Wide-band coronagraph with sinusoidal phase in the angular direction

Ourui Ma; Qing Cao; Fanzhen Hou

doi:10.1364/OE.20.010933

1. Introduction

The direct imaging of exoplanets will play an important role in space exploration. As we know, nearby starlight is 10¹⁰ times brighter than visible reflected light from an Earth-like planet. In the near-infrared band, the starlight is 10⁴ brighter than the light of a young giant planet. To achieve high-contrast imagers, nearby glaring starlight must be removed. Recently, various coronagraph designs have been proposed to solve this problem. As results, high-contrast imagers have been achieved through numerical simulations and experimental demonstrations [1–6]. Interferometric coronagraph relies on interferometric combination of discrete beams devided from the entrance pupil [7–9]. Phase-induced amplitude apodization coronagraph (PIAA) makes a phase aberration to produce an exit pupil which greatly reduces the intensity of the Point Spread Function (PSF) wings [10].

Another approach employs modified Lyot concept [2] with amplitude- and phase-mask. The band-limited mask coronagraph (BLM) operates on the intensity of light in the image plane [11–12]. Compared with amplitude mask coronagraph, phase mask coronagraph allows a smaller inner working angel (IWA) [13]. Four-quadrant phase mask coronagraph (FQPM), angular groove phase mask coronagraph (AGPM), vortex phase mask coronagraph (VPM), and eight-octant phase mask coronagraph (EOPM) have been designed and discussed [14–23]. Unfortunately, chromatic limitations always exist in these designs.

In this paper, we shall suggest a new design of coronagraph with a sinusoidal phase mask (SPM). The mask phase transmission is described by an alternatively sinusoidal and uniform function. Through analytical derivation and numerical tests, we prove that SPM has a extinction of the star light. We also check the performance of the inner working angle and chromatism. It provides that, SPM gives a close detection of exoplanets and can work in broad bandwidth

The paper is organized as follows. In Section 2, we shall describe a method for designing the wide-band coronagraph and show the analytical derivation of SPM. In Section 3, we shall check the performance of SPM by using numerical tests and compare it with VPM and FQPM. In Section 4, we shall conclude this paper.

2. Sinusoidal phase mask

The coronagraph is depicted in Fig. 1 . The aperture stop function circ(.) is defined as

c i r c (r / R_{A S}) = {\begin{matrix} 1 & r \leq R_{A S} \\ 0 & r > R_{A S} \end{matrix},

where r = (x² + y²)^1/2 and R_AS is the radius of the aperture stop, which is just the radius of the lens L1. Consider the initial plane wave U(x, y) = A₀ traveling along the optical axis. One can get the complex amplitude of the field U(x′, y′) in the focal plane [24–28] (see Appendix A)

U (x', y') = t (r', θ) {(i λ_{0} f)}^{- 1} \exp (i k r'^{2} / 2 f) \cdot 2 π A_{0} R_{A S}^{2} \frac{J_{1} (a r')}{a r'},

where r′ = (x′² + y′²)^1/2, θ is the angular coordinate in the transverse plane(x′, y′), k = 2π/λ₀, λ₀ is the designed wavelength of light, a = kR_AS/f, J₁(.) is the first-order Bessel function of the first kind, and t(r′, θ) is the transmission function of the mask. We consider an angle phase mask, for which t(r′, θ) is only related to the angular coordinate θ. That is to say, t(r′, θ) = t(θ). Then one can further get

U (x', y') = A_{1} t (θ) \exp (i k r'^{2} / 2 f) \frac{J_{1} (a r')}{a r'},

where A₁ = 2πA₀R_AS²/(iλ₀f). Through the Fourier transform of lens L2, one can get the complex amplitude U(x′′, y′′) in cylindrical coordinates just before LS [24–28] (see Appendix B)

U (x'', y'') = A_{2} \sum_{n = - \infty}^{\infty} {(- i)}^{n} C_{n} H_{n} {\frac{J_{1} (a r')}{a r'}} e^{i n φ},

where A₂ = −2πA₀R_AS²/(λ₀f)², ϕ is the angular coordinate in the transverse plane(x′′, y′′), and the coefficient C_n is given by

Fig. 1 Set-up of the proposed coronagraphic system; it is composed of three imaging lenses (L1, L2, L3), aperture stop (AS), Lyot stop (LS), the mask, and an detecting system like CCD camera. All lenses have the same focal lengths f.

