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Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 2 — Jan. 27, 2014
  • pp: 1884–1895
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Wide-band six-region phase mask coronagraph

Fanzhen Hou, Qing Cao, Minning Zhu, and Ourui Ma  »View Author Affiliations


Optics Express, Vol. 22, Issue 2, pp. 1884-1895 (2014)
http://dx.doi.org/10.1364/OE.22.001884


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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

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

1. D. Mawet, E. Serabyn, K. Liewer, Ch. Hanot, S. McEldowney, D. Shemo, and N. O’Brien, “Optical Vectorial Vortex Coronagraphs using Liquid Crystal Polymers: theory, manufacturing and laboratory demonstration,” Opt. Express 17(3), 1902–1918 (2009). [CrossRef] [PubMed]

6

6. M. P. Cagigal, V. F. Canales, P. J. Valle, and J. E. Oti, “Coronagraphic mask design using Hermite functions,” Opt. Express 17(22), 20515–20520 (2009). [CrossRef] [PubMed]

] and on-sky application [7

7. E. Serabyna and D. Mawet, “Phase mask coronagraphy at JPL and Palomar,” EPJ Web of Conferences 16, 03004 (2011). [CrossRef]

].

Coronagraphs include so many categories, such as the interferometric coronagraph relying on interferometric combination of the discrete beams divided from the entrance pupil [8

8. P. Baudoz, Y. Rabbia, and J. Gay, “Achromatic interfero coronagraph,” Astron. Astrophys. Suppl. Ser. 141(2), 319–329 (2000). [CrossRef]

10

10. O. Guyon and M. Shao, “The pupil-swapping coronagraph,” Publ. Astron. Soc. Pac. 118(844), 860–865 (2006). [CrossRef]

]; 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

11. O. Guyon, “Phase-induced amplitude apodization of telescope pupils for extrasolar terrestrial planet imaging,” Astron. Astrophys. 404(1), 379–387 (2003). [CrossRef]

]. 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

12. O. Guyon, P. M. Hinz, E. Cady, R. Belikov, and F. Martinache, “High performance Lyot and PIAA coronagraphy for arbitrarily shaped telescope apertures,” 2014 sprint arXiv:1305.6686, accepted to ApJ.

], 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

13. Mawet, L. Pueyo, A. Carlotti, B. Mennesson, E. Serabyn, and J. K. Wallace, “Ring-Apodized Vortex coronagraphs for obscured telescopes. I. Transmissive ring apodizers,” 2013 sprint arXiv:1309.3328, accepted to ApJS.

].

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

14. O. Guyon, E. A. Pluzhnik, M. J. Kuchner, B. Collins, and S. T. Ridgway, “Theoretical limits on extrasolar terrestrial planet detection with coronagraphs,” Astrophys. J. 167(1Suppl.), 81–99 (2006). [CrossRef]

24

24. C. C. Shih and P. V. Estates, “Phase mask with continuous azimuthal variation for a coronagraph imaging system,” U.S. Patent 7,773,307 (2010).

], 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

25. D. Mawet, P. Riaud, O. Absil, and J. Surdej, “Annular groove phase mask coronagraph,” Astrophys. J. 633(2), 1191–1200 (2005). [CrossRef]

27

27. N. Murakami, S. Hamaguchi, M. Sakamoto, R. Fukumoto, A. Ise, K. Oka, N. Baba, and M. Tamura, “Design and laboratory demonstration of an achromatic vector vortex coronagraph,” Opt. Express 21(6), 7400–7410 (2013). [CrossRef] [PubMed]

]. 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

28. D. Mawet, L. Pueyo, A. Carlotti, B. Mennesson, E. Serabyn, J. Wallace, and P. Baudoz, “The multistage and ring-apodized vortex coronagraph: two simple, small-angle coronagraphic solutions for heavily obscured apertures,” Proc. SPIE 8864, 886411 (2013). [CrossRef]

]. Last year, the sinusoidal phase mask (SPM) was proposed by three of the current authors [29

29. O. Ma, Q. Cao, and F. Hou, “Wide-band coronagraph with sinusoidal phase in the angular direction,” Opt. Express 20(10), 10933–10943 (2012). [CrossRef] [PubMed]

].

