Digital coronagraph algorithm

Pedro J. Valle; Antonio Fuentes; Vidal F. Canales; Miguel A. Cagigas; Isidro Villo-perez; Manuel P. Cagigal

doi:10.1364/OSAC.1.000625

1. Introduction

Direct imaging of star faint companions is a topic of great interest. Coronagraphy stands out among several strategies investigated so far. The development of entrance pupil apodizers, coronagraphic masks and Lyot stops has led to an extremely high star light extinction [1–6]. However, the theoretical high extinction of these instruments is ruined when the light wavefront is distorted by the atmosphere random fluctuations. In such a case, high wavefront compensation of atmospheric distortion by means of extreme adaptive optics (ExAO) [7,8] or the use of space based devices [9] is required to approach the theoretical performance. In this context, we propose the Digital Coronagraph Algorithm (DCA) that processes detected non-coronagraphic images. DCA emulates the steps of the optical coronagraph (OC) but, unlike OC, DCA starts from intensity images. The algorithm manipulates such images to achieve the same contrast and inner working angle as the OC. It is worth noting that DCA, like OC, can be combined with other post-processing techniques, as the Angular Differential Imaging (ADI) [10], the Locally Optimized Combination of Images (LOCI) [11] or the Karhunen-Loève Image Projection (KLIP) [12] algorithms.

This simple technique presents several advantages like no coronagraph or extra device has to be manufactured, different coronagraphic masks and Lyot stops can be tested on each target, pointing errors are avoided, wavefront aberrations due to alignment errors are canceled out, it is not necessary to send any coronagraph or device to space and finally coronagraphic processing of archived images can be performed. On the other hand, the main drawback comes from the limited dynamic range of the scientific camera. However, observations to be processed by DCA can be easily designed.

The scheme of this paper is as follows. In section 2, DCA is introduced and is related to the OC. Section 3 explains how DCA affects the stellar companion. In the next section, numerically calculated contrast curves for both techniques are compared. Section 5 shows the result of a simple laboratory experimental checking. In section 6, the effect of noise is analyzed by numerical simulation. In the last section, the main conclusions are outlined.

2. Theoretical framework

The behavior of a standard coronagraph is well known [13]. We will consider the Lyot coronagraph set-up depicted at Fig. 1

Fig. 1 Lyot coronagraph. (a) Entrance pupil, (b) coronagraphic plane, and (c) Lyot.

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. The optical field at the entrance pupil (a) is Fourier transformed by the first lens to produce a field distribution at the coronagraphic plane (b). The field is multiplied by the coronagraphic mask to cancel out the central peak. In the next step, the masked field is Fourier transformed by a second lens to yield the field at the Lyot stop plane (plane c). A Lyot stop is used to cancel out the side peaks due to the diffraction by the mask border. The third lens produces the final image at the detection plane (plane d), with a clear reduction of the star light.

Next, we will compare the behavior of optical and digital coronagraphs at the same planes (for this aim, we use a one-dimensional scheme). Let us assume a circular clear pupil and a centered star. The field distribution at the OC coronagraphic plane, Fig. 2(b)

Fig. 2 One-dimensional summary for the OC (left) and the DCA (right). Entrance pupil plane (a) is omitted. At the coronagraphic plane, b is the APSF, b1 the coronagraphic mask, b2 the masked APSF, b0 the star PSF and bS the sign map. At the Lyot plane, c is the field, c1 the Lyot stop and c2 the filtered field. The coronagraphic image d is at final plane (unit intensity corresponds to the non-coronagraphic star image peak).

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-left, is:

A (r) = J_{1} (π r) / (π r)

where J₁ is the first order Bessel function of the first kind and r the distance to the center (axis of the optical system). In this case, it coincides with the telescope amplitude point spread function, APSF, which quasiperiodically takes positive and negative values. The period is related to λ/D, where λ is the detected wavelength and D the aperture size. DCA emulates the coronagraphic process on non-coronagraphic telescope images. It starts from the intensity image, (Fig. 2b0-right), which has the same periodicity as the field but only takes positive values. In order to emulate the process that takes place in a coronagraph, the first step is to convert this intensity to a field. This task requires the sign distribution across the detection plane for the optical system in use, defined as:

S (r) = A P S F (r) / | A P S F (r) |,

which for a circular aperture yields S(r) = J₁(πr)/∣J₁(πr)∣ and resembles a series of annuli that take values of 1 or −1, as is shown in Fig. 3

Fig. 3 Left: sign map given by Eq. (2) for a clear pupil (white regions represent value + 1 and black ones value −1). Right: super-Gaussian Lyot stop given by Eq. (6); pupil border is shown in green.