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C_{n} = \frac{1}{2 π} \int_{0}^{2 π} t (θ) e^{- i n θ} d θ .

Accordingly, the relation

\sum_{n = - \infty}^{\infty} {| C_{n} |}^{2} = 1

also holds.

We now discuss the n-th order Hankel transform H_n(.) of Eq. (4). When n is a nonzero even number, it can be proven that [17, 29–30] (see also Appendix C)

H_{n} {\frac{J_{1} (a r')}{a r'}} = {\begin{matrix} f_{n} (r''), & r'' \geq R_{A S} \\ 0 & r'' < R_{A S} \end{matrix},

where r′′ = (x′′² + y′′²)^1/2. When the radius R_LS of the LS is smaller than R_AS, the light field f_n(r′′) is completely blocked. When n is an odd or zero, there are still values inside the LS. Therefore it is desirable that C_{2q + 1} (q is an integer) and C₀ can be designed equal to zero for the designed wavelength λ₀.

We now discuss how to get the relation C_{2q + 1}(λ) = 0. For a wavelength λ which is different from λ₀, the transmission function of mask should be written as t(θ, λ). Accordingly, the coefficient C_n(λ) is also related to the wavelength λ. Using Eq. (5), one can get

\begin{array}{l} C_{2 q + 1} (λ) = \frac{1}{2 π} [\int_{0}^{π} t (θ, λ) e^{- i (2 q + 1) θ} d θ + \int_{π}^{2 π} t (θ, λ) e^{- i (2 q + 1) θ} d θ] \\ = \frac{1}{2 π} [\int_{0}^{π} t (θ, λ) e^{- i (2 q + 1) θ} d θ + e^{- i (2 q + 1) π} \int_{0}^{π} t (θ + π, λ) e^{- i (2 q + 1) θ} d θ] . \end{array}

From Eq. (8) one can prove that the coefficient C_{2q + 1}(λ) is always equal to zero, provided that t(θ + π, λ) = t(θ, λ), where θ is from 0 to π. That is to say, the relation C_{2q + 1}(λ) = 0 holds for any double periodic function t(θ, λ) in the range of 0<θ≤2π.

We now further discuss the coefficient C₀(λ). This coefficient may be expressed as a Taylor expansion in the vicinity of λ₀:

C_{0} (λ) = C_{0} (λ_{0}) + \frac{\partial C_{0} (λ)}{\partial λ} |_{λ_{0}} (λ - λ_{0}) + \frac{1}{2} \frac{\partial^{2} C_{0} (λ)}{\partial λ^{2}} |_{λ_{0}} {(λ - λ_{0})}^{2} + \cdot \cdot \cdot .

For nearly achromatic coronagraph, the condition C₀(λ)≈0 needs to be satisfied for a wide band of wavelengths. So we have the two conditions:

{\begin{matrix} C_{0} (λ_{0}) = 0 \\ \frac{\partial C_{0} (λ)}{\partial λ} |_{λ_{0}} = 0. \end{matrix}