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
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.
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'2/2f) 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

18. G. Foo, D. M. Palacios, and G. A. Swartzlander Jr., “Optical vortex coronagraph,” Opt. Lett. 30(24), 3308–3310 (2005). [CrossRef] [PubMed]

]. More information can be found in Appendix A and B of [29

29. O. Ma, Q. Cao, and F. Hou, “Wide-band coronagraph with sinusoidal phase in the angular direction,” Opt. Express 20(10), 10933–10943 (2012). [CrossRef] [PubMed]

]. In our system, the aperture stop function circ(r/RAS) is defined as:
circ(r/RAS)={1rRAS0r>RAS,
(1)
where r=(x2+y2)1/2 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:
t(θ,λ)=exp[i(λ0/λ)G(θ)],
(2)
where G(θ), 0θ<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

29. O. Ma, Q. Cao, and F. Hou, “Wide-band coronagraph with sinusoidal phase in the angular direction,” Opt. Express 20(10), 10933–10943 (2012). [CrossRef] [PubMed]

, 30

30. Q. Cao, M. Gruber, and J. Jahns, “Generalized confocal imaging systems for free-space optical interconnections,” Appl. Opt. 43(16), 3306–3309 (2004). [CrossRef] [PubMed]

] as:
U(x'',y'')=An=(i)nCn(λ)Hn{J1(ar')ar'}einϕ,
(3)
where r=(x2+y2)1/2, A=2πA0RAS2/(λ0f)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)=12π02πt(θ,λ)einθdθ=12π02πei(λ0/λ)G(θ)einθdθ.
(4)
Accordingly,

n=|Cn|2=1.
(5)

The total complex amplitude distribution U(x,y) at the LS plane can be considered as the total contribution of an infinite number of components, see in Eq. (3). |Cn|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 Hn{J1(ar)/(ar)} of Eq. (3) can be proved to be (See Appendix):
Hn{J1(ar')ar'}={fn(r'')r''RAS0r''<RAS,
(6)
where r=(x2+y2)1/2, and fn(r) can be found in Eq. (17) in the Appendix, and more information can be found in [29

29. O. Ma, Q. Cao, and F. Hou, “Wide-band coronagraph with sinusoidal phase in the angular direction,” Opt. Express 20(10), 10933–10943 (2012). [CrossRef] [PubMed]

]. 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")=Aq=(i)2q+1C2q+1H2q+1{J1(ar')ar'}ei(2q+1)ϕ+AC0H0{J1(ar')ar'},
(7)
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), C2q+1 in the first component of Eq. (7) becomes zero [29

29. O. Ma, Q. Cao, and F. Hou, “Wide-band coronagraph with sinusoidal phase in the angular direction,” Opt. Express 20(10), 10933–10943 (2012). [CrossRef] [PubMed]

], thus we further simplify U(x,y)as:
U(x",y")=AC0H0{J1(ar')ar'}.
(8)
Where C0 is defined by Eq. (4), and also can be expressed as follows:

C0(λ)=12π02πcos[(λ0/λ)G(θ)]dθ+i12π02πsin[(λ0/λ)G(θ)]dθ.
(9)

In order to finally eliminate the starlight field U(x",y")inside the LS, here we need 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 2Bθ<π and 2B+πθ<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 0θ<B and G(θ)=f(θB) for Bθ<2B 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:
G(θ)={f(θ)0θ<Bf(θB)Bθ<2B02Bθ<πf(θπ)πθ<π+Bf(θπB)π+Bθ<π+2B0π+2Bθ<2π.
(10)
Figure 2
Fig. 2 The phase function in the angular direction. The function G(θ) is a double periodic function in the range of 0<θ2π, and f(θ) is an arbitrary function with θ ranging from 0 toB.
shows the graph of function G(θ).By using Eq. (9) and Eq. (10), one can get:
C0(λ)=1π[0Beiλ0λf(θ)dθ+B2Beiλ0λf(θB)dθ+2Bπdθ]=1π[0Beiλ0λf(θ)dθ+0Beiλ0λf(θ)dθ+π2B]=1π0B2cos[λ0λf(θ)]dθ+12Bπ.
(11)
Then one can further get:

C0(λ)λ=1π0Bf(θ)2λ0λ2sin[λ0λf(θ)]dθ.
(12)

As C0(λ0)=0, there is no energy within the LS for the designed λ0 in theory. However, if λλ0, there will be a small amount of energy of the starlight remaining inside the LS. For getting an achromatic image when λ is near the central wavelength λ0, we further need [C0(λ)/λ]π|λ0=0. Then one can get:

{0Bcos[f(θ)]dθ=Bπ20Bf(θ)sin[f(θ)]dθ=0.
(13)

Thus, any f(θ) satisfies Eq. (13) can be used to form a phase function G(θ), 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
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 λ0. Note that the function G(θ) is a double periodic function in the range of 0<θ2π.
, the mask has six level regions in the angular direction. Here, B is given by π/4, and f(θ)=π. It can be easily proved that B and f(θ) satisfy Eq. (13). One can get the transmission function of SLPM:

t(θ,λ)={eiλ0λπ0θ<π4,πθ<5π41π2θ<π,3π2θ<2πeiλ0λππ4θ<π2,5π4θ<3π2.
(14)

Substituting the values of B, f(θ), λ and λ0 into Eq. (11), one can numerically calculate the coefficients C0(λ). C0(λ) can be analytically expressed as:

C0(λ)|SLPM=cos2(λ02λπ).
(15)

Figure 4
Fig. 4 The acquired Lyot-stop image of |U(x,y)|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.
shows the intensity distribution of the monochromatic (λ0) incident starlight in front of the LS, this numerical result is calculated by using the 4096 × 4096 Fourier transform algorithm according to |FFT{SLPM×[J1(r)/r]}|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(x,y), but it can be compensated at the LS plane. Thus, the total complex amplitude distribution at the LS plane is equivalent to FFT{FFT[U(x,y)]×SLPM}, and U(x,y) is a circ function in our case, and the Fourier transform of the circ function is J1(r)/r.

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 RLS is chosen to be smaller than RAS, 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 C2q+1(λ)=0 and C0(λ0)=0. The starlight outside the LS is represented by function fn(r), where n=2q(q0).

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

18. G. Foo, D. M. Palacios, and G. A. Swartzlander Jr., “Optical vortex coronagraph,” Opt. Lett. 30(24), 3308–3310 (2005). [CrossRef] [PubMed]

]. Besides, fabrication limits, residuals from the adaptive optics (ground-based telescope) [31

31. J. R. P. Angel, “Ground-based imaging of extrasolar planets using adaptive optics,” Nature 368(6468), 203–207 (1994). [CrossRef]

, 32

32. C. Marois, J. Veran, and C. Correia, “A Fresnel propagation analysis of NFIRAOS/IRIS high-contrast exoplanet imaging capabilities,” Proc. SPIE 8447, 844726 (2012). [CrossRef]

] 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
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).
, the throughput of planet light will be at its maximum when the planet light is projected on the π/4, 3π/4, 5π/4 and 7π/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 (3π/4 and 7π/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.

As we can see in Fig. 5, when θ=π/2, θ=π, θ=3π/2, and θ=2π, 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(θ)=π/4 or f(θ)=3π/4 instead of π. Unfortunately, if f(θ) is smaller than π, the second one of Eq. (13) cannot be satisfied. Accordingly, it is impossible to use f(θ)=π/4 or f(θ)=3π/4 instead of π.

Figure 6
Fig. 6 Comparison between the peak throughput of SLPM (FQPM) (green curve), SPM (red curve), and VPM2 (blue curve), for RLS=RAS. It is pointed out that d is the diameter of aperture stop, where d=2RAS.
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 λ/d.

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

33. D. Mawet, P. Riaud, J. Baudrand, P. Baudoz, A. Boccaletti, O. Dupuis, and D. Rouan, “The four-quadrant phase-mask coronagraph: white lightlaboratory results with an achromatic device,” Astron. Astrophys. 448(2), 801–808 (2006). [CrossRef]

]. The hybrid coronagraphs and the second generation of vector vortex coronagraphs both have very good performance of achromatism [34

34. R. Galicher, P. Baudoz, and J. Baudrand, “Multi-stage four-quadrant phase mask: achromatic coronagraph for space-based and ground-based telescopes,” Astron. Astrophys. 530, A43 (2011). [CrossRef]

36

36. D. Mawet, E. Serabyn, D. Moody, B. Kern, A. Niessner, A. Kuhnert, D. Shemo, R. Chipman, S. McClain, and J. Trauger, “Recent results of the second generation of vector vortex coronagraphs on the high-contrast imaging testbed at JPL,” Proc. SPIE 8151, 81511D (2011). [CrossRef]

].