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-left. Usual pupil geometries include central obscurations and in these cases the APSF slightly changes. The product of the root square of the image intensity and the sign map, Eq. (2), reproduces the field:

A (r) \propto \sqrt{I (r)} S (r)

Note that the fields described by Eqs. (1) and (3) are identical (as depicted in Figs. 2(b) left and right), which means that the star field at the coronagraphic plane can be obtained from a non-coronagraphic image if the sign map is known. From this point, the rest of the process that takes place in an OC can be emulated by numeric Fourier transforms and digital masks. Figure 2 shows that all the functions are identical for OC (left) and DCA (right).

The preceding paragraphs discuss the technique for monochromatic wavelength, but most actual imaging systems work on a limited wavelength band. An advantage of the digital coronagraph is that, unlike the optical coronagraph, it does not introduce any additional dispersion. Of course, the detected image I(r) presents a combination of images that correspond to different wavelengths, so that the sign map shown in Fig. 3 must be an average of the sign maps given by Eq. (3) for each wavelength. Hence, we have to define the corresponding sign map for each wavelength band.

3. Companion behavior

So far, we have only considered the reference centered star, therefore the behavior of any companion must be analyzed. Let us consider independently the effect of DCA on both signals (Fig. 4

Fig. 4 Left column: field and log-intensity profiles at the same planes of Fig. 2 for the host star (black line) and the companion (blue line) in a digital coronagraph. Right column: outline of the dissimilar spatial spread of both contributions at the Lyot plane. The companion is 10⁶ times fainter than the host star and is placed at 13 λ/D.

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, black line is the star and blue line the companion). At the entrance pupil plane, an out of center companion is described by a tilted wavefront [Fig. 4(a)] while in the coronagraphic plane the companion intensity is I_C(r-r₀), where r₀ is the distance from the companion to the star, which intensity is I_S(r) (Fig. 4(b), with r₀ = 13 λ/D and a companion 10⁶ times fainter than the star). The detected image would be described by I(r) = I_S(r) + I_C(r-r₀). When we multiply the square root of the image intensity by the sign map given by Eq. (2), we obtain:

A (r) \propto \sqrt{I (r)} S (r) = \sqrt{I_{S} (r) + I_{C} (r - r_{0})} S (r) .

It must be noted that the sign map S(r) affects both the star and companion intensities. Since coronagraphy is only applied for detecting faint companions, i.e. I_C(r-r₀) < I_S(r), in order to understand Eq. (4), we approximate:

A (r) \approx \sqrt{I_{S} (r)} S (r) + I_{C} (r - r_{0}) S (r) / (2 \sqrt{I_{S} (r)}) .

Equation (5) shows two contributions, the first one is the host star APSF. The second one is due to the companion signal and it can greatly differ from the companion APSF since the centers of symmetry of I_C(r-r₀) and S(r) are a distance apart. Let us analyze the behavior of both contributions at the Lyot plane [Fig. 4(c)]. Here, the star contribution is the one expected in the standard coronagraph. However, the signal from the companion is very different. Figure 4(e) shows that its 2-dimensional distribution spreads out of the pupil border, while the star field is localized at the pupil border [Fig. 4(f)]. It can be understood that the sign map plays the role of a diffraction grating for the companion. Hence, for DCA a new Lyot stop that blocks the host star signal but not the companion signal, located outside the pupil border, is required [as the one shown with red line in Fig. 4(c)]. With a suitable design of Lyot stop, final host star intensity can be deeply reduced while the companion intensity is moderately affected [compare Fig. 4(d) with Fig. 4(b)].

4. Simulation analysis

To analyze the technique performance we first compare the DCA and OC contrast curves in a telescope with a central obscuration of radius 0.33 that of the pupil (as Hubble Space Telescope). The transmission function of the coronagraphic mask is M(r) = 1-G(r), where G(r) is a Gaussian function whose 1/e half-width is 5λ/D. For the Lyot stop we choose the super-Gaussian profile, given by:

S G (ρ; n) = exp (- {| \frac{ρ - ρ_{d}}{σ} |}^{n}),

where σ is the half-width, ρ is the radial coordinate and ρ_d is the radius of the super-Gaussian annulus. The n value was set to 6 because in our simulations it yields the best performance [14]. We use two different super-Gaussian Lyot stops: the first one with ρ_d inside the telescope annular pupil is used for the OC and the second one with ρ_d outside the pupil for the DCA (Fig. 3 right). In an OC, the contrast radial dependence can be defined as [15]:

c (r) = \frac{I (r)}{{| M (r) |}^{2} I_{S} (0)},

where I(r) is the intensity at the radial coordinate in the final image, I_S(0) is the unmasked peak stellar intensity and |M(r)|² is the mask intensity transmission. However, for the DCA, the companion peak intensity will depend on its position. Then, we will use the expression:

c' (r) = \frac{I (r)}{{| M (r) |}^{2} I_{C} (r)},

where I_C(r) is the intensity of a star with the same intensity as the host star but placed at a distance r from it. Figure 5

Fig. 5 Contrast curve for the OC (red line) and for the DCA (blue line) with the same coronagraphic mask. Star intensity radial dependence is also shown (black line).