We now present the new design of sinusoidal phase mask (SPM). We assume a mask with achromatic optical path difference (OPD), so that the phase offset for any point of the focal plane mask goes as the inverse of the wavelength. As shown in Fig. 2 , the phase mask has alternatively sinusoidal and uniform functions in the angular direction. G(θ) is the phase distribution in the angular direction for the designed wavelength λ₀, where t(θ) = e^iG(θ). By taking the chromatic property into account, one can get the transmission function of the mask:

t (θ, λ) = {\begin{matrix} e^{i \frac{λ_{0}}{λ} h \sin (b θ)} & 0 \leq θ < \frac{2 π}{b} \\ 1 & \frac{2 π}{b} \leq θ < π, \frac{2 π}{b} + π \leq θ < 2 π \\ e^{i \frac{λ_{0}}{λ} h \sin [b (θ - π)]} & π \leq θ < \frac{2 π}{b} + π \end{matrix},

where h>0. There are two periods in the range of 2π, so b must be greater than 2.This property can be easily seen from Fig. 2. As we state above, the condition C_{2q + 1}(λ) = 0 holds because there are two periods in the range of 2π. By use of Eq. (5), Eq. (10) and the formula:

{\begin{matrix} J_{0} (x) = \frac{1}{2 π} \int_{0}^{2 π} e^{- i x \sin θ} d θ \\ J_{0}' (x) = - J {}_{1}{(x)} \end{matrix},

one can get

Fig. 2 (a) Surface profile of the SPM composed of sinusoidal region and flat region. With the thickness of mask increasing, the color changes from blue to red. (b) The corresponding phase distribution in the angular direction for the designed wavelength λ₀. Note that the function G(θ) is a double periodic function in the range of 0<θ≤2π. Also note that the unit of h is radian.

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{\begin{matrix} J_{0} (h) = 1 - \frac{b}{2} \\ J_{1} (h) = 0 \end{matrix} .

From the relation J₁(h), one can get the value of h, which can be equal to the second zero-point of the first-order Bessel function. Here we use five significant figures. That is to say h = 3.8317. After getting the value of h, one can further get the value of b, which is given by b = 2.8055. Substituting the values of h, b, λ and λ₀ into Eqs. (11) and (5), one can numerically calculate the coefficients C₀(λ) and C_2m(λ). In particular, by use of Eq. (13), the value of C₀(λ) can be analytically expressed as:

C_{0} (λ) |_{S P M} = \frac{2}{b} J_{0} (\frac{λ_{0}}{λ} h) + 1 - \frac{2}{b} .

After getting the values of the coefficients of C_2m(λ), one can further get the field U(x′′,y′′)|_SPM, which is expressed as:

U (x'', y'') |_{S P M} = A_{2} \sum_{m = - \infty}^{\infty} {(- i)}^{2 m} C_{2 m} H_{2 m} {\frac{J_{1} (a r')}{a r'}} e^{i 2 m φ},

where m is a integer. The analytical solution of the 2m-th order Hankel transform is shown in Appendix C.

3. Numerical tests

To check the performance of the sinusoidal phase mask coronagraph at the designed wavelength λ₀, we numerically calculate the Lyot-stop image which is acquired at the Lyot-stop plane using Eqs. (5) and (15). The calculation is shown in Fig. 3(a) . The infinite sum in Eq. (15) is truncated from m = −15 to m = 15. R_AS is chosen to be 1 and the factor a is chosen to be 1, where a = kR_AS/f. To make sure that the result is accurate, we calculate the finite sum of |C_2m|² with m chosen from −15 to 15, which is 0.9996. Namely,

Fig. 3 Acquired Lyot-stop image of |U(x′′,y′′)|². (a) The calculation by use of Eqs. (5) and (15). (b) The simulation by use of 2-D FFT. The color represents relative intensity. When the value of relative intensity goes from 0 to 1, the color will change from blue to red. The diameter of the “dark” circular region in the middle is 1.

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\sum_{m = - 15}^{15} {| C_{2 m} |}^{2} = 0.9996.

This value implies that the calculation is accurate to 10⁻³ contrast level. In addition, the calculated image is almost the same as the image calculated with m chosen from −10 to 10. This result also shows that the calculation is accurate.

