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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 10 — May. 7, 2012
  • pp: 10933–10943
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Wide-band coronagraph with sinusoidal phase in the angular direction

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


Optics Express, Vol. 20, Issue 10, pp. 10933-10943 (2012)
http://dx.doi.org/10.1364/OE.20.010933


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Abstract

We suggest a new phase mask coronagraph that can work in a wide band of wavelengths. The phase mask has alternatively sinusoidal and uniform functions in the angular direction. We compare it with the four-quadrant phase mask coronagraph and vortex phase mask coronagraph. Through numerical tests, we find that the new mask gives a deep extinction of star light and has a small inner working angle. It is also shown that this mask has a better performance in chromatism than the others for a wide band of wavelengths.

© 2012 OSA

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

1. J. T. Trauger and W. A. Traub, “A laboratory demonstration of the capability to image an Earth-like extrasolar planet,” Nature 446(7137), 771–773 (2007). [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]

]. Interferometric coronagraph relies on interferometric combination of discrete beams devided from the entrance pupil [7

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

9

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

]. 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

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

].

Another approach employs modified Lyot concept [2

2. M. B. Lyot, “A study of the solar corona and prominences without eclipses,” Mon. Not. R. Astron. Soc. 99, 580–594 (1939).

] with amplitude- and phase-mask. The band-limited mask coronagraph (BLM) operates on the intensity of light in the image plane [11

11. M. J. Kuchner and W. A. Traub, “A coronagraph with a band-limited mask for finding terrestrial planets,” Astrophys. J. 570(2), 900–908 (2002). [CrossRef]

12

12. M. J. Kuchner, J. Crepp, and J. Ge, “Eight-order image masks for terrestrial planet finding,” Astrophys. J. 628(1), 466–473 (2005). [CrossRef]

]. Compared with amplitude mask coronagraph, phase mask coronagraph allows a smaller inner working angel (IWA) [13

13. 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]

]. 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

14. 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]

23

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

]. 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

Cn=12π02πt(θ)einθdθ.
(5)

Accordingly, the relation
n=|Cn|2=1
(6)
also holds.

We now discuss the n-th order Hankel transform Hn(.) of Eq. (4). When n is a nonzero even number, it can be proven that [17

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

, 29

29. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions 9th ed. (Dover Publications, 1970), 487.

30

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

] (see also Appendix C)
Hn{J1(ar')ar'}={fn(r''),r''RAS0r''<RAS,
(7)
where r′′ = (x′′2 + y′′2)1/2. When the radius RLS of the LS is smaller than RAS, the light field fn(r′′) is completely blocked. When n is an odd or zero, there are still values inside the LS. Therefore it is desirable that C2q + 1 (q is an integer) and C0 can be designed equal to zero for the designed wavelength λ0.

C2q+1(λ)=12π[0πt(θ,λ)ei(2q+1)θdθ+π2πt(θ,λ)ei(2q+1)θdθ]=12π[0πt(θ,λ)ei(2q+1)θdθ+ei(2q+1)π0πt(θ+π,λ)ei(2q+1)θdθ].
(8)

From Eq. (8) one can prove that the coefficient C2q + 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 C2q + 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:

C0(λ)=C0(λ0)+C0(λ)λ|λ0(λλ0)+122C0(λ)λ2|λ0(λλ0)2+.
(9)

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:

{C0(λ0)=0C0(λ)λ|λ0=0.
(10)

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
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.
, 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) = eiG(θ). By taking the chromatic property into account, one can get the transmission function of the mask:
t(θ,λ)={eiλ0λhsin(bθ)0θ<2πb12πbθ<π,2πb+πθ<2πeiλ0λhsin[b(θπ)]πθ<2πb+π,
(11)
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 C2q + 1(λ) = 0 holds because there are two periods in the range of 2π. By use of Eq. (5), Eq. (10) and the formula:
{J0(x)=12π02πeixsinθdθJ0'(x)=J(x)1,
(12)
one can get

{J0(h)=1b2J1(h)=0.
(13)

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 C2m(λ). In particular, by use of Eq. (13), the value of C0(λ) can be analytically expressed as:

C0(λ)|SPM=2bJ0(λ0λh)+12b.
(14)

