1. Introduction

The detection of exoplanets plays an important role in space exploration. In general, the light directly from a star is much brighter than the light reflected from its orbiting planet. At visible spectrum, the starlight could be 10¹⁰ times brighter than the portion reflected from an Earth-like planet. For the infrared region this ratio varies within the range of about 10⁴−10⁷. In order to obtain high-contrast images of such planets, the nearby starlight must be effectively suppressed. During the past decades, a number of stellar coronagraph concepts have been suggested. And many high-contrast images have been obtained through numerical simulations, experimental demonstrations [1–6] and on-sky application [7].

Coronagraphs include so many categories, such as the interferometric coronagraph relying on interferometric combination of the discrete beams divided from the entrance pupil [8–10]; the pupil apodization coronagraph consisting of the amplitude apodization; the pupil plane phase apodization and the phase induced amplitude apodization coronagraph (PIAA) and so on [11]. Some hybrid coronagraphs (almost all of the new coronagraphs are hybrid coronagraphs) also have very good performance, such as the Phase-Induced Amplitude Apodization complex mask coronagraph (PIAACMC) which uses beam remapping for lossless apodization [12], and the ring-apodized vortex coronagraph (RAVC) which combines a vortex phase mask in the image plane of a high-contrast instrument with a single pupil-based amplitude ring apodizer [13].

Since the phase mask coronagraph has such advantages as small inner working angle (IWA), high throughput and direct imaging, it has been widely studied in recent years. Impressive designs, such as the four-quadrant phase mask coronagraph (FQPM), the vortex phase mask coronagraph (VPM) and the eight-octant phase mask coronagraph (EOPM) had already been widely studied and developed [14–24], and are even operating on sky on the best telescopes in the world. The annular groove phase mask coronagraph (AGPM), as a particular realization of the vector vortex coronagraph (VVC), is currently operated in the mid-IR [25–27]. These coronagraphs have a very good performance in obtaining high-contrast images. The Jet Propulsion Laboratory has recently demonstrated reaching 10⁻⁹ raw contrast level in the visible spectrum on the High Contrast Imaging Testbed (HCIT) [28]. Last year, the sinusoidal phase mask (SPM) was proposed by three of the current authors [29].

In this paper, we shall propose an achromatic six-region phase mask coronagraph. According to our results based on simulation under ideal conditions, a single piece of the mask has a better performance of achromatism. In Section 2, we give an analytical derivation of the six regions of the achromatic phase mask and discuss why we choose six regions for the mask. In Section 3, we design a specific six-level phase mask (SLPM), as an example of the proposed six-region phase mask, to numerically analyse its performance of the elimination of starlight, inner working angle and achromatism. And in Section 4, we make a conclusion and discussion.

2. A six-region phase mask for an achromatic coronagraph

Figure 1 shows the set-up of the coronagraph, the phase mask is put at the focal plane $\left(x\prime ,y\prime \right)$, and the system reimages the entrance pupil at the Lyot stop (LS). In order to make the full use of the caliber of L1, the aperture stop (AS) is pressed close to L1 (AS diameter equals to the diameter of L1). This operation causes an extra phase factor $exp(ikr{\text{'}}^2/2f)$ comparing to an ordinary 4F system at the focal plane $\left(x\prime ,y\prime \right)$. The phase factor can be compensated at the LS plane by setting the distance between L2 and LS to be 2f [18]. More information can be found in Appendix A and B of [29]. In our system, the aperture stop function $circ\left(r/R_{AS}\right)$ is defined as:

 

Fig. 1 The set-up of the coronagraph system, which is composed of three imaging lenses (L1, L2, L3), an aperture stop (AS), a Lyot stop (LS), a phase mask, and a detecting receiver like a CCD camera, with all lenses having the same focal length $f$.