In order to further check the above-mentioned result, we perform the 2D-FFT simulation. We use the 4096 × 4096 Fourier transform algorithm, and the result is shown in Fig. 3(b). One can see that, the results of Fig. 3(a) and Fig. 3(b) are the same. In particular, there is almost no field distribution in the circular region. So SPM diffracts almost all the on-axis light away from the optical axis. If the radius R_LS is chosen to be smaller than R_AS, then the star light is completely blocked after travelling through the Lyot stop. That is to say, the throughput of on-axis star light is zero. However, technology limits of the fabrication of phase mask will lead to phase error. The finite sizes of phase mask and of lens L2 will lead to truncation error in the spatial frequency distribution. All the above-mentioned limits will lead to the nonzero values of the field in the nearby inner area of the circle r′′ = R_AS. So R_LS should be chosen to be slightly less than R_AS in the actual situation.

With deep extinction of the star light, coronagraphic throughput of the planet near the star should be large enough, in order that we can observe it easily. Inner working angel (IWA) is the angular distance at which the throughput of planet is half of the maximal throughput. It plays an important role in the design of a coronagraph. With a small IWA, we can detect exoplanets at a close distance near the star. Because of the irregular surface profile of SPM, the throughput of planet which is imaged at different angular region of mask is different. Apparently, peak throughput of planet occurs when the light of planet is imaged at the angle bisector of the flat region of SPM.

Figure 4 shows the peak throughputs of SPM, FQPM and VPM2 (the topological charge of VPM is 2). In the simulation, we use the 1024 × 1024 Fourier transform algorithm. The transform range includes more than 70 sidelobes. One can see that, the peak throughput of SPM is worse than the other two designs, when the angular separation is less than 1.6λ/d. However, when angular separation is greater than 1.6λ/d, SPM has a better performance than VPM2 in peak throughput. Also, we can see that SPM has a 1.1λ/d IWA. This property is better than the designs of a phase-induced amplitude apodization coronagraph, of a band-limited mask coronagraph, and of a visible nulling coronagraph [13]. It is pointed out that the stellar angular diameter will affect the useful throughput of coronagraph, which we don’t further discuss here.

Fig. 4 Comparison between the peak throughput of SPM (blue curve), FQPM (green curve), and VPM2 (red curve), for R_LS = R_AS. It is pointed out that the pitch multiplicity of VPM2 is chosen equal to two [23], and d is the diameter of aperture stop, where d = 2R_AS.

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As we know, one can employ wide-band light with a lower chromatism. Accordingly, low chromatic coronagraph needs less time to image the exoplanets. Therefore, it is desired that the coronagraph mask can be designed efficient for a wide band of wavelengths. It has been proved that the throughput of SPM for on-axis light is zero because of the zero values of C₀ and C_{2q + 1} for the designed wavelength λ₀. But when wavelength changes in the vicinity of λ₀, C₀(λ) is no longer equal to zero. As we state in section 2, the C₀(λ) of a SPM is given by Eq. (14). It can be proven that |C₀(λ)|² of the FQPM and VPM2 can be written as:

{\begin{matrix} {| C_{0} (λ) |}_{F Q P M}^{2} = \cos^{2} (\frac{λ_{0}}{2 λ} π) \\ {| C_{0} (λ) |}_{V P M 2}^{2} = \frac{\sin^{2} (\frac{λ_{0}}{λ} π)}{{(\frac{λ_{0}}{λ} π)}^{2}} \end{matrix} .

Figure 5 shows the value |C₀(λ)|² of SPM with the wavelength changing from 450nm to 650nm compared with those of FQPM and VPM2, where λ₀ = 550nm. One can see that SPM has a smaller value of |C₀(λ)|² than the others for a bandwidth as large as 200nm. Especially when the wavelength lies between 500nm to 600nm, the value |C₀(λ)|² of SPM is more than 10²-10⁸ times smaller than those of the other two designs. These values explicitly show that SPM has a better performance in chromatism. Wide-band performance is the largest advantage of the suggested SPM.