The bandwidth of the SLPM comes from its inherent characteristic ([C0(λ)/λ]|λ0=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 C0(λ0) and C2q+1(λ). But when the wavelength changes in the vicinity of λ0, C0(λ) is no longer equal to zero. The order of magnitude of |C0(λ)|2 can determine the order of magnitude of |U(x,y)|2 within the LS in the simulation. The C0(λ) of SLPM is given by Eq. (15) which has been stated in section 2. The |C0(λ)|2 of SLPM, SPM (b = 2.8055, h = 3.8317), and the single FQPM can be respectively written as:

{|C0(λ)|SLPM2=[cos2(λ02λπ)]2|C0(λ)|SPM2=[2bJ0(λ0λh)+12b]2|C0(λ)|FQPM2=[cos(λ02λπ)]2.
(16)

Figure 7
Fig. 7 Comparison between the value |C0(λ)|2 of the SPM (red curve), the SLPM (blue curve), and the single FQPM (black curve) for designed wavelength λ0=550nm.
shows the value |C0(λ)|2 of the SLPM with the wavelength ranging from 450 nm to 650 nm compared with the SPM and the single FQPM, whereλ0=550 nm. The SLPM has a value |C0(λ)|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 |C0(λ)|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.

Figure 8
Fig. 8 The value |C0(λ)|2 of the SLPM with different central wavelength. λ0=450nm (blue curve), λ0=550nm (red curve) and λ0=650nm (black curve).
shows the different value |C0(λ)|2 of the SLPM with λλ0ranges from −100 nm to 100 nm. The blue curve (λ0=450 nm), the red curve (λ0=550nm)), and the black curve (λ0=650 nm)) represent different central wavelengths correspondingly. There is a tending that by choosing a longer central wavelength λ0, the value of the |C0(λ)|2tends to decrease, and then a better performance of achromatism can be obtained.

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 λ/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.

Appendix

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 when r1 [29

29. O. Ma, Q. Cao, and F. Hou, “Wide-band coronagraph with sinusoidal phase in the angular direction,” Opt. Express 20(10), 10933–10943 (2012). [CrossRef] [PubMed]

, 37

37. A. Carlotti, G. Ricort, and C. Aime, “Phase mask coronagraphy using a Mach-Zehnder interferometer,” Astron. Astrophys. 504(2), 663–671 (2009). [CrossRef]

]:
H2m{J1(r')r'}=fn(r'')=0J2m(r''r')J1(r')dr'=Γ(m+1)r''2Γ(2)Γ(m)×F21(1+m,1m;2;1r''2),r''1
(17)
where F21 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. Equation (17) can be further written as
H2m{J1(r')r'}=mr''2n=0m1(1+m)n(1m)n2nn!1r''2n,
(18)
where (x)n is the Pochhammer symbol, and it is defined by:
(x)n=x(x+1)(x+2)(x+n-1).
(19)
So one can further get:
H2m{J1(r')r'}=mr''2n=0m1Γ(m+n+1)Γ(1+m)(1)nΓ(m)Γ(mn)Γ(n+2)Γ(n+1)1r''2n=mr''2n=0m1(1)nΓ(m+n+1)Γ(m)Γ(1+m)Γ(mn)Γ(n+2)Γ(n+1)1r''2n=mr''2n=0m1(1)n(m+n)!(m1)!m!n!(mn1)!(n+1)!1r''2n,
(20)
where the property of Г(m)=(m1)! has been used. In particular, the first three transforms are simply given by:

H2{J1(r')r'}=1r''2H4{J1(r')r'}=2r''2+3r''4H6{J1(r')r'}=3r''212r''4+10r''6.
(21)

Acknowledgments

We are indebted to the two reviewers for their comments and suggestions for improving the paper. We also thank ZiChen Cao and YaJun Ge for many fruitful discussions.

References and links

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D. Mawet, E. Serabyn, K. Liewer, R. Burruss, J. Hickey, and D. Shemo, “The vector vortex coronagraph: laboratory results and first light at Palomar observatory,” Astrophys. J. 709(1), 53–57 (2010). [CrossRef]

5.

G. A. Swartzlander Jr, E. L. Ford, R. S. Abdul-Malik, L. M. Close, M. A. Peters, D. M. Palacios, and D. W. Wilson, “Astronomical demonstration of an optical vortex coronagraph,” Opt. Express 16(14), 10200–10207 (2008). [CrossRef] [PubMed]

6.

