Wide-band six-region phase mask coronagraph - OSA Publishing

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Abstract: An achromatic six-region phase mask coronagraph, used for the detection of ...... curve), and the single FQPM (black curve) for designed wavelength 0.
Wide-band six-region phase mask coronagraph Fanzhen Hou, Qing Cao,* Minning Zhu, and Ourui Ma Department of Physics, Shanghai University, 99 Shangda Road, Baoshan District, Shanghai 200444, China * [email protected]

Abstract: An achromatic six-region phase mask coronagraph, used for the detection of exoplanets, is proposed. The mask has six regions in angular direction and could work in wideband. Furthermore, a six-level phase mask, as an example of the six-region phase mask, is theoretically investigated. According to numerical simulations, this specific mask has a deep elimination of starlight, good performance of achromatism and small inner working angle. As a single phase mask, the ratio of the remaining starlight of the six-level phase mask to the total incident starlight is less than 0.001 when the wavelength is between 500 nm and 600 nm. © 2014 Optical Society of America OCIS codes: (110.6770) Telescopes; (100.2980) Image enhancement; (070.6110) Spatial filtering; (350.1260) Astronomical optics; (110.2970) Image detection systems.

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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

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Received 11 Dec 2013; revised 8 Jan 2014; accepted 13 Jan 2014; published 21 Jan 2014 13 January 2014 | Vol. 22, No. 1 | DOI:10.1364/OE.22.001884 | OPTICS EXPRESS 1885

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 highcontrast 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

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 .

Figure 1 shows the set-up of the coronagraph, the phase mask is put at the focal plane ( x′, y′) , 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 '2 / 2 f ) comparing to an ordinary 4F system at the focal plane ( x′, y ′) . 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 ( r / RAS ) is defined as:

1 r ≤ RAS circ( r RAS ) =  , 0 r > RAS where r = ( x 2 + y 2 )

1/2

(1)

and RAS is the radius of the aperture stop. Assuming that the phase of

the mask only varies with the angular coordinate θ , thus the transmission function of the phase mask can be defined as t (θ ) , and θ is the angular coordinate on the focal plane

( x′, y′) . By taking the bandwidth of the incident starlight into account (assuming the mask with achromatic optical path difference), one can get:

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Received 11 Dec 2013; revised 8 Jan 2014; accepted 13 Jan 2014; published 21 Jan 2014 13 January 2014 | Vol. 22, No. 1 | DOI:10.1364/OE.22.001884 | OPTICS EXPRESS 1886

t (θ , λ ) = exp i (λ0 / λ )G (θ ) ,

(2)

where G (θ ) , 0 ≤ θ < 2π , is the phase of t (θ , λ ) , and λ0 is the central wavelength of the wideband light. 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 ( x, y ) = A0 . The complex amplitude U ( x′′, y ′′) in cylindrical coordinates on the LS can be derived [29, 30] as: ∞  J ( ar ')  inφ U ( x '', y '') = A  ( −i ) n Cn (λ ) H n  1 e ,  ar '  n =−∞

where r ′ = ( x′2 + y ′2 )

1/2

(3)

, A = −2π A0 RAS 2 / (λ0 f ) 2 , φ is the angular coordinate on the

transverse plane ( x′′, y ′′) , a = kRAS / f , k = 2π / λ0 , and Cn is given by Cn ( λ ) =

1 2π





0

t (θ , λ )e − inθ dθ =

1 2π





0

ei(λ0 / λ )G (θ ) e − inθ dθ .

(4)

Accordingly, ∞

C

n =−∞

2 n

= 1.

(5)

The total complex amplitude distribution U ( x′′, y ′′) at the LS plane can be considered as 2

the total contribution of an infinite number of components, see in Eq. (3). Cn 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 {J 1 ( ar′) / ( ar′)} of Eq. (3) can be proved to be (See Appendix):  J ( ar ')   f n ( r '') r '' ≥ RAS Hn  1 , = r '' < RAS  ar '   0

(6)

where r ′′ = ( x′′2 + y ′′2 ) , and f n ( r′′) can be found in Eq. (17) in the Appendix, and more 1/2

information can be found in [29]. Thus, this even portion of starlight is completely blocked when the radius RLS of the LS is smaller than RAS. Therefore, when r ′′ < RAS , there remains only the odd and zero portion of starlight, and U ( x′′, y′′) can be simplified as: ∞

U ( x ", y " ) = A  ( −i ) q =−∞

2 q +1

 J ( ar ' )  i ( 2 q +1)φ  J ( ar ' )  C2 q +1 H 2 q +1  1 + AC0 H 0  1 e  , (7)  ar '   ar ' 

where q is an arbitrary integer. When the transmission function t (θ , λ ) is π-periodic in the range of 2π (namely, double periodic in the angular direction), C2 q +1 in the first component of Eq. (7) becomes zero [29], thus we further simplify U ( x′′, y ′′) as:  J ( ar ' )  U ( x ", y " ) = AC0 H 0  1 .  ar ' 

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(8)

Received 11 Dec 2013; revised 8 Jan 2014; accepted 13 Jan 2014; published 21 Jan 2014 13 January 2014 | Vol. 22, No. 1 | DOI:10.1364/OE.22.001884 | OPTICS EXPRESS 1887

Where C0 is defined by Eq. (4), and also can be expressed as follows: 1 2π 1 2π (9) cos  (λ0 / λ )G (θ )dθ + i sin (λ0 / λ )G (θ )dθ .  0 2π 2π 0 In order to finally eliminate the starlight field U ( x ", y " ) inside the LS, here we need C0 (λ ) =

C0 ( λ0 ) = 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 (θ ) , say 2B ≤ θ < π and 2 B +π ≤ θ < 2π , where B is an arbitrary number and 0 < 2B < π . According to Eq. (9), C0

of this region of integration becomes a real number when G (θ ) = 0 . In order to also make image(C0 ) = 0 in other regions of integration to further simplify Eq. (9), we set

G (θ ) = f (θ ) for 0 ≤ θ < B and G (θ ) = − f (θ − B ) for B ≤ θ < 2 B in each π-period,

f (θ ) 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 C0 . Therefore, with three regions in each period, a six-region phase mask is created, and the

six-region phase function G (θ ) can be expressed as:

0≤θ < B f (θ )   − f (θ − B ) B ≤ θ < 2B   0 2B ≤ θ < π . G (θ ) =  ( θ − π ) π ≤θ