3 Circular prolate spheroidal functions as apodization functions solution to the coronagraphic problem

Considering for example Eq. (6), the coronagraphic effect appears as a subtraction of two wave amplitudes in the relay pupil plane. Though the coronagraphic effect is expected in the final focal plane D, it is here more convenient to consider the relay pupil plane. Ideally, we seek for an exact wave subtraction, within the entire exit pupil, or at least, the best subtraction possible.

The integral in Eq. (7) or Eq. (8) must be proportional to $\Phi(r)$, to permit an exact subtraction of the two wavefronts inside the aperture in the relay pupil plane: the apodization function $\Phi(r)$ must be the eigenfunction of this kernel.

The solutions $\Phi(r)$ are given by the circular prolate functions, which are the eigenfunctions of this eigenvalue problem. We give a brief presentation of them in Appendix B.

For the apodization function $\Phi(r)$ (normalized circular prolate function):

$$\Phi(r)=\frac{\Theta(r)}{\Theta(0)},$$ (9)

associated with the eigenvalue $\Lambda$ (positive number between 0 an 1), the integral in Eq. (8) can be expressed using Eq. (B.3):

$${\left( 2~\pi \right) }^2 \int _{0}^{\frac{D}{2}}\xi ~\Phi (\xi )~{K_0}(\xi ,r)~{\rm d}\xi=\Lambda~\Phi(r).$$ (10)

Equations (7) and (8) then find a much simpler expression:

$${{\Psi}^-_{\rm C}}(r) = \Phi (r)~ \Pi \left(\frac{r}{D}\right) - \epsilon ~\Lambda ~ \Phi (r) ,$$ (11)

$${{\Psi }^+_{\rm C}}(r) = \Pi \left(\frac{r}{D}\right)~\Phi (r)~\left( 1 - \epsilon ~\Lambda \right).$$ (12)

Depending on the eigenvalues and on $\epsilon$, we discuss the solutions to the problem in the following subsections for Roddier & Roddier and Lyot. A lot of valuable information can be derived analytically (throughput, impulse response, residual star intensity etc.) from the prolate properties.

3.1 Circular prolate solution for Roddier & Roddier coronagraphy ($\epsilon={\mathsfsl 2)}$

Figure 1: Roddier & Roddier coronagraphy with prolate entrance pupil apodization. A) Eigenvalue $\Lambda$ as a function of the mask size a with the solution for R&R $\Lambda =1/2$. B) Corresponding circular prolate apodization, in amplitude. C) Illustration of the PSF for a planet, assuming the coronagraphic effect negligible, for a point source, sufficiently well off-axis. The grey tint rectangle shows the size of the mask, at the same scale.

For Roddier & Roddier's coronagraphy, the $\pi$ phase mask imposes $\epsilon=2$. The amplitude inside the exit pupil (Eq. (12)) becomes:

$${{\Psi }^+_{\rm C}}(r) = \Pi \left(\frac{r}{D}\right)~\Phi (r)~\left( 1 - 2 ~\Lambda \right).$$ (13)

An exact solution exists (total extinction) for the eigenvalue $\Lambda =1/2$ and a unique mask size and apodization (eigenfunction) corresponds to this eigenvalue. From the inverse function of the eigenvalue curve, we obtain the mask diameter a=1.06/D, as illustrated in Fig. 1.

The transmission of the corresponding apodization is represented in amplitude in Fig. 1. This solution is in excellent agreement with the solution obtained numerically by Guyon & Roddier (2000). A comparison between the analytical and numerical solution is made in Soummer et al. (2002a). It is also necessary to know the energy throughput of the entrance pupil. Using the orthogonality relation for circular prolate functions (Eq. (B.5)), the throughput (normalized to the throughput of the raw circular aperture) finds a simple expression:

$$T = { \frac{4}{\pi ~D^2} }~\int _{0}^{\frac{D}{2}} {\left( ... ...\pi ~D^2} }~{ \frac{2~\pi ~\Lambda }{ {\Theta }^2 (0)} }\cdot$$ (14)

At first sight, this expression seem to depend on the telescope diameter. However, considering Eq. (C.1) and knowing that $\varphi(0)=0$ (Slepian 1964), we can show that $\Theta(0)\propto \frac{1}{D}$ and the relative intensity transmission is independent of the telescope diameter.