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As the starlight from far away can be considered as plane waves, traveling along the optical axis in our system, one can define the pupil incident light as $U\left(x,y\right)=A_0$. The complex amplitude $U\left(x\prime \prime ,y\prime \prime \right)$ in cylindrical coordinates on the LS can be derived [29, 30] as:

The total complex amplitude distribution $U\left(x\prime \prime ,y\prime \prime \right)$ at the LS plane can be considered as the total contribution of an infinite number of components, see in Eq. (3). ${\left|C_n\right|}^2$ as weight factors regulate the percentage of the starlight intensity of different order. When n is a nonzero even number, the n-th order Hankel transform $H_n\left\{J_1\left(ar\prime \right)/\left(ar\prime \right)\right\}$ of Eq. (3) can be proved to be (See Appendix):

In order to finally eliminate the starlight field $U\left(x",y"\right)$inside the LS, here we need $C_0\left({\lambda }_0\right)=0$. Since we need to preserve the planet light not impacted on the mask as far as possible, we put a zero-phase region in each period of the $G\left(\theta \right)$, say $2B\le \theta <\pi$ and $2B\text{+}\pi \le \theta <2\pi$, where B is an arbitrary number and $0<2B<\pi$. According to Eq. (9), $C_0$ of this region of integration becomes a real number when $G\left(\theta \right)=0$. In order to also make $image(C_0)=0$ in other regions of integration to further simplify Eq. (9), we set $G\left(\theta \right)=f\left(\theta \right)$ for $0\le \theta <B$ and $G(\theta )=-f\left(\theta -B\right)$ for $B\le \theta <2B$ in each π-period, $f\left(\theta \right)$ is an arbitrary function with θ ranging from 0 to B. Thus, two reverse phase regions and one zero-phase region in each π-period can successfully remove the imaginary part from $C_0$. Therefore, with three regions in each period, a six-region phase mask is created, and the six-region phase function $G\left(\theta \right)$can be expressed as:

 

Fig. 2 The phase function in the angular direction. The function $G\left(\theta \right)$ is a double periodic function in the range of $0<\theta \le 2\pi$, and $f\left(\theta \right)$ is an arbitrary function with θ ranging from 0 to$B$.

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As $C_0\left({\lambda }_0\right)=0$, there is no energy within the LS for the designed ${\lambda }_0$ in theory. However, if $\lambda \ne {\lambda }_0$, there will be a small amount of energy of the starlight remaining inside the LS. For getting an achromatic image when $\lambda$ is near the central wavelength ${\lambda }_0$, we further need $\left[\partial C_0\left(\lambda \right)/\partial \lambda \right]{\pi |}_{{\lambda }_0}=0$. Then one can get:

Thus, any $f(\theta )$ satisfies Eq. (13) can be used to form a phase function $G\left(\theta \right)$, then one can get a achromatic six-region phase mask in theory.

3. Numerical simulations

We now design a six-level phase mask (SLPM) as an instance of the six-region phase mask we analyzed in section 2. As shown in Fig. 3, the mask has six level regions in the angular direction. Here, $B$ is given by $\pi /4$, and $f\left(\theta \right)=\pi$. It can be easily proved that $B$ and $f\left(\theta \right)$ satisfy Eq. (13). One can get the transmission function of SLPM:

 

Fig. 3 (a) Surface profile of the SLPM composed of six level regions. Different color represents different region. (b) The corresponding phase distribution in the angular direction for the designed wavelength ${\lambda }_0$. Note that the function $G\left(\theta \right)$ is a double periodic function in the range of $0<\theta \le 2\pi$.

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Substituting the values of $B$, $f\left(\theta \right)$, $\lambda$ and ${\lambda }_0$ into Eq. (11), one can numerically calculate the coefficients $C_0\left(\lambda \right)$. $C_0\left(\lambda \right)$ can be analytically expressed as:

Figure 4 shows the intensity distribution of the monochromatic (${\lambda }_0$) incident starlight in front of the LS, this numerical result is calculated by using the 4096 × 4096 Fourier transform algorithm according to $\left|FFT\right\{SLPM\times \left[J_1\left(r\prime \right)/r\prime \right]\}{|}^2$. As we have mentioned above, when an aperture stop presses close to L1, a phase factor will be attached to the complex amplitude distribution $U\left(x\prime ,y\prime \right)$, but it can be compensated at the LS plane. Thus, the total complex amplitude distribution at the LS plane is equivalent to $FFT\left\{FFT\left[U\left(x,y\right)\right]\times SLPM\right\}$, and $U\left(x,y\right)$ is a circ function in our case, and the Fourier transform of the circ function is $J_1\left(r\prime \right)/r\prime$.