M. P. Cagigal, V. F. Canales, P. J. Valle, and J. E. Oti, “Coronagraphic mask design using Hermite functions,” Opt. Express 17(22), 20515–20520 (2009). [CrossRef] [PubMed]

7.

E. Serabyna and D. Mawet, “Phase mask coronagraphy at JPL and Palomar,” EPJ Web of Conferences 16, 03004 (2011). [CrossRef]

8.

P. Baudoz, Y. Rabbia, and J. Gay, “Achromatic interfero coronagraph,” Astron. Astrophys. Suppl. Ser. 141(2), 319–329 (2000). [CrossRef]

9.

N. Baba, N. Murakami, T. Ishigaki, and N. Hashimoto, “Polarization interferometric stellar coronagraph,” Opt. Lett. 27(16), 1373–1375 (2002). [CrossRef] [PubMed]

10.

O. Guyon and M. Shao, “The pupil-swapping coronagraph,” Publ. Astron. Soc. Pac. 118(844), 860–865 (2006). [CrossRef]

11.

O. Guyon, “Phase-induced amplitude apodization of telescope pupils for extrasolar terrestrial planet imaging,” Astron. Astrophys. 404(1), 379–387 (2003). [CrossRef]

12.

O. Guyon, P. M. Hinz, E. Cady, R. Belikov, and F. Martinache, “High performance Lyot and PIAA coronagraphy for arbitrarily shaped telescope apertures,” 2014 sprint arXiv:1305.6686, accepted to ApJ.

13.

Mawet, L. Pueyo, A. Carlotti, B. Mennesson, E. Serabyn, and J. K. Wallace, “Ring-Apodized Vortex coronagraphs for obscured telescopes. I. Transmissive ring apodizers,” 2013 sprint arXiv:1309.3328, accepted to ApJS.

14.

O. Guyon, E. A. Pluzhnik, M. J. Kuchner, B. Collins, and S. T. Ridgway, “Theoretical limits on extrasolar terrestrial planet detection with coronagraphs,” Astrophys. J. 167(1Suppl.), 81–99 (2006). [CrossRef]

15.

D. Rouan, P. Riaud, A. Boccaletti, Y. Clénet, and A. Labeyrie, “The four-quadrant phase-mask coronagraph,” Publ. Astron. Soc. Pac. 112(777), 1479–1486 (2000). [CrossRef]

16.

L. Abe, F. Vakili, and A. Boccaletti, “The achromatic phase knife coronagraph,” Astron. Astrophys. 374(3), 1161–1168 (2001). [CrossRef]

17.

D. Rouan, J. Baudrand, A. Boccaletti, P. Baudoz, D. Mawet, and P. Riaud, “The four quadrant phase mask coronagraph and its avatars,” C. R. Phys. 8(3-4), 298–311 (2007). [CrossRef]

18.

G. Foo, D. M. Palacios, and G. A. Swartzlander Jr., “Optical vortex coronagraph,” Opt. Lett. 30(24), 3308–3310 (2005). [CrossRef] [PubMed]

19.

J. H. Lee, G. Foo, E. G. Johnson, and G. A. Swartzlander Jr., “Experimental verification of an optical vortex coronagraph,” Phys. Rev. Lett. 97(5), 053901 (2006). [CrossRef] [PubMed]

20.

G. A. Swartzlander Jr., “The optical vortex coronagraph,” J. Opt. A, Pure Appl. Opt. 11(9), 094022 (2009). [CrossRef]

21.

N. Murakami, R. Uemura, N. Baba, J. Nishikawa, M. Tamura, N. Hashimoto, and L. Abe, “An eight-octant phase mask coronagraph,” Publ. Astron. Soc. Pac. 120(872), 1112–1118 (2008). [CrossRef]

22.

N. Murakami, J. Nishikawa, K. Yokochi, M. Tamura, N. Baba, and L. Abe, “Achromatic eight-octant phase mask coronagraph using photonic crystal,” Astrophys. J. 714(1), 772–777 (2010). [CrossRef]

23.

G. A. Swartzlander Jr., “Achromatic optical vortex lens,” Opt. Lett. 31(13), 2042–2044 (2006). [CrossRef] [PubMed]

24.

C. C. Shih and P. V. Estates, “Phase mask with continuous azimuthal variation for a coronagraph imaging system,” U.S. Patent 7,773,307 (2010).

