaces of the compensating glass plates. This impairs the contrast of the interferograms. In Fig. 4(d) we can see the dramatic loss of resolution in the same plane when the phase distortion plate is inserted into the system.
In order to take the interferogran, the final objective must be focused on the pupil plane. In the Fig. 4(e) we show the interferogram obtained without the PDP. By looking at its border we can distinguish the direct and inverted pupils. The interference patterns formed by straight fringes appear where both overlap. The biphase plate causes a discontinuity in the fringes across the interferogram in a near horizontal line (its orientation is irrelevant). Finally, the pupil plane in presence of the same aberrations as in Fig. 4(d) is shown in Fig. 4(f). The fringes are not longer straight, but the interferogram retains a certain symmetry. Several spots appear in both interferograms caused by defects in the optical surfaces. There are two of them near the center that are caused by only one defect in the first objective. We take advantage of this by using the middle point between them to define the inversion center of the interferogram.
4. Image retrieving
4.1 Interferogram capture and correction
The interferograms in Fig. 4 have a good SNR; however, we are interested in the behavior of the proposed method for low illumination levels and how this limitation can be improved by combining a set of interferograms. In order to verify the noise reduction by combining several interferograms with different wavefront distortions $\Delta$, we took a series of 11 of them, interspersing between each shot a PDP displacement across the input pupil. The exposure time of the camera was set in such a way that about a mean of 190 photons were detected per pixel. However, the standard deviation of the number of photons throughout the series is different for each pixel, according to its visibility and ranging from 10 to 90 photons. Note that a pixel with zero visibility presents a constant classical intensity equal to $\bar I$, but a standard deviation of 14 photons is expected from quantum noise. A central square region of the interferograms was selected to numerically process them avoiding peripheral defects of the images. The influence of the remaining defects was minimized by normalizing each interferogram by the system transmittance, which was estimated from the mean value of all the interferograms.
When we used the procedure of section 2, we detected slight asymmetries in the interferograms; for example, there is a subtle curvature in the fringes of the Fig. 4(e). If ignored, a poor removal of the twin image occurs when we apply the Eq. (5). This is due to residual aberrations originated inside the interferometer: a lack of flatness of its optical surfaces and/or aberrations in the inverter system. It can be modeled by making $\mu _1$ and $\mu _2$ also dependent on the position in the pupil. In spite of that, the equations of section 2 remain valid, provided that $\Delta$ includes the odd part of $\mu _1-\mu _2$ and $\mu$ is its even part:
4.2 Processing algorithms and results
After that correction, we have applied the Eq. (5) to each interferogram, and then we have performed a Fourier transform to obtain a real function that corresponds to an aberrated image. One of these is shown in the Fig. 5(a), where the negative values of the calculated intensity where set to zero. As explained above, this image only contains third and higher odd aberrations, so its resolution is much better than that of a conventional optical image (for instance, Fig. 4(d)) which contains all aberration orders. In Fig. 5(b) the series of computed images was averaged following the shift and add method (and then truncated when negative). The image is remarkably improved and no halo has appeared, but each source is still a few pixels wide, so the diffraction limit was not achieved. The numerical adaptive optics procedure explained in section 2 was applied to the Fig. 5(a) to obtain 5(c). For that, the rightmost point source was used as the guide star. In spite of that source is one of the weakest of our object (responsible of about 14 photons per pixel in the interferogram), we obtain a resolution similar to that of Fig. 5(b), where the complete series was used.
Fig. 5. (a) Low resolution image numerically recovered from a single interferogram with odd aberrations. (b) Improved image from the series of recovered images by the shift-and-add method. (c) Improved image obtained by applying a numerical method inspired in adaptive optics to the same interferogram than that used in (a). A 20 pixel wide disk around the faint rightmost point source of (a) was used to obtain the aberration $\Delta (\boldsymbol {\rho },t)$. A gamma correction of 1/2.4 was applied to all images to highlight the faintest details.
On the other hand, in order to apply Eq. (6b) we must take into account that the number of detected photons per pixel ($K_{ij}$) is a random variable (even for a fixed turbulence) due to the quantum or photon noise. Its expected value and its variance are equal and they are proportional to the classical irradiance $I$. Moreover, the calculation of $(\Delta I)^2=I^2+{\bar I}^2-2I\bar I$ is involved in our method, as shown in the Eq. (6b). For that, $I^2$ must be obtained from the expected value of $K_{ij}(K_{ij}-1)$ [1, Chapter 9] instead of $K_{ij}^2$. In other words, the photon noise generates a bias if the captured image is directly used in the calculation of the squared terms of the equations. This bias hinders the convergence of the subsequent numerical method.