 

Fig. 4 The acquired Lyot-stop image of ${\left|U\left(x\prime \prime ,y\prime \prime \right)\right|}^2$. The simulation is under ideal condition. The color represents relative starlight intensity. The radius of the “dark” circular region in the middle is 1.

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Theoretically, for on-axis monochromatic light, there is no energy remaining inside the circular region, as one can see from Fig. 4. This means the SLPM diffracts the on-axis monochromatic light away from the optical axis. If the radius R_LS is chosen to be smaller than R_AS, then the starlight is completely blocked after travelling through the LS. Comparing to our analytical solution, the starlight intensity distributed inside the LS is 0, since $C_{2q+1}\left(\lambda \right)=0$ and $C_0\left({\lambda }_0\right)=0$. The starlight outside the LS is represented by function $f_n\left(r\prime \prime \right)$, where $n=2q\left(q\ne 0\right)$.

As the numerical simulations are implemented under ideal conditions, we haven’t taken in to consideration the wavefront aberrations and phase errors. But in practical applications, the finite size of the phase mask and the lens L2 will lead to truncation error to the spatial frequency distribution [18]. Besides, fabrication limits, residuals from the adaptive optics (ground-based telescope) [31, 32] and all the other optics in the instruments may also introduce phase errors. These errors can bring nonzero noise to the field inside the LS.

In order to detect light from the orbiting planet, coronagraphic throughput of the planet near the star should be large enough. Inner working angle (IWA), which usually is the minimal angular distance at which the throughput of planet is half of the maximal throughput (we choose the peak throughput but the throughput in this paper), plays an important role in the design of a coronagraph. The smaller IWA is, the closer exoplanets near the star one can detect. Because the SLPM has six angular regions, the throughput of planet light depends on which region of the mask it is imaged on. The simulating result is shown in Fig. 5, the throughput of planet light will be at its maximum when the planet light is projected on the $\pi /4$, $3\pi /4$, $5\pi /4$ and $7\pi /4$ region of the SLPM, the same as the result of single FQPM. But the abrupt transitions of angular phase as shown in Fig. 3 will cause uncontrollable errors, it is necessary for us to choose the zero-phase region ($3\pi /4$ and $7\pi /4$) of the SLPM. Since the SLPM has such loss of discovery space, it’s necessary to rotate the coronagraph for the best view of detecting the planet in practical applications.

 

Fig. 5 Numerical result of the throughput of the planet light in the range of 2π of different angular separation (λ/d). The y-axis (throughput) is the ratio of the planet light within the LS and the total amount of planet light, while the x-axis is the angular coordinate of the SLPM surface (rad).

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As we can see in Fig. 5, when $\theta =\pi /2$, $\theta =\pi$, $\theta =3\pi /2$, and $\theta =2\pi$, the phase transitions will strongly attenuated the planet light throughput. Thus, one may want to build a six-region mask with smaller phase-shift. For example, one could hope to use $f(\theta )=\pi /4$ or $f(\theta )=3\pi /4$ instead of $\pi$. Unfortunately, if $f(\theta )$ is smaller than $\pi$, the second one of Eq. (13) cannot be satisfied. Accordingly, it is impossible to use $f(\theta )=\pi /4$ or $f(\theta )=3\pi /4$ instead of $\pi$.

Figure 6 shows the peak throughputs (assuming that the telescope orientation is optimal) of the SLPM (FQPM), SPM and the single VPM2 (the topological charge of VPM is 2). One can see that, as a single phase mask, the peak throughput of the SLPM are quite satisfactory, and also with a small IWA less than one $\lambda /d$.

 

Fig. 6 Comparison between the peak throughput of SLPM (FQPM) (green curve), SPM (red curve), and VPM2 (blue curve), for $R_{LS}=R_{AS}$. It is pointed out that d is the diameter of aperture stop, where $d=2R_{AS}$.

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Achromatic coronagraphs are essential when we aim at detecting and characterizing exoplanets. Therefore, it is necessary to design a wideband coronagraph mask. By applying the birefringent principle, which uses two kinds of phase plate materials, one can increase the bandwidth [33]. The hybrid coronagraphs and the second generation of vector vortex coronagraphs both have very good performance of achromatism [34–36].