25.

D. Mawet, P. Riaud, O. Absil, and J. Surdej, “Annular groove phase mask coronagraph,” Astrophys. J. 633(2), 1191–1200 (2005). [CrossRef]

26.

A. Niv, G. Biener, V. Kleiner, and E. Hasman, “Polychromatic vectorial vortex formed by geometric phase elements,” Opt. Lett. 32(7), 847–849 (2007). [CrossRef] [PubMed]

27.

N. Murakami, S. Hamaguchi, M. Sakamoto, R. Fukumoto, A. Ise, K. Oka, N. Baba, and M. Tamura, “Design and laboratory demonstration of an achromatic vector vortex coronagraph,” Opt. Express 21(6), 7400–7410 (2013). [CrossRef] [PubMed]

28.

D. Mawet, L. Pueyo, A. Carlotti, B. Mennesson, E. Serabyn, J. Wallace, and P. Baudoz, “The multistage and ring-apodized vortex coronagraph: two simple, small-angle coronagraphic solutions for heavily obscured apertures,” Proc. SPIE 8864, 886411 (2013). [CrossRef]

29.

O. Ma, Q. Cao, and F. Hou, “Wide-band coronagraph with sinusoidal phase in the angular direction,” Opt. Express 20(10), 10933–10943 (2012). [CrossRef] [PubMed]

30.

Q. Cao, M. Gruber, and J. Jahns, “Generalized confocal imaging systems for free-space optical interconnections,” Appl. Opt. 43(16), 3306–3309 (2004). [CrossRef] [PubMed]

31.

J. R. P. Angel, “Ground-based imaging of extrasolar planets using adaptive optics,” Nature 368(6468), 203–207 (1994). [CrossRef]

32.

C. Marois, J. Veran, and C. Correia, “A Fresnel propagation analysis of NFIRAOS/IRIS high-contrast exoplanet imaging capabilities,” Proc. SPIE 8447, 844726 (2012). [CrossRef]

33.

D. Mawet, P. Riaud, J. Baudrand, P. Baudoz, A. Boccaletti, O. Dupuis, and D. Rouan, “The four-quadrant phase-mask coronagraph: white lightlaboratory results with an achromatic device,” Astron. Astrophys. 448(2), 801–808 (2006). [CrossRef]

34.

R. Galicher, P. Baudoz, and J. Baudrand, “Multi-stage four-quadrant phase mask: achromatic coronagraph for space-based and ground-based telescopes,” Astron. Astrophys. 530, A43 (2011). [CrossRef]

35.

J. Trauger, D. Moody, B. Gordon, J. Krist, and D. Mawet, “A hybrid Lyot coronagraph for the direct imaging and spectroscopy of exoplanet systems: recent results and prospects,” Proc. SPIE 8151, 81510G (2011). [CrossRef]

36.

D. Mawet, E. Serabyn, D. Moody, B. Kern, A. Niessner, A. Kuhnert, D. Shemo, R. Chipman, S. McClain, and J. Trauger, “Recent results of the second generation of vector vortex coronagraphs on the high-contrast imaging testbed at JPL,” Proc. SPIE 8151, 81511D (2011). [CrossRef]

37.

A. Carlotti, G. Ricort, and C. Aime, “Phase mask coronagraphy using a Mach-Zehnder interferometer,” Astron. Astrophys. 504(2), 663–671 (2009). [CrossRef]

OCIS Codes
(070.6110) Fourier optics and signal processing : Spatial filtering
(100.2980) Image processing : Image enhancement
(110.2970) Imaging systems : Image detection systems
(110.6770) Imaging systems : Telescopes
(350.1260) Other areas of optics : Astronomical optics

ToC Category:
Imaging Systems

History
Original Manuscript: December 11, 2013
Revised Manuscript: January 8, 2014
Manuscript Accepted: January 13, 2014
Published: January 21, 2014

Citation
Fanzhen Hou, Qing Cao, Minning Zhu, and Ourui Ma, "Wide-band six-region phase mask coronagraph," Opt. Express 22, 1884-1895 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-2-1884