The Fig. 6(b) shows the result of the left-hand side of Eq. (6a) when applied to one interferogram (Fig. 6(a)). As expected, a periodic structure can be seen because the object is a collection of a few points and the wavy appearance of the fringes in Fig. 6(a) due to the aberration $\Delta$ is removed. However, the pattern is restricted to an elliptical central region, since the term $\cos ^22\mu$ depends on the position and cancels in an annular region. Now, the time dependence of $\hat \mu (t)$ is an advantage because each interferogram presents different useful regions. In the Fig. 6(c) we see a weighted average of the squared visibility modulus of all interferograms, being the weight $\cos ^22\mu$ to reduce the influence of the noisy regions of each interferogram:
Fig. 6. (a) Interferogram in the plane of the pupil for a particular aberrated input wavefront (b) Squared modulus of the visibility obtained from this interferogram by applying Eq. (6a). (c) Weighted average of the squared modulus of the visibility of the series of eleven interferograms.
Figure 7(a) shows the experimental autocorrelation of the object, obtained by computing the Fourier transform of the averaged visibility module (Fig. 6(c)), and then removing the white noise. This noise appears evenly distributed throughout the Fourier plane as a consequence of the spatial randomness of the photodetection [1, Chapter 9]. Figure 7(a) can be compared with the expected autocorrelation of the object for a similar resolution, as obtained by numerical simulation from the microscopic image of the artificial object (Fig. 7(b)). Although some noise still shows up in our method, and the central peak is higher than expected, the width of the peaks is around one pixel like in the simulation. This proofs that the turbulence effect was fully removed achieving the maximum resolution for the used aperture. The object was recovered from the experimental autocorrelation by using the Fienup’s iterative method [28] which takes advantage of the fact that the intensity of the object can not be negative. Specifically, by starting with an aberrated image, 4 cycles were applied, each including 25 iterations of the hybrid input-output algorithm (with $\beta =0.7$) and 5 iterations of the error-reduction one. The result can be seen in Fig. 7(c), where a much cleaner image is obtained than in Fig. 5, and the resolution is about one pixel as in the autocorrelation. Consequently, the weak source close to the brightest one is clearly resolved, even better than in the conventional image without the PDP (Fig. 4(b)). This shows that the static aberrations of the optical system have also been corrected. We must highlight again that this procedure could be applied to a single shot interferogram if there were no residual aberrations in the interferometer.
Fig. 7. (a) Autocorrelation of the object, obtained as the Fourier Transform of Fig. 6(c). (b) Simulation of the autocorrelation of the object (Fig. 4(a)) for a resolution similar to that of (a). (c) Recovered image by applying the Fienup’s procedure to (a). A gamma correction of 1/2.4 was applied to all images to highlight the faintest details.
4.3 Discussion
In view of the above results, we are going to discuss several interesting features and limitations of our proposal.
The weakest sources of our object provide about 14 photons each per pixel in the camera, which is enough to detect them from a single interferogram as we have seen. Let us translate this to a more realistic situation: this number of photons could be provided by a Sun-type star of magnitude 9.8 in a 13 nm spectral range centered around 800 nm on a pupil area of ($r_0 / 2) ^ 2$ = 0.01 m$^2$ (to be conjugated to a pixel) for a time of 10 ms (within a coherence time). This magnitude ($\sim$ 10) roughly matches the limiting sensitivity of speckle interferometry. This coincidence could be more general because both methods need a similar spectral filtering and the light of a star must be distributed in the same amount of pixels. In speckle interferometry, the short exposure image of a star contains $(D/r_0)^2$ speckles, so the number of pixels must to be at least four times larger to resolve its pattern. This quantity coincides with that necessary to sample the pupil in our method and obtain an object field equal to the seeing. The sensitivity of our method can be improved by combining a very large number of interferograms using some of the above methods, but a more detailed study is needed, mainly if the average number of photons per pixel is less than one.
Like any interferometric system, ours must be protected from vibrations, but it is not an especially difficult task because our interferometer is quite compact. The vibrations of the telescope itself due to wind forces can alter the interferogram; for example, a torsion of the structure that changes the orientation of the telescope will change the fringe spacing. However, it will also move the image on a conventional non-interferometric telescope with similar resolution. If the vibration frequency is much less than the inverse of the coherence time ($\sim$100 Hz), its effects will be corrected simultaneously with that of atmospheric turbulence. Also, some vibration modes can cause defocus in a conventional telescope or other even aberrations, to which our system is insensitive. Therefore, this system could be even more robust against vibrations than conventional ones.
