1. Introduction

Light from distant stars and galaxies travels through the near perfect vacuum of space for millions of years, carrying information about the distant parts of the universe, until the last fraction of a second when it plunges through the Earth’s atmosphere before reaching our eyes or telescopes on the ground. Atmospheric turbulence causes degradation or aberration of images, such that increasing the aperture of a telescope does not lead to improvement of resolution. Placing the telescope in space above the atmosphere, or at least at as high an altitude as possible on top of mountains, alleviates the problem, but at considerable cost and constraint. The concept of adaptive optics (AO), put forward at least several decades ago [1,2], is based on sensing the wavefront and adjusting the optical system in real time to compensate for any distortion. The development of wavefront sensors and compensators with sufficient levels of performance were crucial to implementation of the AO in large telescope systems that began in the 1990s [3]. Remarkable astronomical images have been obtained using ground-based telescopes with AO compensation of atmospheric turbulence, whose quality can surpass even space-based telescopes under favorable conditions [4]. The principle of AO is also applied to other imaging systems, most notably in ophthalmic imaging [5] as well as in laser beam forming and remote sensing [6].

In a typical AO system, a guide star of sufficient brightness is used for measuring the aberration. A common method of wavefront sensing is the Shack–Hartmann sensor, consisting of a lenslet array and a CCD camera underneath it [7]. Distortions of the wavefront lead to shifts of lenslet focal spots, proportional to the local slope of the wavefront. The information is used to compute the necessary displacement of a deformable mirror or other spatial light modulator (SLM). The wavefront sensing, wavefront modulation, and control subsystems form a closed loop to reach a configuration that minimizes the aberration in the image of the full field. Specifications of the AO system are governed by the key parameters of atmospheric turbulence [8]. The Fried parameter is the effective spatial scale of the turbulent eddies, such that it, rather than the telescope aperture, limits the image resolution. The isoplanatic angle is the maximum angular distance between the guide star and the object under observation, which is limited by the distribution of turbulent layers over a range of altitudes. The Greenwood frequency represents the inverse time scale of the turbulence, and therefore dictates the response time of the AO feedback system. Thus, the number of sensor subapertures limits the spatial bandwidth of measurable aberration, while the number and types of actuators in the modulator limit the accuracy and bandwidth of corrections. The speed and accuracy of computation and optomechanical feedback are critical aspects of such an AO system.

We have recently introduced AO principles based on the ability of digital holography (DH) to measure and manipulate optical phase profiles [9]. They replace the wavefront sensor and corrector hardware components with numerical processing by DH [10,11] for wavefront measurement and compensation. The principle of aberration compensation is a well-known characteristic of holography, as demonstrated in the early years of conventional holography [12]. Phase conjugation by nonlinear optics is a dynamic holography process and has been an active area of research and development for applications in AO [13]. More generally speaking, these processes are based on the decoding of an encoded image using its correlation with the key, such as in the Vander Lugt matched filter [14]. In context of AO, the true image is encoded by the aberration, and the aberration key is extracted using the guide star information. DH is particularly versatile and flexible in extracting and manipulating the optical phase profiles, as demonstrated in digital holographic microscopy [15–17]. Compensation of low-order aberrations, including tilt, spherical aberration, and astigmatism, have been demonstrated either by double exposure of the field with and without the specimen, or by assuming a portion of the object field to be flat [18]. Automatic compensation of higher-order terms of the Zernike polynomials has been demonstrated, and a concept of numerical parametric lens has been introduced which can shift, magnify, and compensate for aberrations [19].

Almost all objects of telescopic observations, such as stars and planets, are incoherent sources. A number of techniques have been put forward for holography of incoherent sources since the early years of conventional holography, including the grating achromatic interferometer [20], the triangular interferometer [21], and conoscopic holography [22]. Experimental demonstrations of most of these techniques, however, have been limited to reconstruction of not more than a few point sources. The main difficulty is that although it is possible to generate self-interference of elementary objects, such as point sources, often in the form of Fresnel zone patterns, the interference patterns are incoherent with respect to each of the other point sources. As soon as the number of point sources increases beyond a few, or for objects of extended surfaces, the intensity of interference patterns adds incoherently, rapidly washing out the fringes. On the other hand, in DH, together with the phase shifting technique [23], it is possible to extract the complex amplitude from the interference pattern of each point source. The complex amplitudes then add coherently among all the point sources.