For the solution of Roddier & Roddier, we find $T_{RR}~=~72.6\%$

In the focal plane B (the phase mask plane), the amplitude of the wave (before the phase mask) can be expressed analytically, using the invariance of the circular prolate functions to a finite HT (Eq. (B.1)):

$${{\Psi }^-_{\rm B}}(r) = \frac{1}{\imath \lambda f}~ {{ \in... ...~\frac{r}{\lambda~ f}~\xi \right)~2~\pi ~\xi ~ {\rm d}\xi }}.$$ (15)

We obtain the following wave amplitude before the phase mask:

$${{\Psi }^-_{\rm B}}(r) = \frac{1}{\imath \lambda f}~{ \frac... ...{\frac{1}{2}}~\Phi \left(\frac{r~D}{a~\lambda ~f}\right)\cdot$$ (16)

Reminding the notation of Eq. (4), we can stress that the wave amplitude is simply homothetic from a plane to another (invariance of the circular prolate to a finite HT):

$${\Psi }^-_{\rm B}(r) = { \frac{D}{d} }~{\Lambda }^{\frac{1}{2}}~\Phi \left(\frac{r~D}{d}\right)\cdot$$ (17)

Moreover, the amplitude inside the phase mask $(0~\leq~r~\leq~d/2)$ in plane B is exactly homothetic to the amplitude inside the entrance pupil in plane A (this point was discussed in details for the rectangular aperture, Aime et al. 2002).

This expression can also be used as a good approximation for the planet response. Indeed, for an off-axis point source, situated at a distance well outside the coronagraphic mask, the extinction can be neglected and the planet focal intensity is similar to the apodized aperture without coronagraph:

$$I_{\rm p}(r) ={\left( \frac{D}{d} \right) }^2~\Lambda ~{\Phi ^2\left((r)\frac{~D}{d}\right)}~\cdot$$ (18)

This intensity for the planet (apodization alone), normalized to the impulse response maximum of the unapodized aperture is represented in logarithmic scale in Fig. 1:

$${\tilde{I}_{\rm p}}(r) =\left({\frac{4}{\pi D^2}}\right)^2~ I_{\rm p}(r).$$ (19)

The grey tint rectangle represent the size of the phase mask.

Note that the loss of resolution due to the apodization of the entrance pupil is very low ($\sim$5%): the first zero of the impulse response is at $1.28\lambda/D$, instead of $1.22\lambda/D$ for the classical unapodized Airy's pattern (the apodization required for Roddier & Roddier is very light), and the equivalent widths are almost the same to 1% precision.

Figure 2:  Approximated solutions for Lyot coronagraphy with prolate apodization. The solutions are given for three arbitrary eigenvalues close to 1: $\Lambda=0.9~,\;\Lambda=0.99~,\;\Lambda=0.999$. For each solution, we give the radial apodization in amplitude (left), and the residual star intensity in the final focal plane (in log scale). The grey tint rectangles correspond to the mask size at the same scale. Note that the mask is always smaller than the central core.

For this total extinction solution (Roddier & Roddier), the wave amplitude in plane A is the truncated central part of the prolate solution $\Phi(r)$ (Eq. (2)). Considering Eq. (11), it appears that in plane C, the wave before the Lyot stop is the complementary external part of the same prolate $\Phi(r)$ ( $\epsilon~ \Lambda=1$ for R&R). The conservation of energy imposes to the star energy (collected in plane A), to be totally rejected outside the pupil aperture in plane C (and eliminated by the Lyot stop).

This physical requirement can be checked mathematically, using the circular prolates properties, following Frieden (1971). Using Eqs. (B.7) and (B.8), with $\Lambda =1/2$, we can write:

$$\int _{0}^{\frac{D}{2}}r~\vert{\Phi (r)}\vert^2~ {\rm d}r =... ...c{1}{2} \int _{0}^{\infty }r~\vert{\Phi (r)}\vert^2~ {\rm d}r,$$ (20)

and straightforwardly:

$$\int _{0}^{\frac{D}{2}}r~\vert{\Phi (r)}\vert^2~{\rm d}r = \int _{\frac{D}{2}}^{\infty }r~\vert{\Phi (r)}\vert^2~{\rm d}r,$$ (21)

which is the expression of energy conservation for circular prolate functions: the energy in the central part and complementary external part of the prolate wave are equal (if $\Lambda =1/2$, i.e. R&R coronagraphy).