The bandwidth of the SLPM comes from its inherent characteristic ($\left[\partial C_0\left(\lambda \right)/\partial \lambda \right]|{}_{{\lambda }_0}=0$), which means that just one kind of material can meet this requirement. It has been proved that the throughput of SLPM for on-axis starlight is zero is caused by the zero values of $C_0({\lambda }_0)$ and $C_{2q+1}(\lambda )$. But when the wavelength changes in the vicinity of ${\lambda }_0$, $C_0\left(\lambda \right)$ is no longer equal to zero. The order of magnitude of ${\left|C_0\left(\lambda \right)\right|}^2$ can determine the order of magnitude of ${\left|U\left(x\prime \prime ,y\prime \prime \right)\right|}^2$ within the LS in the simulation. The $C_0\left(\lambda \right)$ of SLPM is given by Eq. (15) which has been stated in section 2. The ${\left|C_0\left(\lambda \right)\right|}^2$ of SLPM, SPM (b = 2.8055, h = 3.8317), and the single FQPM can be respectively written as:

Figure 7 shows the value ${\left|C_0\left(\lambda \right)\right|}^2$ of the SLPM with the wavelength ranging from 450 nm to 650 nm compared with the SPM and the single FQPM, where${\lambda }_0=550$ nm. The SLPM has a value ${\left|C_0\left(\lambda \right)\right|}^2$ nearly the same as that of SPM, and also it has a better performance than the single FQPM (monochromatic FQPM), as shown in Fig. 7. At the wavelength between 500 nm and 600 nm, the value ${\left|C_0\left(\lambda \right)\right|}^2$ is less than 10⁻³. Thus, one can say that the total rejection is almost 100% within this range, and the change of the total rejection is very tiny, when the wavelength varies. It is worth mentioning that the index of refraction of the material (SiO₂) used in making the phase mask, merely changes by 2.21 × 10⁻³, with a corresponding phase variation no more than 1.5 × 10⁻² rad. This value is so small that we ignored this factor in the achromatic simulation.

 

Fig. 7 Comparison between the value ${\left|C_0(\lambda )\right|}^2$ of the SPM (red curve), the SLPM (blue curve), and the single FQPM (black curve) for designed wavelength ${\lambda }_0=550nm$.

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Figure 8 shows the different value ${\left|C_0\left(\lambda \right)\right|}^2$ of the SLPM with $\lambda -{\lambda }_0$ranges from −100 nm to 100 nm. The blue curve (${\lambda }_0=450$ nm), the red curve (${\lambda }_0=550$nm)), and the black curve (${\lambda }_0=650$ nm)) represent different central wavelengths correspondingly. There is a tending that by choosing a longer central wavelength ${\lambda }_0$, the value of the ${\left|C_0\left(\lambda \right)\right|}^2$tends to decrease, and then a better performance of achromatism can be obtained.

 

Fig. 8 The value ${\left|C_0(\lambda )\right|}^2$ of the SLPM with different central wavelength. ${\lambda }_0=450nm$ (blue curve), ${\lambda }_0=550nm$ (red curve) and ${\lambda }_0=650nm$ (black curve).

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4. Conclusion and discussion

We have proposed an achromatic six-region coronagraph that could work in wideband. The mask phase should satisfy the requirements of both double periodic and Eq. (13). We also have proposed the SLPM, a special example of the six-region coronagraph. Through theoretical analysis and numerical simulations, we find that the SLPM has an IWA less than one $\lambda /d$, which enables a close detection of exoplanets. However, the most important point is that the SLPM, as a single phase mask, has a good performance of achromatism. This property is due to its inherent characteristic, which means one can manufacture the SLPM by using only one kind of material.

All of the simulations in this paper are implemented under ideal conditions. The capabilities of the coronagraph to cope with obscured pupil, low-order aberrations, numerical noise and the errors from optical devices are not concerned. The segmented coronagraph will yield attenuation of off-axis signal (companion, disks) along the transitions (similarly to the FQPM and 8-octant phase mask). All the above problems must be considered when testing the concept in the lab, and installing it onto a real telescope to do science observations. Accordingly, these problems will be studied in our future work.