DH can be carried out using incoherent sources in the same manner as using coherent sources, provided that the object and reference arms are maintained at equal path lengths within the margin of the coherence length [24]. But in such a system, the observable depth is limited to the coherence length. Conventional or analog holography of objects larger or deeper than coherence length using an incoherent source presents a particular challenge, as mentioned earlier. On the other hand, a DH adaptation of the triangular interferometer and conoscopic holography, for example, has been shown to overcome the limitations of the analog versions [25,26]. The optical scanning holography is a unique approach to incoherent holography, where a three-dimensional (3D) Fresnel zone pattern is scanned over the object volume [27]. A significant new development is the Fresnel incoherent correlation holography (FINCH), where two copies of the object wave are superposed with an extra phase curvature introduced on one of them, so that every point source on the object produces the Fresnel zone interference pattern [28]. The complex field amplitude is extracted by quadrature shifting of the relative phase between the two copies. FINCH has been demonstrated to be capable of generating color holograms of objects illuminated with arc lamps or holograms of fluorescence [29,30].

In a recent paper [31], we have introduced a new method of AO to measure and compensate aberrations based on incoherent digital holography (IDH) with a modified FINCH interferometer. As with the coherent holographic AO system, the hologram of a guide star contains sufficient information to compensate the effect of an aberration on the full-field hologram. Reconstruction from an uncompensated full-field hologram yields distorted images comparable to direct nonholographic images formed through the same aberrating medium, while the compensated holographic images have much of the aberrations removed. In this paper, we provide full descriptions of the theoretical simulation and experimental studies of the method—incoherent digital holographic adaptive optics (IDHAO)—that we have carried out. The IDHAO process is seen to be robust and efficient under various ranges of parameters, such as aberration types and strength. Furthermore, low-image signals buried under noise can be extracted by IDHAO to yield high-quality images with good contrast and resolution, both for point-like sources and continuous extended objects illuminated with common incoherent light sources. We begin with a theoretical description of image reconstruction in incoherent DH and aberration compensation and AO in IDH in Section 2, followed by simulation studies in Section 3 that visualize theoretical predictions. A series of experiments described in Section 4 clearly confirm expected characteristics and capabilities of the IDHAO system. Theoretical and experimental results are summarized and discussed in Section 5, particularly in view of potential astronomical and other imaging applications. We conclude in Section 6.

2. Theory

Theoretical description of IDHAO is developed by considering propagation of spherical waves from object source points through a variation of a FINCH interferometer, followed by integration of the interference intensities over the source points. A complex hologram is obtained by invoking a phase-shifting procedure, leading to an expression of the holographic image as a convolution of object intensity with a complex point spread function (PSF). Even in the presence an of aberrating layer, under a specific condition, one obtains a holographic image as a convolution but with distorted PSF. A straightforward procedure is described to compensate for the aberration by combining holograms of the full field and a guide star.

A. IDH of Basic Configuration

Consider the essential configuration of IDH, Fig. 1, based on a variation of the FINCH interferometer, where the pixel-sharing SLM is replaced with a modified Michelson interferometer. The beam-splitter (BS) produces two copies of the spherical wave emanating from each point source of an object, and the two mirrors (${\mathrm{M}}_A$ and ${\mathrm{M}}_B$), equidistant from BS, add different curvatures to the spherical waves. Two copies of the same optical field are always coherent with each other. Therefore, when they are superposed at the camera plane, they produce Fresnel zone-like interference patterns. Specifically, consider a point source of strength $I_o={|E_o|}^2$ on ${\mathrm{\Sigma }}_o(x_o,y_o)$-plane a distance $z_o$ from the curved mirrors. First, consider IDH imaging without the aberrator $\mathrm{\Psi }$ in place. The field reflecting from ${\mathrm{M}}_A$ and arriving at the camera plane ${\mathrm{\Sigma }}_c(x_c,y_c)$ a distance $z_c$ from the mirror is, under Fresnel or paraxial approximation [10]:

 

Fig. 1. Basic configuration of IDH. S, source point; $\mathrm{\Psi }$, phase aberrator; BS, beam splitter; ${\mathrm{M}}_A$ (${\mathrm{M}}_B$), curved mirror with focal length $f_A$ ($f_B$).

