Appendix B: Circular prolate functions

Circular prolate functions where invented independently by Slepian (1964) and Heurtley (1964) and have found very interesting applications in Optics, for example for the study of confocal laser modes or wave aberrations. Theses applications of prolate functions in Optics are detailed in the excellent review paper by Frieden (1971). The short presentation of the mathematical properties of circular functions is drawn from Frieden's paper. However, notations have been adapted to fit the present application.

Circular prolate functions are the circular analogy to the linear prolate function, known since Niven (1880) and re-discovered by Slepian in 1959 (Slepian & Pollak 1961). Their properties are strictly analogous to those of the linear functions. In the circular case, Hankel Transforms (HT) replace the linear Fourier Transforms (FT). Circular prolate function are defined for any order N of HT, but in the present application we will only need to consider the HT of zero order, involving the Bessel J₀ function that appears in Fourier Optics for Hankel Transforms. We will omit the subscripts N and n in the notations for clarity (the prolate function for n=0 is the only one with maximum at origin).

Circular prolate functions are defined by their invariance to a finite Hankel Transform (the finite HT of a function is the HT of the truncated function):

$$\int _{0}^{\frac{D}{2}}{{\Theta }}(\xi )~{{J}_0}(2~\pi ~r~\x... ...bda }^{\frac{1}{2}}~{{\Theta }}\left(\frac{r~D}{a}\right)\cdot$$ (B.1)

The eigenvalue $\Lambda$ and the prolate function $\Theta(r)$ depend on the so-called prolateness parameter $c=\frac{\pi~D~a}{2}$, that we omit in the notations, for clarity.

Reciprocally, the circular have the following property for the infinite HT:

 

$$\int _{0}^{\infty }{{\Theta }}(\xi )~{{J}_0}(2~\pi ~r~\xi )~2~\pi ~\xi ~{\rm d}\xi \frac{D}{a}~{\Lambda }^{-\frac{1}{2}}~{{\Theta }}\left(\frac{r~D}{a}\right)$$ =

$$\quad \; (0 \leq r \leq a/2),$$

$$0 \; (r>a/2).$$ = (B.2)

These two properties are the most interesting properties for coronagraphy: the HT of a truncated central part of the prolate function gives the prolate function itself (to normalisation factors). The HT of the entire prolate function gives the same central truncated part. A heuristic illustration of these two properties (Eqs. (B.1) and (B.2)) is given in Fig. B.1.

Figure B.1: Illustration of the properties of prolate function. Top: invariance to the finite Hankel Transform (Eq. (B.1)); the HT of the truncated prolate gives the prolate itself. Bottom: the HT of the entire prolate gives the truncated prolate (Eq. (B.2)).

Performing a finite HT on both side of Eq. (B.1), we have:

 

$${\left( 2~\pi \right) }^2~\int _{0}^{\frac{D}{2}}\xi ~{{\Theta }}(\xi )~{{K}_0}(\xi ,r)~{\rm d}\xi = \Lambda ~{{\Theta }}(r)$$ (B.3)

$${{K}_0}(\xi ,r) = \int _{0}^{\frac{a}{2}}\rho ~{{J}_0}(2~\pi ~\rho ~\xi )~{{J}_0}(2~\pi ~\rho ~r)~{\rm d}\rho.$$ (B.4)

The circular prolate function appear to be the eigenfunctions of this integral equation, with eigenvalue $\Lambda$ (we give the kernel expression in Eqs. (A.8) and (A.9)).

A given eigenvalue correspond to a unique mask size a and a unique prolate function, as illustrated in Fig. B.2 (usually this curve is drawn as a function of the prolateness parameter $c=\frac{\pi~D~a}{2}$, but for clarity we consider it as a function of the mask size (the telescope diameter D is a constant).

Figure B.2: Eigenvalue $\Lambda$ as a function of the mask size a, and corresponding prolate functions of zero order $\Theta (r)/\Theta (0)$ (normalized). The closest to 1 the eigenvalue, the stronger the apodization. Three examples are given for three eigenvalues. The corresponding apodization functions are given in amplitude.

The double orthogonality property of the linear prolates also exists for the circular prolates of orders mand n:

 

$$\int _{0}^{\frac{D}{2}}r~{{\Theta }_n}(r)~{{\Theta }_m}(r)~{\rm d}r = {{\Lambda }_n}~{{\delta }_{m n}}$$ (B.5)

$$\int _{0}^{\infty }r~{{\Theta }_n}(r)~{{\Theta }_m}(r)~{\rm d}r = {{\delta }_{m n}}$$ (B.6)

where ${{\delta }_{m n}}$ is the Kronecker delta.

This property is useful for the computation of the transmissions of the prolate apodizations (with m=n=0).

An other fundamental property is the fractional energy within $0\leq r \leq r_0$ which finds a simple expression:

$$e = \frac{\int _{0}^{\frac{a}{2}}r~\vert{{{\Theta }_{N,n}}(... ...{{{\Theta }_{N,n}}(r)}\vert^2~{\rm d}r} = {{\Lambda }_{N,n}}.$$ (B.7)

Moreover, the maximum fractional energy correspond to N=0 and n=0 (the case considered in the present study), and we have (Frieden 1971):

$$e_{{\rm max}}=\Lambda.$$ (B.8)

This is why we consider only for apodization the functions corresponding to N=0 and n=0. ( $\Theta_0(r)$is also the only one with its maximum at the origin).