Fig. 5 Comparison among the value |C₀(λ)|² of SPM (blue curve), FQPM (green curve), and VPM2 (red curve), for designed wavelength λ₀ = 550nm.

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4. Conclusions

We have suggested a method to design an achromatic coronagraphic mask. The property of two periods in the angular range of 2π and Eq. (10) are the main conditions. Our new sinusoidal phase mask coronagraph that uses a SPM with sinusoidal function in the angular direction may have a completely extinction of the star light. It is shown that SPM has an IWA as small as 1.1λ/d, representing a close detection of exoplanets. This property is very similar to other phase mask coronagraphs. The most important point is that SPM has a low chromatism compared with FQPM and VPM. With such advances, the sinusoidal phase mask coronagraph can work in a broad bandwidth. It should be also applicable to the near-infrared band in principle. We expect that it could be widely used for direct detection of the earthlike exoplanets.

The authors are indebted to the two reviewers and the editor for their comments and suggestions for improving the paper.

Appendix A

Here, we use matrix optics to get the complex amplitude U(x′, y′) [24–26]. In matrix optics, a specific transfer is denoted by a specific ABCD matrix, which has the form

[\begin{matrix} A & B \\ C & D \end{matrix}],

where A-D are the four matrix elements. In Ref [25], the location and angle of matrix are swapped. That is to say, A, B, C, D in Ref [25]. correspond to D, C, B, A in this paper, respectively. The expression of matrix in this paper is used in most papers. Transfer matrix M₁ from the transverse plane (x, y) to (x′, y′) in Fig.1 can be easily determined to be

M_{1} = [\begin{matrix} 1 & f \\ 0 & 1 \end{matrix}] [\begin{matrix} 1 & 0 \\ - f^{- 1} & 1 \end{matrix}] = [\begin{matrix} 0 & f \\ - f^{- 1} & 1 \end{matrix}] .

According to the Collins formula [25,26], one can get the complex amplitude U(x′, y′) in the focal plane (x′, y′)

\begin{array}{l} U (x', y') = t (r', θ) (^{i λ B) - 1} \exp (i k f) \iint A_{0} c i r c (r / R_{A S}) \\ \times \exp {\frac{i k}{2 B} [A (x^{2} + y^{2}) + D (x'^{2} + y'^{2}) - 2 (x x' + y y')]} d x d y . \end{array}

We now remove the constant phase factor exp(ikf). Taking the values of A-D into Eq. (A2) and using the polar coordinates

{\begin{matrix} x = r \cos φ & x' = r \cos θ \\ y = r \sin φ & y' = r \sin θ \end{matrix}

one can further get [27,28]

\begin{array}{l} U (x', y') = t (r', θ) (^{i λ f) - 1} \exp (i k r'^{2} / 2 f) \\ \times \iint A_{0} c i r c (r / R_{A S}) \exp [- \frac{i k}{f} (x x' + y y')] d x d y \\ = t (r', θ) (^{i λ f) - 1} \exp (i k r'^{2} / 2 f) \\ \times \int_{0}^{\infty} A_{0} r c i r c (r / R_{A S}) {\int_{0}^{2 π} \exp [- \frac{i k}{f} r r' \cos (φ - θ)] d φ} d r . \end{array}