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For an extended object illuminated by incoherent light, the source points are incoherent with each other. Therefore, the total intensity is the sum or integral of the point-source interference intensities $g(x_c;x_o)$ contributed by all the source points:

B. IDH with Aberration

Now, consider IDH imaging in the presence of the phase aberrator $\mathrm{\Psi }(x^{\prime })$ at the ${\mathrm{\Sigma }}^{\prime }(x^{\prime },y^{\prime })$ plane. The field component $A$ arriving at the output plane ${\mathrm{\Sigma }}_c(x_c,y_c)$ is now

C. IDHAO by Compensation of Aberration

The last integral is turned into a convolution if we set $z^{\prime }=0$, as described below, so that ${\alpha }_A={\alpha }_B=\alpha$. Then

For comparison, one can also calculate the direct image formed when one of the mirrors ${\mathrm{M}}_B$ focuses the image on the camera plane, while the other mirror is blocked, so that

3. Simulation

We have carried out a series of simulation studies to characterize the predictions of the theory. The IDHAO is implemented experimentally using the optical configuration depicted in Fig. 2, and the simulation assumes a similar configuration. The objective lens ${\mathrm{L}}_o$ of focal length $f_o$ forms an intermediate image in front of the interferometer. The phase aberrator $\mathrm{\Psi }$ is placed close to the lens ${\mathrm{L}}_o$. The relay lens ${\mathrm{L}}_a$ of focal length $f_a$ is used to image the phase aberrator $\mathrm{\Psi }$ onto the mirrors, achieving the requirement of $z^{\prime }=0$. The imaging lens ${\mathrm{L}}_c$ is used in combination with ${\mathrm{L}}_o$ to adjust the magnification and resolution of the system [33,34]. The interference filter (IF) helps improve the interference contrast. One of the mirrors, ${\mathrm{M}}_A$, is planar and piezo-mounted for phase-shifting process, while the other mirror, ${\mathrm{M}}_B$, is curved with focal length $f_B$.

 

Fig. 2. Optical configuration used in simulation and experiments. ${\mathrm{M}}_A$, piezo-mounted plane mirror; ${\mathrm{M}}_B$, curved mirror with focal length $f_B$; L’s, lenses; IF, interference filter.

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A. IDH

First, a basic IDH process, without any aberration, is illustrated in Fig. 3 where we use the pattern of the Big Dipper as our object field $I_o$ [Fig. 3(a)]. Each frame is 256 pixels and 10 mm across. The assumed optical configuration is such that the light from each point source arrives on the mirrors as a plane wave. The mirror ${\mathrm{M}}_B$ has focal length $f_B=3000\mathrm{mm}$, the imaging lens ${\mathrm{L}}_c$ is absent, and the camera is at $z_c=z_4+z_5=1000\mathrm{mm}$ from the mirrors, so that the holographic image distance is $z_i=2000\mathrm{mm}$. The wavelength is set $\lambda =633\mathrm{nm}$. One of the four phase-shifted interference pattern, $h_{\phi }$, on the camera plane is shown in Fig. 3(b), which we refer to as the raw interferograms. A combination of four such interferograms yields the complex hologram $H_0$, whose amplitude and phase profiles are shown in Figs. 3(c) and 3(d), respectively. In most of the illustrations in this paper, complex images are shown with a pair of amplitude and phase images. The amplitude images are rendered with the Matlab definition of “jet” color map, while for the phase images, the blue-white-red color map corresponds to the range $[-\pi ,\pi ]$. Numerical propagation of the complex hologram to an appropriate distance results in the reconstructed image $I_o^{\prime }$ in Figs. 3(e) and 3(f).

 

Fig. 3. Simulation of IDH process: (a) Object field, $I_o$; (b) one of raw interferograms $h_{\phi }$ on the camera plane; (c) amplitude; (d) phase of the complex hologram $H_0$; (e) amplitude; and (f) phase of reconstructed image $I_o^{\prime }$.

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B. IDHAO

The adaptive optics or aberration compensation process by IDHAO is illustrated in Fig. 4, where we start with the same input image $I_o$ in Fig. 4(a). The assumed phase aberration profile $\mathrm{\Psi }$, Fig. 4(b), consists of two Zernike polynomial terms, $\mathrm{\Psi }=a_{\mathrm{\Psi }}(Z_3^{+1}+Z_5^{-1})$ with $a_{\mathrm{\Psi }}=-0.5$. The raw interferogram $g_{\phi }$ for the guide star and $h_{\phi }$ for full-field illumination are shown in Figs. 4(c) and 4(f), respectively. A set of four quadrature phase-shifted interferograms combine to produce the complex holograms $G_{\mathrm{\Psi }}$ in Figs. 4(d) and 4(e) for the guide star and $H_{\mathrm{\Psi }}$ in Figs. 4(g) and 4(h) for the full field. Figure 4(i) is the expected direct image $I_B$ through the aberrator without any holographic process, while Figs. 4(j) and 4(k) show the reconstruction $I_{\mathrm{\Psi }}$ from the hologram $H_{\mathrm{\Psi }}$. Both $I_B$ and $I_{\mathrm{\Psi }}$ show serious degradation due to the aberration. The compensated hologram ${\widehat{H}}_{\mathrm{\Psi }}$ is shown in Figs. 4(l) and 4(m). Note that the vortex-like structures in Fig. 4(h) have been replaced by more concentric ones in Fig. 4(m). Aberration-compensated image ${\widehat{I}}_{\mathrm{\Psi }}$ is obtained in Figs. 4(n) and 4(o) by numerical propagation from the compensated hologram ${\widehat{H}}_{\mathrm{\Psi }}$, or equivalently by correlation of $H_{\mathrm{\Psi }}$ and $G_{\mathrm{\Psi }}$. The compensated image ${\widehat{I}}_{\mathrm{\Psi }}$ is seen to have removed much of the distortions seen in the direct image $I_B$ or the uncompensated holographic image $I_{\mathrm{\Psi }}$, resulting in well-focused image of the point sources of $I_o$.