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References

  1. D. Mawet, E. Serabyn, K. Liewer, Ch. Hanot, S. McEldowney, D. Shemo, N. O’Brien, “Optical Vectorial Vortex Coronagraphs using Liquid Crystal Polymers: theory, manufacturing and laboratory demonstration,” Opt. Express 17(3), 1902–1918 (2009). [CrossRef] [PubMed]
  2. J. T. Trauger, W. A. Traub, “A laboratory demonstration of the capability to image an Earth-like extrasolar planet,” Nature 446(7137), 771–773 (2007). [CrossRef] [PubMed]
  3. E. Serabyn, D. Mawet, R. Burruss, “An image of an exoplanet separated by two diffraction beamwidths from a star,” Nature 464(7291), 1018–1020 (2010). [CrossRef] [PubMed]
  4. D. Mawet, E. Serabyn, K. Liewer, R. Burruss, J. Hickey, D. Shemo, “The vector vortex coronagraph: laboratory results and first light at Palomar observatory,” Astrophys. J. 709(1), 53–57 (2010). [CrossRef]
  5. G. A. Swartzlander, E. L. Ford, R. S. Abdul-Malik, L. M. Close, M. A. Peters, D. M. Palacios, D. W. Wilson, “Astronomical demonstration of an optical vortex coronagraph,” Opt. Express 16(14), 10200–10207 (2008). [CrossRef] [PubMed]
  6. M. P. Cagigal, V. F. Canales, P. J. Valle, J. E. Oti, “Coronagraphic mask design using Hermite functions,” Opt. Express 17(22), 20515–20520 (2009). [CrossRef] [PubMed]
  7. E. Serabyna and D. Mawet, “Phase mask coronagraphy at JPL and Palomar,” EPJ Web of Conferences 16, 03004 (2011). [CrossRef]
  8. P. Baudoz, Y. Rabbia, J. Gay, “Achromatic interfero coronagraph,” Astron. Astrophys. Suppl. Ser. 141(2), 319–329 (2000). [CrossRef]
  9. N. Baba, N. Murakami, T. Ishigaki, N. Hashimoto, “Polarization interferometric stellar coronagraph,” Opt. Lett. 27(16), 1373–1375 (2002). [CrossRef] [PubMed]
  10. O. Guyon, M. Shao, “The pupil-swapping coronagraph,” Publ. Astron. Soc. Pac. 118(844), 860–865 (2006). [CrossRef]
  11. O. Guyon, “Phase-induced amplitude apodization of telescope pupils for extrasolar terrestrial planet imaging,” Astron. Astrophys. 404(1), 379–387 (2003). [CrossRef]
  12. O. Guyon, P. M. Hinz, E. Cady, R. Belikov, and F. Martinache, “High performance Lyot and PIAA coronagraphy for arbitrarily shaped telescope apertures,” 2014 sprint arXiv:1305.6686, accepted to ApJ.
  13. Mawet, L. Pueyo, A. Carlotti, B. Mennesson, E. Serabyn, and J. K. Wallace, “Ring-Apodized Vortex coronagraphs for obscured telescopes. I. Transmissive ring apodizers,” 2013 sprint arXiv:1309.3328, accepted to ApJS.
  14. O. Guyon, E. A. Pluzhnik, M. J. Kuchner, B. Collins, S. T. Ridgway, “Theoretical limits on extrasolar terrestrial planet detection with coronagraphs,” Astrophys. J. 167(1Suppl.), 81–99 (2006). [CrossRef]
  15. D. Rouan, P. Riaud, A. Boccaletti, Y. Clénet, A. Labeyrie, “The four-quadrant phase-mask coronagraph,” Publ. Astron. Soc. Pac. 112(777), 1479–1486 (2000). [CrossRef]
  16. L. Abe, F. Vakili, A. Boccaletti, “The achromatic phase knife coronagraph,” Astron. Astrophys. 374(3), 1161–1168 (2001). [CrossRef]
  17. D. Rouan, J. Baudrand, A. Boccaletti, P. Baudoz, D. Mawet, P. Riaud, “The four quadrant phase mask coronagraph and its avatars,” C. R. Phys. 8(3-4), 298–311 (2007). [CrossRef]
  18. G. Foo, D. M. Palacios, G. A. Swartzlander., “Optical vortex coronagraph,” Opt. Lett. 30(24), 3308–3310 (2005). [CrossRef] [PubMed]
  19. J. H. Lee, G. Foo, E. G. Johnson, G. A. Swartzlander., “Experimental verification of an optical vortex coronagraph,” Phys. Rev. Lett. 97(5), 053901 (2006). [CrossRef] [PubMed]