Using the formula

{\begin{matrix} \exp [- i 2 π r ρ \cos (φ - θ)] = \sum_{k = - \infty}^{\infty} {(- i)}^{k} J_{k} (2 π r ρ) \exp [- i k (φ - θ)] \\ \int_{0}^{ρ} r J_{0} (r) d r = ρ J_{1} (ρ) \end{matrix},

one can finally get

\begin{array}{l} U (x', y') = t (r', θ) (^{i λ f) - 1} \exp (i k r'^{2} / 2 f) \\ \times \sum_{n = - \infty}^{\infty} \int_{0}^{\infty} A_{0} r c i r c (r / R_{A S}) J_{n} (\frac{k}{f} r r') {\int_{0}^{2 π} {(- i)}^{n} \exp [- i n (φ - θ)] d φ} d r \\ = t (r', θ) (^{i λ f) - 1} \exp (i k r'^{2} / 2 f) \\ \times \sum_{n = - \infty}^{\infty} \int_{0}^{\infty} A_{0} r c i r c (r / R_{A S}) J_{n} (\frac{k}{f} r r') {(- i)}^{n} \exp (i n θ) [\int_{0}^{2 π} \exp (- i n φ) d φ] d r \\ = t (r', θ) (^{i λ f) - 1} \exp (i k r'^{2} / 2 f) \cdot 2 π A_{0} \int_{0}^{\infty} r c i r c (r / R_{A S}) J_{0} (\frac{k}{f} r' r) d r \\ = t (r', θ) (^{i λ_{0} f) - 1} \exp (i k r'^{2} / 2 f) \cdot 2 π A_{0} R_{A S}^{2} \frac{J_{1} (\frac{k R_{A S}}{f} r')}{\frac{k R_{A S}}{f} r'} . \end{array}

Appendix B

The transfer matrix M₂ from the transverse plane (x′, y′) to the transverse plane (x′′, y′′) in Fig. 1 can be determined to be [24–26]

M_{2} = [\begin{matrix} 1 & 2 f \\ 0 & 1 \end{matrix}] [\begin{matrix} 1 & 0 \\ - f^{- 1} & 1 \end{matrix}] [\begin{matrix} 1 & f \\ 0 & 1 \end{matrix}] = [\begin{matrix} - 1 & f \\ - f^{- 1} & 0 \end{matrix}] .

According to the Collins formula [25,26], one can get the complex amplitude U(x′′, y′′)

\begin{array}{l} U (x'', y'') = (^{i λ B) - 1} \exp (i 3 k f) \iint U (x', y') \\ \times \exp {\frac{i k}{2 B} [A (x'^{2} + y'^{2}) + D (x''^{2} + y''^{2}) - 2 (x' x'' + y' y'')]} d x' d y' . \end{array}

We now remove the constant phase factor exp(i3kf). Taking the values of A-D and Eq. (3) into Eq. (B2), one can further get

\begin{array}{l} U (x'', y'') = (^{i λ f) - 1} \iint A_{1} t (θ) \frac{J_{1} (a r')}{a r'} \exp (i k r'^{2} / 2 f) \\ \times \exp {\frac{i k}{2 f} [- (x'^{2} + y'^{2}) - 2 (x' x'' + y' y'')]} d x' d y' \\ = A_{1} (^{i λ f) - 1} \iint t (θ) \frac{J_{1} (a r')}{a r'} \exp [- \frac{i k}{f} (x' x'' + y' y'')] d x' d y' . \end{array}

Note that the phase factor exp(ikr′²/2f) does not exist any longer, because it has been offset by the phase factor exp(-ikr′²/2f). Eq. (B3) is just the 2-D Fourier transform of t(θ)J(ar′)/ar′.

Using Eq. (A4), it can be further expressed by the infinite sum of Hankel transform [27,28]

\begin{array}{l} U (x'', y'') = A_{1} (^{i λ f) - 1} \int_{0}^{\infty} r' \frac{J_{1} (a r')}{a r'} {\int_{0}^{2 π} t (θ) \exp [- \frac{i k}{f} r' r'' \cos (θ - ϕ)] d θ} d r' \\ = A_{1} (^{i λ f) - 1} \sum_{n = - \infty}^{\infty} \int_{0}^{\infty} r' \frac{J_{1} (a r')}{a r'} J_{n} (\frac{k}{f} r' r'') \\ \times {\int_{0}^{2 π} {(- i)}^{n} t (θ) \exp [- i k (θ - ϕ)] d θ} d r' \\ = A_{1} (^{i λ f) - 1} \sum_{n = - \infty}^{\infty} \int_{0}^{\infty} r' \frac{J_{1} (a r')}{a r'} J_{n} (\frac{k}{f} r' r'') \\ \times (^{- i) n} \exp (i n ϕ) {\int_{0}^{2 π} t (θ) \exp (- i n θ) d θ} d r' \\ = A_{2} \sum_{n = - \infty}^{\infty} {(- i)}^{n} C_{n} H_{n} {\frac{J_{1} (a r')}{a r'}} \exp (i n ϕ), \end{array}