 

Fig. 4. Simulation of the AO process by IDH: (a) the object field $I_o$, (b) the assumed phase aberration $\mathrm{\Psi }=a_{\mathrm{\Psi }}(Z_3^{+1}+Z_5^{-1})$ with $a_{\mathrm{\Psi }}=-0.5$, (c) one of the guide star interferograms $g_{\phi }$, (d) and (e) the guide star complex hologram $G_{\mathrm{\Psi }}$, (f) one of the full-field interferograms $h_{\phi }$, (g) and (h) the full-field complex hologram $H_{\mathrm{\Psi }}$, (i) direct image $I_B$ without holographic process, (j) and (k) IDH image $I_{\mathrm{\Psi }}$ without aberration compensation, (l) and (m) complex hologram ${\widehat{H}}_{\mathrm{\Psi }}$ with aberration compensation (n) and (o) IDH image ${\widehat{I}}_{\mathrm{\Psi }}$ with aberration compensation. (Note the somewhat irregular arrangement of subpanel labels.)

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C. Parametric Behavior of IDHAO

Behavior of IDHAO is simulated as several different parameters are varied, including the resolution, aberration strength, and aberration type. IDHAO is seen to be effective under a wide range of these parameters, as well as under increasing noise added to the phase profile. Aberration compensation is also expected to be equally effective for extended continuous objects as well as point-source objects.

As shown in Fig. 5(a), the resolution of the IDHAO-reconstructed images is found to be consistent with the expected behavior of $\delta x\propto \lambda f_B/a$, where $a=10\mathrm{mm}$ is the frame size and the mirror ${\mathrm{M}}_A$ is planar, while the focal length of ${\mathrm{M}}_B$ is varied over a range. In Fig. 5(b), an aberration $\mathrm{\Psi }=a_{\mathrm{\Psi }}(Z_3^{+1}+Z_5^{-1})$ with $a_{\mathrm{\Psi }}=-0.5$ is assumed. Evidently, the resolution of the reconstructed image is not significantly affected by the aberration when compensated by IDHAO, and behaves in an otherwise similar manner as in Fig. 5(a). The image resolution is also expected to depend on the arrangement of the imaging lens ${\mathrm{L}}_c$ and the camera with respect to the overlap of the reflections from the two mirrors ${\mathrm{M}}_A$ and ${\mathrm{M}}_B$, though we did not include the imaging lens in our simulation [33,34].

 

Fig. 5. Resolution of reconstructed images ${\widehat{I}}_{\mathrm{\Psi }}$ versus focal length: (a) without aberration $a_{\mathrm{\Psi }}=0.0$ and (b) with aberration $a_{\mathrm{\Psi }}=-0.5$. The focal length of ${\mathrm{M}}_B$ is varied as $f_B=(\mathrm{i})5000$, (ii) 3000, (iii) 2000, and (iv) 1000 mm, while $z_i=f_B-z_c$.

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The behavior of the IDHAO with increasing strength of aberration is illustrated in Fig. 6. The aberration is varied for (a) $a_{\mathrm{\Psi }}=-0.1$, (b) $a_{\mathrm{\Psi }}=-0.2$, (c) $a_{\mathrm{\Psi }}=-0.5$, and (d) $a_{\mathrm{\Psi }}=-1.0$ of the same type $\mathrm{\Psi }=a_{\mathrm{\Psi }}(Z_3^{+1}+Z_5^{-1})$ as above. Other parameters are as in Fig. 4. The aberration phase profiles are displayed on the left of each row. Then, $I_{\mathrm{\Psi }}$ is the uncompensated holographic image and ${\widehat{I}}_{\mathrm{\Psi }}$ is the holographic image compensated by IDHAO, while the direct image $I_B$ is also shown for comparison. That is, $I_{\mathrm{\Psi }}$ or $I_B$ are the “before” images, while ${\widehat{I}}_{\mathrm{\Psi }}$ is the “after” image. It is seen that the compensated image maintains reasonable quality for even a high level of aberration, whereas both the uncompensated and direct images deteriorate quite rapidly. For the strongest aberration in Fig. 6(d), the aberration completely obliterates the uncompensated image $I_{\mathrm{\Psi }}$ and the direct image $I_B$, though the latter develops a numerical artifact that makes the obliteration less obvious. However, the compensated image ${\widehat{I}}_{\mathrm{\Psi }}$ still maintains good resolution and contrast. On the other hand, note that while the compensated image ${\widehat{I}}_{\mathrm{\Psi }}$ maintains relatively sharp central peaks, the peak height does decrease and broader halo develops around each peak as the aberration increases.