3.2 Residual amplitude for Lyot coronagraphy ( $\epsilon={\mathsfsl 1}$)

Figure 3: Lyot coronagraphy with increasing mask size (and increasing eigenvalues and apodization strength). Top: residual energy for the star, normalized to the entrance pupil transmission. Center: illustration of the loss of transmission for the approximate solutions due to the increasing strength of the apodization. Bottom: illustration of the loss of resolution due to the apodization. The full line represents the position of the first zero of the PSF and can be compared to the half width of the mask (Dashed line). The three values at a=1.96, 2.90 and 3.74 correspond to the examples shown in Fig. 2.

For Lyot coronagraphy, the opaque mask imposes $\epsilon=1$. The amplitude in the relay pupil plane (Eq. (12)) becomes:

$${{\Psi }^+_{\rm C}}(r) = \Pi \left(\frac{r}{D}\right)~\Phi (r)~\left( 1 - ~\Lambda \right),$$ (22)

and there is no relevant exact solution: the trivial solution $\Lambda=1$ would correspond to an infinite size opaque mask (Fig. B.2).

However, approximate solutions can be obtained for eigenvalues $\Lambda$ close to 1 and finite mask size: taking advantage of the rapid saturation of the eigenvalue curve (Fig. B.2), we can choose a mask size a corresponding to an eigenvalue close to 1, so that the residual amplitude (Eq. (22)) remains close to zero. A corresponding prolate apodization exists but is no longer an exact solution and a residual amplitude exists.

The surprising result is that the residual amplitude in the pupil plane (Eq. (22)) is itself proportional to the initial prolate function, to the factor $(1-\Lambda)$. The overall effect of Lyot coronagraphy with prolate apodization is here simply an attenuation coefficient for the on-axis point source.

The residual star intensity is then itself apodized. Moreover, in plane D, the energy is maximally concentrated within a surface equivalent to the coronagraphic mask (prolate fundamental property Eqs. (B.7) and (B.8) (Slepian 1964; Frieden 1971). This is an important difference with classical Lyot coronagraphy, for which the residual pupil intensity in plane C is known to be maximum at the edge of the pupil (opposite effect to apodization). This is why prolate apodizations are the optimal solution for Lyot coronagraphy, in terms of maximum residual star energy concentration.

Following Eq. (22), the residual intensity for the on-axis star, is then simply reduced by the factor $(1-\Lambda)^2$ compared to the intensity with the apodizer alone (without the mask) Eq. (17):

$$\vert{{{\Psi }_{\rm D}}(r)}\vert^2 = (1-\Lambda)^2 ~{\left(... ...\right) }^2~\Lambda ~{\Phi ^2\left(\frac{r~D}{d}\right)}~\cdot$$ (23)

Examples of the apodized pupil and normalized star residual intensity are given in Fig. 2, for the arbitrary eigenvalues $\Lambda=0.9$, $\Lambda=0.99$, $\Lambda=0.999$: the mask sizes are represented as grey tint rectangles and the intensity curves are normalized to the maximum of the intensity for the unapodized pupil:

$$\vert{\tilde{\Psi} _{\rm D}(r)}\vert^2={\left( \frac{4}{\pi ~D^2} \right) }^2 \times \vert{{{\Psi }_{\rm D}}(r)}\vert^2.$$ (24)

An important point to notice is that with entrance pupil prolate apodization, the Lyot mask size is always smaller than the core of the diffraction pattern. This can be seen in Fig. 2, and also in Fig. 3 (bottom). This is a very important difference with classical Lyot coronagraphy for which the mask extends over several Airy rings.

The integrated residual energy is simply reduced by the factor $(1-\Lambda)^2$ compared to the throughput Tof the apodizer alone (Eq. (14)):

$$E ={\frac{4}{\pi ~D^2} }~{ \frac{2~\pi ~\Lambda (1-\Lambda)^2}{ {\Theta }^2 (0)}}= (1-\Lambda)^2 \times T.$$ (25)

The closest to 1 the eigenvalue lie, the higher the coronagraphic star energy rejection (Fig. 3 top). However, a tradeoff exists: the transmission of the entrance pupil (Fig. 3 center) and the angular resolution (Fig. 3 bottom) decrease with $\Lambda$. The three examples of Fig. 2 are indicated in these figures with the dotted lines.