We must remember that the larger the telescope, the greater the phase $\alpha$ of the wavefront aberration to be corrected. This is a serious scalability problem for deformable mirrors that are commonly used in adaptive optics as they must increase their stroke along with the number of actuators. As in diffractive adaptive optics systems that lack range limitation since the phase is wrapped [29], our algorithms work independently of the phase of turbulence since they also work in module $2\pi$. Moreover, the coherence length is also not an issue, since it must be linearly increased with the diameter of the telescope and is therefore always greater than the wavefront distortion. Therefore, there is no practical limitations to the aberrations that our system can manage even for big telescopes.
The last algorithm presented in the section 4.2 does not need a guide star to recover the image, which is a very particular feature. Note that it is difficult to apply adaptive optics to an extended object, because Shack-Hartmann sensors must correlate in real time the different images of this object provided by each sub-aperture, which is much more demanding than locating centroids corresponding to a guide star. A radial shearing interferometer was proposed as an wavefront sensor in adaptive optics systems for extended objects [30]. However, this system is not appropriate for this task, since, being similar to ours, it loses visibility at the periphery of the pupil if the object is resolved, which prevents measuring the turbulence to be corrected. In principle, it could be used to directly retrieve object information from the interferogram in some similar way as just explained. However, a lot of light is wasted in this design to maintain good interference contrast, and the maximum spatial frequency that could be recovered is less than half that of our system. Since the Fienup algorithm can work without a guide star, it could be advantageously applied to obtain high resolution images of bright and extended objects, for which there is no clear alternative method. Solar astronomy, Solar System planets or even their satellites are posible science cases. In some of them, a field aperture must be inserted in the focal plane to restrict the object field within the isoplanatic angle. Then a scanning and a numerical stiching of the corrected images will be needed.
4.4 Combination with other techniques
Pupil inversion interferometry provides a way to measure the modulus of visibility without being affected by atmospheric turbulence, while aperture masking interferometry does the same with the phase of visibility. This suggests combining them, taking advantage of the fact that masking interferometry uses only a small part of the pupil. So, a holed mirror located in a conjugate plane of the pupil can redirect this discarded part of the wavefront to an inversion interferometer where the visibility modulus can be obtained in broad regions of the pupil. The subsequent numerical processing should integrate both types of data to recover the intensity using much more information than the masking interferometry method alone.
Pupil inversion interferometry can be also combined with partial or simplified adaptive optics. The latter only has to compensate for local wavefront distortions resulting in an increased effective Fried’s parameter. This, along with a proper object field restriction, will widen the interference fringes. Therefore a smaller pupil image is possible on the sensor while still keeping the thinnest fringe wider than twice the pixel pitch of the camera. In other words, we can sample the pupil more coarsely but still above the Nyquist rate. As each pixel in the interferogram plane conjugates now with a larger surface of the input pupil, more photons are detected per pixel, which improves the SNR. In this way, weaker sources can be detected in exchange for a reduced object field. Particularly, lets us consider an adaptive system that corrects the turbulence of each pair of diametrically opposed mirrors of a segmented telescope in such a way that the pupil inversion interference pattern of the guide star be a constant in these mirrors. Since a piston phase is allowed between neighboring mirrors, this task is much easier than a full correction. Indeed, the correction of each pair is independent of the rest of them, a parallel computing can be applied, and a moderate range deformable mirror can be used. Consequently, the system works as if the Fried parameter increases to the diameter of a segment of the mirror. Each mirror pair can also be thought of as a synthetic aperture system taking a particular point in the $u-v$ plane. The complete telescope acts as a parallelized system that accesses many of these points simultaneously.
5. Conclusions
We have presented of a pupil inversion interferometer based on a Mach-Zehnder configuration, that could be incorporated to a astronomical telescope. It enables us to recover a perturbed complex visibility function of an incoming wavefront from a single shot interferogram. The phase of this function is affected by the odd part of the atmospheric turbulences, whilst its modulus is not perturbed at all; in particular, the retrieved visibility is insensitive to defocus and astigmatism generated by the atmosphere. Our optical configuration has several advantages: it prevents the loss of light, since both output ports of the interferometer can be used; it can work with unpolarized sources, because the inverter optical system is refractive; and it is a static system, because no moving mirrors are needed to take two phase-shifted consecutive interferograms. Instead, a $\pi /2$ phase plate placed in a conjugate of the pupil, inside one of the arms of the interferometer enable us to obtain the same information in only one interferogram. In this way, the frame rate of the camera is no longer a critical feature, which often conflicts with its sensitivity.