 

Fig. 6. IDHAO versus strength of aberration. In each row the phase aberration $\mathrm{\Psi }$, the uncompensated image $I_{\mathrm{\Psi }}$, the compensated image ${\widehat{I}}_{\mathrm{\Psi }}$, and the direct image $I_B$ are shown. Strength of the aberration $\mathrm{\Psi }=a_{\mathrm{\Psi }}(Z_3^{+1}+Z_5^{-1})$ is varied as (a) $a_{\mathrm{\Psi }}=-0.1$, (b) $a_{\mathrm{\Psi }}=-0.2$, (c) $a_{\mathrm{\Psi }}=-0.5$, and (d) $a_{\mathrm{\Psi }}=-1.0$.

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The IDHAO process is equally effective for various types of aberrations, as shown in Fig. 7. In particular, the effectiveness of IDHAO is not contingent upon the symmetry of the aberration profile. Note that the uncompensated images $I_{\mathrm{\Psi }}$ and $I_B$ show similar patterns of distortion for each type of aberration, and the compensation removes most of the distortions in ${\widehat{I}}_{\mathrm{\Psi }}$.

 

Fig. 7. IDHAO versus various types of aberration: (a) $\mathrm{\Psi }=(-0.5)\times Z_1^{+1}+(-0.5)\times Z_2^0$, (b) $\mathrm{\Psi }=(-0.5)\times Z_1^{+1}+(-0.5)\times Z_3^{+1}$, (c) $\mathrm{\Psi }=(-0.5)\times Z_3^{+3}+(-0.5)\times Z_5^{-5}$, and (d) $\mathrm{\Psi }=(-0.7)\times Z_2^0+(-0.5)\times Z_5^{-5}$.

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Next, in Fig. 8, a gray scale photographic image is used as the model object, most other parameters being the same as before. Because of the large number of nonzero object pixels, the computation is substantially longer compared to the preceding simulations, where the object consisted of seven illuminated pixels—the IDHAO process needs to be calculated for each source point before integrating over all pixels. However, the IDHAO process is seen to behave as expected and is able to remove much of the distortion and mostly recover the resolution loss due to the assumed phase aberration.

 

Fig. 8. IDHAO of extended gray-scale object: (a) the assumed phase aberration $\mathrm{\Psi }=(0.5)\times (Z_5^{+3}+Z_4^{-2})$, (b) object $I_o$, (c) uncompensated image $I_{\mathrm{\Psi }}$, and (d) compensated image ${\widehat{I}}_{\mathrm{\Psi }}$.

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Lastly, we look at the behavior of IDHAO in the presence of noise. In Fig. 9, the noise is simulated by adding random phase noise on the complex holograms $G_{\mathrm{\Psi }}$ and $H_{\mathrm{\Psi }}$. A phase of $a_{\mathrm{\Phi }}$ times a random number in the range [0,1] is added to each pixel in the holograms, with $a_{\mathrm{\Phi }}=5$ for rows (a) and (c) or $a_{\mathrm{\Phi }}=10$ for rows (b) and (d). The first two rows (a) and (b) assume no aberration, $a_{\mathrm{\Psi }}=0$, while the lower two rows (c) and (d) assume $a_{\mathrm{\Psi }}=0.2$, in $\mathrm{\Psi }=a_{\mathrm{\Psi }}(Z_3^{+1}+Z_5^{-1})$. The phase noise in each case is significant enough to almost obscure the underlying Fresnel ring patterns, though not completely. Reconstructed images $I_{\mathrm{\Psi }}$ without compensation suffers, and, especially when a small amount of aberration is present, the noise almost completely obliterates the signal. On the other hand, the compensated images ${\widehat{I}}_{\mathrm{\Psi }}$ degrade only mildly even when both phase noise and aberration are present. This behavior is remarkable but also reasonable, in view of the fact that ${\widehat{I}}_{\mathrm{\Psi }}$ is the correlation of $H_{\mathrm{\Psi }}$ with $G_{\mathrm{\Psi }}$, meaning that any noise or aberration in $H_{\mathrm{\Psi }}$ that has low correlation with $G_{\mathrm{\Psi }}$ does not have significant contribution to the final image ${\widehat{I}}_{\mathrm{\Psi }}$.