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compares contrast values of the OC, Eq. (7), and the DCA, Eq. (8). It can be seen that the DCA contrast slightly improves that of the OC except for distances larger than 25 λ/D. Let us note that these contrast values could even be enhanced using other coronagraphic masks like vortex or phase quadrant.

5. Checking the laboratory images

In a second stage to check the DCA technique we have carried out a simple experiment with laboratory generated images. The set-up is sketched in Fig. 6

Fig. 6 The beam is divided by a beam splitter (BS1). In the arm that corresponds to the star, the light reaches a second beam splitter (BS2) with the help of a flat mirror (FLAT 1). In the other arm, the light goes through a phase element (PE) and a neutral filter (NF) and is reflected by FLAT2 to BS2. Both beams are collimated by the spherical mirror SPHE 1 and focused by SPHE 2 to the scientific camera

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. It consists on a pigtail laser (Thorlabs LPS-635-FC) emitting at 635 μm with a core diameter of 4.5 μm and a numerical aperture of NA = 0.12. A beam splitter (BS1) divides the beam so that one beam is used to simulate the host star and the other beam, once it has traversed a phase element (PE) to lose its coherence and a neutral filter (NF) to reduce its amplitude, is deviated by mirror FLAT2 to play the role of a faint companion. A second beam splitter (BS2) recombines light of companion (coming from FLAT2) and host star (coming from FLAT1). A collimating mirror (spherical mirror of 1000 mm focal length, SPHE1) and a telescope (spherical mirror of 500 mm focal length, SPHE2) are used to form the final image, which is acquired by a camera Photometrics CH350/L that is based on a back-thinned CCD (16-bits).

Figure 7

Fig. 7 Left: Experimental image of a host star and a companion (encircled) placed at a distance equivalent to 8 λ/D with an intensity 1000 times fainter than that of the host star (the selected scale saturates the central core). Right: the same image after DCA, where the companion stands out over the background.

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shows the experimental image corresponding to a star plus a companion placed at a distance of 8 λ/D and whose intensity is 1000 times fainter than that of the host star. The companion peak cannot be distinguished from the background (Fig. 7-left). To apply DCA, we have deduced the sign map directly from the science image. This sign map is similar to that shown in Fig. 3-left and is equivalent to that given by Eq. (2). The image that results when DCA is applied with a coronagraphic mask of radius 6 λ/D and a super-Gaussian Lyot stop is shown in Fig. 7-right. It can be noticed that the peak is clearly outlined, with an intensity higher than five times the background typical deviation. Hence, DCA efficiently cancels the star light background, which increases the companion detectability.

6. Noise analysis

To this point, we have considered non aberrated fields or noiseless images. The first limitation when dealing with actual astronomy images comes from the dynamic range of the detection camera. Objects whose intensity is outside the camera dynamic range will be lost. This limitation can be overcome in the case of future images planned to be DCA processed. The saturated region corresponding to the star core pixels can be selectively switch off or blocked using a hard-edge mask during the exposition time lowering the detectable light level. The second limitation comes from the electronic, Poisson and speckle noises involved in the detection process. Objects with intensity lower than 5 times the noise typical deviation will also be lost. When using a low electronic noise camera, according to Soummer et al. [16], the noise in the focal plane image is completely dominated by the speckle pinning contribution, which is removed in the OC images while the noise from the intensity random term remains. Now, we perform a similar analysis for the DCA. For a real pupil function A, the optical field on the pupil will be A exp(iϕ), where ϕ is the pupil wavefront error of the incoming wave. The intensity of the coronagraphic image for the OC and for the DCA will be:

I_{OC} = {| \bar{\bar{\bar{A e^{i ϕ}} M} L} |}^{2},

I_{DCA} = {| \bar{\bar{| \bar{A e^{i ϕ}} | S M} L} |}^{2},

where the upper bars stand for the Fourier transforms, M is the coronagraphic mask, L the Lyot stop and S the sign distribution map given by Eq. (2) (coordinate dependence has been omitted for shortness). Ground-based telescopes assisted by ExAO may reach Strehl values of 0.9. In that case, we can approximate exp(iϕ) ≈1 + iϕ in Eqs. (9) and (10). For the OC:

I_{OC} \approx {| \bar{A} M * \bar{L} + i \bar{A ϕ} M * \bar{L} |}^{2},

(* stands for convolution product) which leads to:

I_{OC} \approx {| \bar{A} M * \bar{L} |}^{2} + {| \bar{A ϕ} M * \bar{L} |}^{2} + 2 (\bar{A} M * \bar{L}) Re (\bar{A ϕ} M * \bar{L}),

where Re denotes real part. The first term is deterministic, the second one is the pure random term and the last one the pinned speckle contribution. In the case of the DCA, let us first note that:

{| \bar{A e^{i ϕ}} |}^{2} \approx {| \bar{A} + i \bar{A ϕ} |}^{2} = {| \bar{A} |}^{2} + {| \bar{A ϕ} |}^{2} - 2 \bar{A} Im (\bar{A ϕ}),

where Im denotes imaginary part. The square root of Eq. (13) can be written as its Taylor expansion and consider only those terms up to the first degree. The product of this expansion by the sign map yields:

| \bar{A e^{i ϕ}} | S \approx | \bar{A} | S - I m (\bar{A ϕ}) .

If Eq. (14) is used in Eq. (10), the coronagraphic image for the DCA remains:

I_{DCA} \approx {| \bar{A} M * \bar{L} - Im (\bar{A ϕ}) M * \bar{L} |}^{2},

which finally leads to

I_{DCA} \approx {| \bar{A} M * \bar{L} |}^{2} + {| Im (\bar{A ϕ}) M * \bar{L} |}^{2} - 2 (\bar{A} M * \bar{L}) (Im (\bar{A ϕ}) M * \bar{L}) .

The comparison of Eqs. (12) and (16) shows that the reduction of the deterministic term $\bar{A} M * \bar{L}$ cancels pinned speckled noise in DCA the same way as in OC (last term in both equations). A further analysis requires two remarks. First, light outside the pupil area is blocked by the Lyot stop used in the OC, whilst, as shown in Fig. 4(c), the Lyot stop used in the DCA only allows light that passes outside the pupil area. Secondly, due to the sign function S, a part of the companion light will be superimposed to that from the pupil border [Fig. 4(e)]. Therefore, when the residual light of the star is blocked, a part of the companion light is also blocked. The amount of the companion light blocked will depend on the companion position.

A more detailed comparison between OC and DCA with speckle noise can be carried out by computer simulation. A series of 200 wavefronts that provide Strehl values around 0.9 were used. The entrance pupil, coronagraphic mask and Lyot stop were the same as those used in Fig. 5. From the set of resulting images, we evaluated the corresponding contrast curves shown in Fig. 8

Fig. 8 Contrast curves for the OC (red line) and the DCA (blue line) from a series of 200 wavefronts with average Strehl 0.9. Black line shows the star intensity radial dependence.

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, according to Eqs. (7) and (8). As expected, these curves are higher than those in ideal conditions (Fig. 5) for both techniques. For angular companion distances between 7 and 20 λ/D, both OC and DCA reach similar values, although the DCA curve is slightly better and strongly modulated. For larger distances, the OC faintly improves the DCA. So, we can conclude that star extinction in DCA is of the same order than that in OC when wavefront aberrations are considered. This supports that the noise typical deviation is similar for both OC and DCA coronagraphic images.

7. Conclusions

In conclusion, star light cancelation produced by OC can be reproduced by DCA with high quality telescope images. We only need the APSF sign map, which can be deduced from the telescope parameters or from experimental measurements, and the use of an outer Lyot stop. The advantages of this simple technique are that no coronagraph has to be manufactured, different coronagraphic masks and Lyot stops can be checked, pointing errors are avoided, alignment errors are canceled out, space telescope does not require a coronagraph, it eliminates noise as efficiently as the OC and, finally, coronagraphic processing of archived images can be performed. The main drawback comes from the limited dynamic range of the scientific camera, but it can be overcome by blocking the stellar peak during exposition time. Experimental laboratory results show that unobserved faint companion in direct images can be detected after DCA processing of the detected image. We have analyzed noise effects in a simulation with extremely compensated wavefronts and we have demonstrated that the reduction of theoretical contrast caused by speckle noise is similar in OC and DCA. Finally, it is worth noting that DCA can be combined with other post-processing techniques like ADI, LOCI or KLIP, and be applied to images from the archive generated by the different space telescope programs or by ground-based telescopes with extreme adaptive optics.

Funding

Ministerio de Economía y Competitividad (AYA2016-78773-C2-1-P).

References

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Digital coronagraph algorithm

Abstract

1. Introduction

2. Theoretical framework

3. Companion behavior

4. Simulation analysis

5. Checking the laboratory images

6. Noise analysis

7. Conclusions

Funding

References

Cited By

Figures (8)

Equations (16)

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