where H_n{.} is the n-th order Hankel transform, and C_n is given by Eq. (5).

Appendix C

To increase the readability, we here present a brief derivation for the 2m-th order Hankel transform, though they may be found elsewhere. The nonzero positive even order Hankel transform has an analytical solution [29,30]:

\begin{array}{l} H_{2 m} {\frac{J_{1} (r')}{r'}} = \int_{0}^{\infty} J_{2 m} (r'' r') J_{1} (r') d r' \\ \begin{matrix} = \frac{Γ (m + 1)}{r''^{2} Γ (2) Γ (m)} \times {}_{2}F_{1} (1 + m, 1 - m; 2; \frac{1}{r''^{2}}), & r'' \geq 1 \end{matrix} \end{array}

where ₂F₁ is the hypergeometric function, and Г(m) is the gamma function. In the region of r′′<1, the values of the above-mentioned Hankel transforms are always zero. Eq. (C1) can be further written as

H_{2 m} {\frac{J_{1} (r')}{r'}} = \frac{m}{r''^{2}} \sum_{n = 0}^{m - 1} \frac{{(1 + m)}_{n} {(1 - m)}_{n}}{2_{n} n!} \frac{1}{r''^{2 n}},

where (x)_n is the Pochhammer symbol, and it is defined by

{(x)}_{n} = x (x + 1) (x + 2) \cdot \cdot \cdot (x + n - 1) .

So we can further get

\begin{array}{l} H_{2 m} {\frac{J_{1} (r')}{r'}} = \frac{m}{r''^{2}} \sum_{n = 0}^{m - 1} \frac{\frac{Γ (m + n + 1)}{Γ (1 + m)} \cdot \frac{{(- 1)}^{n} Γ (m)}{Γ (m - n)}}{Γ (n + 2) Γ (n + 1)} \frac{1}{r''^{2 n}} \\ = \frac{m}{r''^{2}} \sum_{n = 0}^{m - 1} {(- 1)}^{n} \frac{Γ (m + n + 1) Γ (m)}{Γ (1 + m) Γ (m - n) Γ (n + 2) Γ (n + 1)} \frac{1}{r''^{2 n}} \\ = \frac{m}{r''^{2}} \sum_{n = 0}^{m - 1} {(- 1)}^{n} \frac{(m + n)! (m - 1)!}{m! n! (m - n - 1)! (n + 1)!} \frac{1}{r''^{2 n}}, \end{array}

where the property of Г(m)=(m-1)! has been used. In particular, the first three transforms are simply given by:

\begin{array}{l} H_{2} {\frac{J_{1} (r')}{r'}} = \frac{1}{r''^{2}} \\ H_{4} {\frac{J_{1} (r')}{r'}} = - \frac{2}{r''^{2}} + \frac{3}{r''^{4}} \\ H_{6} {\frac{J_{1} (r')}{r'}} = \frac{3}{r''^{2}} - \frac{12}{r''^{4}} + \frac{10}{r''^{6}} . \end{array}

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Wide-band coronagraph with sinusoidal phase in the angular direction

Abstract

1. Introduction

2. Sinusoidal phase mask

3. Numerical tests

4. Conclusions

Appendix A

Appendix B

Appendix C

References and links

Cited By

Figures (5)

Equations (33)

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