 

Fig. 9. Effect of noisy hologram on IDHAO: (a) $a_{\mathrm{\Psi }}=0$, $a_{\mathrm{\Phi }}=5$; (b) $a_{\mathrm{\Psi }}=0$, $a_{\mathrm{\Phi }}=10$; (c) $a_{\mathrm{\Psi }}=0.2$, $a_{\mathrm{\Phi }}=5$; (d) $a_{\mathrm{\Psi }}=0.2$, $a_{\mathrm{\Phi }}=10$. In each row the phase of $G_{\mathrm{\Psi }}$ and $H_{\mathrm{\Psi }}$ and the amplitude of $I_{\mathrm{\Psi }}$ and $I^{\prime }$ are shown.

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Note that this simulates noise in the detection system, not in the optical field itself. Noise in the optical field is accounted for as aberration if it is present at a plane conjugate to the interferometer mirrors. Noise, or aberration, away from the conjugate plane is not compensated by the present scheme of IDHAO, unless one sets up a multiconjugate configuration, which may be possible borrowing some of the techniques available in conventional AO systems [3].

4. Experiment

We have carried out a series of experiments to confirm the results of theoretical and simulation studies, and to observe general behavior of the IDHAO process. First, to demonstrate the basic IDH with focusing property, the optical configuration of Fig. 2 is set up using a group of three red LED’s at $z_1=650\mathrm{mm}$ and another red LED closer at $z_1^{\prime }=440\mathrm{mm}$. All three lenses are of focal length $f_o=f_a=f_c=100\mathrm{mm}$, while the curved mirror has $f_B=600\mathrm{mm}$. Various distances are set at $z_2=220\mathrm{mm}$, $z_3=200\mathrm{mm}$, $z_4=200\mathrm{mm}$, and $z_5=140\mathrm{mm}$, $\pm 5\mathrm{mm}$. The Pixelfly CCD camera has a $6.3\mathrm{mm}\times 4.8\mathrm{mm}$ sensor area with $640\times 480$ pixels, which projects to an approximately $60\mathrm{mm}\times 45\mathrm{mm}$ field of view at the object plane at $z_1$ from ${\mathrm{L}}_o$. The plane mirror ${\mathrm{M}}_A$ is piezo-mounted and driven with the sawtooth output of a digital function generator and four phase-shift interferograms are acquired at 20 fps. Hologram acquisition, reconstruction, and aberration compensation are carried out by LabVIEW-based programs. The entire cycle takes a few seconds, including file write/read and without any attempt to optimize the speed.

A. IDH

One of the raw holograms $h_{\phi }$ of the four-LED object field is shown in Fig. 10(a) and the corresponding complex hologram $H_0$ in Fig. 10(b). One may note the slightly higher fringe frequency of the lower left LED, compared to the other three LEDs, due to its closer distance to the objective lens. Numerical propagation of the hologram focuses the LEDs at two separate distances, (c) at $z_i=22000\mathrm{mm}$ and (d) at $z_i=14870\mathrm{mm}$, whose values are consistent with the effective image distances projected to the object space. The image distances can be calculated by using Eq. (8) and geometrical optics consideration, and the approximate $10:1$ lateral magnification corresponds to $100:1$ longitudinal magnification.

 

Fig. 10. IDH without aberration: (a) one of the raw holograms $h_{\phi }$; (b) complex hologram $H_0$; (c) reconstructed image at $z_i=22,000\mathrm{mm}$;, corresponding to $z_1=650\mathrm{mm}$; (d) reconstructed image at $z_i^{\prime }=14,870\mathrm{mm}$; corresponding to $z_1^{\prime }=440\mathrm{mm}$.