The integrated star residual energy is not the most pertinent criterion to evaluate the performance, since the high contrast imaging problem comes from the central source diffracted light. A better criterion is to consider the fractional energy in the residual star diffraction wings, relatively to the instrument throughput:

$${e_{\rm wings}^{(1)}} = \frac{\int _{d/2}^{\infty }{\vert{{{... ... }{\vert{{{\Psi }_{\rm A}}(r)}\vert^2}~2~\pi ~r~{\rm d}r}\cdot$$ (26)

Combining with Eq. (25), we obtain:

$${e_{\rm wings}^{(1)}} = (1-\Lambda)^2~\frac{\int _{d/2}^{\in... ... }{\vert{{{\Psi }_{\rm D}}(r)}\vert^2}~2~\pi ~r~{\rm d}r}\cdot$$ (27)

Using the properties of encircled energy for the prolate functions (Eqs. (B.7) and (B.8)), we obtain simply:

$${e_{\rm wings}^{(1)}} =(1-\Lambda)^3.$$ (28)

This criterion can be used to compare the difference of the performance between the classical Lyot coronagraph and the prolate apodized Lyot coronagraph. For the classical unapodized technique, this fractional energy is computed by a numerical integration:

 

$${e_{\rm wings}^{(2)}} = \frac{4}{\pi \zeta^2 D^2} \left(\int _{0}... ... _{0}^{\frac{d}{2}}2~\pi ~r~{{{\Psi }^{(\rm L)}_{\rm D}}(r)}^2~{\rm d}r\right),$$ (29)

where $\zeta$ is a coefficient for the Lyot Stop diameter reduction. ${{{\Psi }^{(\rm L)}_{\rm C}}(r)}$ and ${{{\Psi }^{(\rm L)}_{\rm D}}(r)}$ are the wave amplitudes for the unapodized Lyot coronagraph. We remind that for classical unapodized Lyot coronagraph, the residual intensity in the exit pupil is concentrated at the edges of the pupil and a reduction of the diameter (Lyot Stop) permits a better rejection. A comparison of ${e_{\rm wings}^{(1)}}$ (Eq. (28)) and ${e_{\rm wings}^{(2)}}$ (Eq. (29)) is made in Fig. 4 for same mask sizes. The comparison is made for the Lyot stop diameters of $100\%$, $90\%$ and $80\%$. For example, with a mask size of $3\lambda/D$, the efficiency of the prolate technique is roughly 10⁴ better than for the classical Lyot technique.

The parameters and performances for these different techniques are summarized in Table 1, including typical values for the residual diffracted light at different positions ( $3\lambda/D$, $5\lambda/D$ and $7\lambda/D$).

Figure 4: Comparison of the fractional wings energy (in log scale) for classical unapodized Lyot and prolate apodized Lyot, as a function of the mask size. Full line (top curve): unapodized Lyot without Lyot stop reduction. The two similar curves underneath correspond to a Lyot Stop reduction of $90\%$ and $80\%$ respectively. The dashed line (bottom curve) corresponds to the prolate apodized coronagraph.

 

 

Table 1: Comparison of the performances for the technique with prolate apodization and the classical unapodized Lyot technique with Lyot Stop (LS) reduction, expressed in percents. In this table, we give the values of the parameters introduced in the text.

Technique	eigenvalue	Throughput	Mask	Residual	Wings	level @	level @	level @
	$\Lambda$	T	a	starlight	$e_{\rm wings}$	$3\lambda/D$	$5\lambda/D$	$7\lambda/D$
R&R Prolate	1/2	72.6%	1.06/D	0	0	0	0	0
Lyot Prolate	0.9	41.6%	1.96/D	10-2	10-3	$3\times10^{-6}$	10-6	10-7
Lyot Prolate	0.99	25.7%	2.90/D	10-4	10-6	$3\times10^{-9}$	$3\times10^{-10}$	10-10
Lyot Prolate	0.999	19.0%	3.74/D	10-6	10-9	$3\times10^{-12}$	$5\times10^{-13}$	10-13
Lyot 100% LS	-	100%	4/D	$4.8\times10^{-2}$	$3.8\times10^{-2}$	$1.6\times10^{-4}$	$3.2\times10^{-4}$	$5\times10^{-5}$
Lyot 90% LS	-	81%	4/D	$1.7\times10^{-2}$	$1.6\times10^{-2}$	$1.6\times10^{-5}$	$6.3\times10^{-5}$	$4.0\times10^{-6}$
Lyot 80% LS	-	64%	4/D	$9.0\times10^{-3}$	$6.9\times10^{-3}$	$1.6\times10^{-5}$	$5.0\times10^{-5}$	$2.5\times10^{-6}$