  20. G. A. Swartzlander., “The optical vortex coronagraph,” J. Opt. A, Pure Appl. Opt. 11(9), 094022 (2009). [CrossRef]
  21. N. Murakami, R. Uemura, N. Baba, J. Nishikawa, M. Tamura, N. Hashimoto, L. Abe, “An eight-octant phase mask coronagraph,” Publ. Astron. Soc. Pac. 120(872), 1112–1118 (2008). [CrossRef]
  22. N. Murakami, J. Nishikawa, K. Yokochi, M. Tamura, N. Baba, L. Abe, “Achromatic eight-octant phase mask coronagraph using photonic crystal,” Astrophys. J. 714(1), 772–777 (2010). [CrossRef]
  23. G. A. Swartzlander., “Achromatic optical vortex lens,” Opt. Lett. 31(13), 2042–2044 (2006). [CrossRef] [PubMed]
  24. C. C. Shih and P. V. Estates, “Phase mask with continuous azimuthal variation for a coronagraph imaging system,” U.S. Patent 7,773,307 (2010).
  25. D. Mawet, P. Riaud, O. Absil, J. Surdej, “Annular groove phase mask coronagraph,” Astrophys. J. 633(2), 1191–1200 (2005). [CrossRef]
  26. A. Niv, G. Biener, V. Kleiner, E. Hasman, “Polychromatic vectorial vortex formed by geometric phase elements,” Opt. Lett. 32(7), 847–849 (2007). [CrossRef] [PubMed]
  27. N. Murakami, S. Hamaguchi, M. Sakamoto, R. Fukumoto, A. Ise, K. Oka, N. Baba, M. Tamura, “Design and laboratory demonstration of an achromatic vector vortex coronagraph,” Opt. Express 21(6), 7400–7410 (2013). [CrossRef] [PubMed]
  28. D. Mawet, L. Pueyo, A. Carlotti, B. Mennesson, E. Serabyn, J. Wallace, P. Baudoz, “The multistage and ring-apodized vortex coronagraph: two simple, small-angle coronagraphic solutions for heavily obscured apertures,” Proc. SPIE 8864, 886411 (2013). [CrossRef]
  29. O. Ma, Q. Cao, F. Hou, “Wide-band coronagraph with sinusoidal phase in the angular direction,” Opt. Express 20(10), 10933–10943 (2012). [CrossRef] [PubMed]
  30. Q. Cao, M. Gruber, J. Jahns, “Generalized confocal imaging systems for free-space optical interconnections,” Appl. Opt. 43(16), 3306–3309 (2004). [CrossRef] [PubMed]
  31. J. R. P. Angel, “Ground-based imaging of extrasolar planets using adaptive optics,” Nature 368(6468), 203–207 (1994). [CrossRef]
  32. C. Marois, J. Veran, C. Correia, “A Fresnel propagation analysis of NFIRAOS/IRIS high-contrast exoplanet imaging capabilities,” Proc. SPIE 8447, 844726 (2012). [CrossRef]
  33. D. Mawet, P. Riaud, J. Baudrand, P. Baudoz, A. Boccaletti, O. Dupuis, D. Rouan, “The four-quadrant phase-mask coronagraph: white lightlaboratory results with an achromatic device,” Astron. Astrophys. 448(2), 801–808 (2006). [CrossRef]
  34. R. Galicher, P. Baudoz, J. Baudrand, “Multi-stage four-quadrant phase mask: achromatic coronagraph for space-based and ground-based telescopes,” Astron. Astrophys. 530, A43 (2011). [CrossRef]
  35. J. Trauger, D. Moody, B. Gordon, J. Krist, D. Mawet, “A hybrid Lyot coronagraph for the direct imaging and spectroscopy of exoplanet systems: recent results and prospects,” Proc. SPIE 8151, 81510G (2011). [CrossRef]
  36. D. Mawet, E. Serabyn, D. Moody, B. Kern, A. Niessner, A. Kuhnert, D. Shemo, R. Chipman, S. McClain, J. Trauger, “Recent results of the second generation of vector vortex coronagraphs on the high-contrast imaging testbed at JPL,” Proc. SPIE 8151, 81511D (2011). [CrossRef]
  37. A. Carlotti, G. Ricort, C. Aime, “Phase mask coronagraphy using a Mach-Zehnder interferometer,” Astron. Astrophys. 504(2), 663–671 (2009). [CrossRef]

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