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B. IDHAO

Phase aberrators are fabricated by coating an optically clear plastic piece with acrylic paint to produce irregular surface profiles of varying degrees. An aberrator is placed just behind the objective lens ${\mathrm{L}}_o$, and the resulting IDH images are shown in the upper row of Fig. 11. The concentric rings of Figs. 10(a) and 10(b) are now replaced with severely distorted patterns in Fig. 11(a) and 11(b). An attempt to numerically focus the images from $H_{\mathrm{\Psi }}$ yields Fig. 11(c) and 11(d) at the two expected image distances. Now a guide star hologram $G_{\mathrm{\Psi }}$ is acquired by aperturing the object field to leave only one of the LEDs on, in this case one belonging to the three-LED group at $z_1=650\mathrm{mm}$, as shown in Fig. 11(e). Aberration compensation using the guide star hologram, as described in theory and simulation, then results in the corrected hologram ${\widehat{H}}_{\mathrm{\Psi }}$ in Fig. 11(f) and the corrected images ${\widehat{I}}_{\mathrm{\Psi }}$ in Fig. 11(g) and 11(h). Note, as in simulation of Fig. 4, that the distorted fringes in Fig. 11(b) are replaced with more concentric ones in Fig. 11(f). Note also that the guide star at $z_1$ is effective in compensating the images at both $z_1$ and $z_1^{\prime }$. Another test, not presented, of using a guide star at $z_1^{\prime }$ was equally effective in compensating the images at the two distances. That is, a guide star can be placed at any distance relative to the object volume. This is a unique property of IDHAO, which may not be available in other existing AO techniques.

 

Fig. 11. IDHAO with a phase aberrator placed just behind the objective lens: (a) one of the raw holograms $h_{\phi }$; (b) complex hologram $H_{\mathrm{\Psi }}$, reconstructed images $I_{\mathrm{\Psi }}$; (c) at $z_i$; (d) at $z_i^{\prime }$; (e) guide star hologram $G_{\mathrm{\Psi }}$ of an LED at $z_1$; (f) corrected hologram ${\widehat{H}}_{\mathrm{\Psi }}$, corrected images ${\widehat{I}}_{\mathrm{\Psi }}$; (g) at $z_i$; and (h) at $z_i^{\prime }$.

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C. Noise

Next, the LED brightness is reduced to various levels to observe effects of noise and low signals on the IDH and IDHAO images. In Fig. 12, all four LEDs are placed at $z_1=650\mathrm{mm}$, which constitutes the object field. One of them, the leftmost one, is also used as the guide star. No deliberate aberrator is in place. In each row of Fig. 12, the first image, column (i), is the reconstructed image $I_{\mathrm{\Psi }}$ without compensation, while columns (ii)–(iv) are the compensated images ${\widehat{I}}_{\mathrm{\Psi }}$ with increasing brightness of the guide star. The rows (a)–(c) represent the object field with decreasing brightness. In all cases, the IDHAO process is seen to be capable of reducing the noise and increasing the contrast. Also, it is significant to note that the contrast enhancement increases as the guide star brightness. The effect is especially marked for the weakest signal in (c), where the uncompensated image is almost buried in the background, but the brighter guide star brings out the signal with high contrast, while maintaining good resolution.

 

Fig. 12. IDHAO versus noise: (a)–(c) varying brightness of the object field, consisting of all four LEDs; (i) IDH image $I_{\mathrm{\Psi }}$ without AO; (ii)–(iv) IDHAO images ${\widehat{I}}_{\mathrm{\Psi }}$, with varying brightness of the guide star.

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D. Extended Objects

In addition to the point-like sources of LEDs, extended objects of finite surfaces are also used to test the IDHAO process. In the top row of Fig. 13, a miniature halogen lamp illuminates a resolution target from behind through a piece of frosted glass, while in the bottom row, a knight chess piece is illuminated with the halogen lamp in front. The objects are at $z_1^{\prime }=450\mathrm{mm}$. In both cases an IF, centered at 600 nm and of linewidth 10 nm, is placed in front of the camera to improve the interference contrast, although we have not noticed significant difference in resulting images. In each case, a raw interferogram $h_{\phi }$ and the complex hologram $H_{\mathrm{\Psi }}$ are shown. Note that for these extended objects, the raw interferograms only have barely recognizable interference contrast. No deliberate phase aberrator is present. An LED placed next to the objects at $z_1^{\prime }=450\mathrm{mm}$ serves as the guide star.

 

Fig. 13. IDH of extended objects, a resolution target (top) and a knight chess piece (bottom), illuminated with a miniature halogen lamp. In each case, a raw hologram $h_{\phi }$ and the amplitude and phase of the complex hologram $H_{\mathrm{\Psi }}$ are shown.

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In Fig. 14, an experiment similar to that of Fig. 12 is carried out by varying the brightness of the halogen lamp and the guide star LED. Again, it is seen that the IDHAO process is quite effective in extracting and strengthening the image signals buried in background noise.

 

Fig. 14. IDHAO versus noise: (a)–(c) varying brightness of the resolution target, (i) IDH image $I_{\mathrm{\Psi }}$ without AO; (ii)–(iv) IDHAO images ${\widehat{I}}_{\mathrm{\Psi }}$, with varying brightness of the guide star.