We propose three different algorithms to post-process the visibility function and remove the remaining turbulence effects from it. Two of them are inspired by the shift-and-add method and adaptive optics; both being very simple. In the third one, the visibility modulus is computed and the photon noise is corrected; then the Fienup’s iterative method is applied to recover the phase of the visibility. The three algorithms have been checked with a series of eleven experimental interferograms obtained from a laboratory system in low light conditions. All of them improve the image, but the third one provides the best image in terms of noise and resolution. In particular, it reaches the diffraction limit of the processed interferogram, also correcting the static aberrations of the optical system.
In short, inversion pupil interferometry is a method to circumvent the atmospheric turbulence in astronomical imaging by a post-detection stage; it should be further explored. The proposed interferometer and algorithms implement this technique in a manner that is economical, simple, efficient, polarization-insensitive, passive, scalable and tolerant to aberrations of both the telescope and the interferometer itself. It is an excellent option to study bright and extended objects, such as the planets of the Solar System or Solar astronomy.
A. Optical image inverter design
A typical optical image inverter with lateral magnification $-1$ can be designed by using two identical lenses with focal distance $f$ and separated a distance $2f$. For this system, it is easy to check that the image position is always at a distance $4f$ from the object position. However we are also interested in obtaining a position of the image (exit pupil) coincident with the one of the object (entrance pupil). This goal is possible if we insert a new lens (field lens) with focal length $f_0$ at the middle of the other two lenses (see Fig. 2). If the telescope pupil is at a distance $s_1=-L$ from the first lens of the image inverter, we obtain conjugates of the pupil after the field lens and the last lens at distances $s'_2=(1/f_0+(L-f)/f^2)^{-1}$ and $s'_3=2f-f^2/f_0-L$, respectively. Next, by imposing that the position of the pupil image meets the pupil itself, $s'_3 = -L-2f$, the relation $f_0=f/4$ is obtained. Remarkably, this condition is independent of the pupil position as it does not include $L$. We can also check analytically that the lateral magnification remains $M=-1$ for any value of $L$. Alternatively, we can verify that a ray entering the system parallel to the optical axis also emerges parallel to it at the same distance but on the opposite side. Now, the image distance of the field lens simplifies to $s'_2=f^2/(3f+L)$, giving a real image ($0<s'_2<f$) since both $f$ and $L$ are positive. We place the biphase plate in this position, so it is conjugated with the pupil plane through the first and second lenses and also through the third one because of the inverter design.
B. Fabrication and test of ion-exchanged glass plates.
Phase plates can be fabricated easily by ion-exchange (IE) in glass. IE is a well-established technique used in fields such as materials processing and integrated optics [31]. It provides important advantages compared to methods based on changing the thickness of the substrate. For example, plates fabricated by IE are monolithic. This ensures that the change of phase is achieved inside the substrate, which provides an excellent uniformity, robustness, and durability of the plate. Moreover, high precision phase shifts in particular regions of the substrate can be achieved through the control of fabrication parameters such as diffusion time ($t$), temperature ($T$), and ionic concentration.
We fabricated a two-region plate which produces a phase step in the incident beam by about $\pi$/2 radians for $\lambda$=589 nm. A standard photolithographic process was used to define the two regions. Plane-round soda-lime glasses of 25.3 mm diameter and 1 mm thickness were coated with a 250 nm aluminium film and about 1.5 $\mu$m of positive photoresist. The photoresist was exposed during 1"s to UV light through a mask consisting of a single razor blade, which divided the round glass into two equal parts. After development and aluminium removal of the exposed region, the semicircular mask was ready to perform the IE. The IE process was made in a 5% mole AgNO$_3/$NaNO$_3$ salt melt at T=340$^\circ$C and for $t$=3.5’. Next we characterized the plate by measuring the effective indices of the waveguide generated in the exchanged region. From these effective indices the refractive index profile of the waveguide was recovered by using the Inverse Wentzel-Kramers-Brillouin method (IWKB) [32]. In particular, the following refractive index profile provided an excellent fitting to the experimental data:
with $\Delta$n=0.07024, $\alpha$=7.33, and d=1.303 $\mu$m. From these parameters the phase step was calculated by the following equation This phase step is 5% less than our goal $\pi$/2, which means an accuracy of $\lambda /80$. A similar performance in the system of Itoh and Ohtsuka [20] would require an accuracy of 2.6 nm in the position of their piezoelectrically driven mirror. Moreover, if even more accuracy were needed, values better than $\lambda /200$ could be achieved by IE.Funding
Ministerio de Economía, Industria y Competitividad, Gobierno de España (AYA2016-78773-C2-2-P); European Regional Development Fund.
Acknowledgments
The authors thank Ana Alonso for the technical help, and Héctor González-Nuñez and Manuel P Cagigal for their useful suggestions.
Disclosures
The authors declare no conflicts of interest.
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