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Finally, Fig. 15 shows that the IDHAO process is effective for a 3D object volume, as we found for LED objects in the experiment of Fig. 11. In this experiment, an LED at $z_1=650\mathrm{mm}$ is turned on in addition to the knight piece at $z_1^{\prime }=450\mathrm{mm}$—the guide star LED is still at the same distance as the knight. In the corrected images of the bottom row, as well as the uncorrected images of the top row, the objects separately come into clear focus at the appropriate distances.

 

Fig. 15. 3D focusing of IDHAO of extended object: (a) uncorrected image $I_{\mathrm{\Psi }}$ reconstructed at $z_i^{\prime }=17,500\mathrm{mm}$, corresponding to $z_1^{\prime }=450\mathrm{mm}$; (b) reconstructed image at $z_i=22,000\mathrm{mm}$, corresponding to $z_1=650\mathrm{mm}$; (c) and (d) corresponding images ${\widehat{I}}_{\mathrm{\Psi }}$ with compensation.

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5. Discussion

The IDHAO is described to be an effective and robust means of compensating optical aberrations in incoherent optical imaging systems. Theoretical framework based on propagation of spherical waves from source points followed by integration of intensity over the object points correctly predicts the possibility of aberration compensation. General behavior of the theoretical model of IDHAO is illustrated by simulation studies. They show robust aberration compensation under various ranges of parameters, including aberration type and strength, as well as in presence of significant noise. Finally, experimental studies corroborate these predictions extremely well. The IDHAO process is seen to be effective for both point-like sources and extended objects. In particular, a brighter guide star is seen to be very effective in bringing out weak signals buried under noise through the fact that the IDHAO signal is proportional to the auto-correlation of the complex guide star hologram.

The main results of these theoretical, simulation, and experimental studies can be summarized as follows.

Other than observation and verifications at several representative values, the study of parametric behavior of IDHAO in this work has been mostly qualitative. Our next task is to establish a quantitative comparison of IDHAO with conventional AO systems, considering such factors as resolution, dynamic range, sensitivity, isoplanatic angle, and operating speed. The resolution is mostly determined by the telescopic input image, as long as the interferometer configuration is optimized. The dynamic range and lateral resolution of aberration compensation in IDHAO are, in principle, unlimited. Strong and weak aberrations are handled in exactly the same manner, assuming that the aberration is not so severe that it cannot be described as a phase profile at a plane conjugate to the interferometer plane—but this is the same assumption taken in a conventional system. The lateral resolution of aberration compensation is all of the CCD pixel resolution, which is a very different situation than in a conventional system, where it is limited by the number of actuators in the deformable mirror. The isoplanatic angle for multilayered turbulence is under the same constraint as with conventional AO systems. As with conventional systems, a multiconjugate configuration is feasible, but at significant cost of complexity. The sensitivity or the achievable Strehl ratio using IDHAO, compared to conventional AO, would be first governed by the efficiency of the IDH holographic image formation, which entails formation of the fringes and computation of the holographic images from the acquired fringes. It is also affected, as we have seen above, by the noise suppression aspect of the guide star correlation. The operating speed may depend, over a large range, on the mode of operation. The IDHAO as described here can be considered an open loop feed forward approach, where the two inputs—the full-field and guide-star holograms—are combined to provide the output. There is no feedback. In that sense, the operating speed is instantaneous, as long as the two holograms are acquired simultaneously, for example by splitting the telescope input beam into two parts with separate interferometers for the full-field and guide-star holograms.

We may also comment on the comparison of the interferometer used in our IDHAO with the FINCH interferometer. In FINCH, a single SLM is used in effect to create a superposition of two lenses. This is done by either using one‐half of the pixel count to create each of the two components [28] or by using polarization dependence of the SLM to create the two curvatures for the two polarization components [33]. The IDHAO utilizes the full aperture in a straightforward manner. The presence of the SLM array structure in the beam path in FINCH can cause diffraction sidelobes, an issue that needs to be dealt with. On the other hand, the FINCH configuration is, in principle, mechanically more compact and stable compared to IDHAO.

6. Conclusions

Theoretical, simulation, and experimental studies show that the IDHAO system has many of the desirable properties of conventional AO systems at substantially reduced complexity, and very likely cost, of the optomechanical system. Wavefront sensing and correction by IDHAO have almost the full resolution of the CCD camera. The dynamic range of deformation measurement is essentially unlimited. One can clearly envision potential applications in astronomical and other imaging systems. Work is in progress to study details of key parameters for astronomical imaging, including resolution, sensitivity, and speed, as well as for ophthalmic and other imaging applications.