After getting the values of the coefficients of C2m(λ), one can further get the field U(x′′,y′′)|SPM, which is expressed as:
U(x'',y'')|SPM=A2m=(i)2mC2mH2m{J1(ar')ar'}ei2mφ,
(15)
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)
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.
. The infinite sum in Eq. (15) is truncated from m = −15 to m = 15. RAS is chosen to be 1 and the factor a is chosen to be 1, where a = kRAS/f. To make sure that the result is accurate, we calculate the finite sum of |C2m|2 with m chosen from −15 to 15, which is 0.9996. Namely,

m=1515|C2m|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 RLS is chosen to be smaller than RAS, 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′′ = RAS. So RLS should be chosen to be slightly less than RAS 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
Fig. 4 Comparison between the peak throughput of SPM (blue curve), FQPM (green curve), and VPM2 (red curve), for RLS = RAS. 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 = 2RAS.
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

13. 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]

]. It is pointed out that the stellar angular diameter will affect the useful throughput of coronagraph, which we don’t further discuss here.

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 C2q + 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:

{|C0(λ)|FQPM2=cos2(λ02λπ)|C0(λ)|VPM22=sin2(λ0λπ)(λ0λπ)2.
(16)

Figure 5
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.
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.

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

24. 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]

26

26. N. Hodgson and H. Weber, Optical Resonators (Springer-Verlag, 1997), Chaps. 1 and 2.

]. In matrix optics, a specific transfer is denoted by a specific ABCD matrix, which has the form

[ABCD],

where A-D are the four matrix elements. In Ref [25

25. S. A. Collins Jr., “Lens-system diffraction integral written in terms of Matrix Optics,” J. Opt. Soc. Am. 60(9), 1168–1177 (1970). [CrossRef]

], the location and angle of matrix are swapped. That is to say, A, B, C, D in Ref [25

25. S. A. Collins Jr., “Lens-system diffraction integral written in terms of Matrix Optics,” J. Opt. Soc. Am. 60(9), 1168–1177 (1970). [CrossRef]

]. 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

M1=[1f01][10f11]=[0ff11].
(A1)

According to the Collins formula [25

25. S. A. Collins Jr., “Lens-system diffraction integral written in terms of Matrix Optics,” J. Opt. Soc. Am. 60(9), 1168–1177 (1970). [CrossRef]

,26

26. N. Hodgson and H. Weber, Optical Resonators (Springer-Verlag, 1997), Chaps. 1 and 2.

], one can get the complex amplitude U(x′, y′) in the focal plane (x′, y′)

U(x',y')=t(r',θ)(iλB)1exp(ikf)A0circ(r/RAS)×exp{ik2B[A(x2+y2)+D(x'2+y'2)2(xx'+yy')]}dxdy.
(A2)

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

{x=rcosφx'=rcosθy=rsinφy'=rsinθ

one can further get [27

27. M. Born and E. Wolf, Principles of Optics 7th ed. (Cambridge University Press, 1999), Chap. 8.

,28

28. J. W. Goodman, Introduction to Fourier Optics 3rd ed. (Roberts and Company Publishers, 2005), Chap. 2.

]

U(x',y')=t(r',θ)(iλf)1exp(ikr'2/2f)×A0circ(r/RAS)exp[ikf(xx'+yy')]dxdy=t(r',θ)(iλf)1exp(ikr'2/2f)×0A0rcirc(r/RAS){02πexp[ikfrr'cos(φθ)]dφ}dr.
(A3)

Using the formula

{exp[i2πrρcos(φθ)]=k=(i)kJk(2πrρ)exp[ik(φθ)]0ρrJ0(r)dr=ρJ1(ρ),
(A4)

one can finally get

U(x',y')=t(r',θ)(iλf)1exp(ikr'2/2f)×n=0A0rcirc(r/RAS)Jn(kfrr'){02π(i)nexp[in(φθ)]dφ}dr=t(r',θ)(iλf)1exp(ikr'2/2f)×n=0A0rcirc(r/RAS)Jn(kfrr')(i)nexp(inθ)[02πexp(inφ)dφ]dr=t(r',θ)(iλf)1exp(ikr'2/2f)2πA00rcirc(r/RAS)J0(kfr'r)dr=t(r',θ)(iλ0f)1exp(ikr'2/2f)2πA0RAS2J1(kRASfr')kRASfr'.
(A5)

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

24. 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]

26

26. N. Hodgson and H. Weber, Optical Resonators (Springer-Verlag, 1997), Chaps. 1 and 2.

]

M2=[12f01][10f11][1f01]=[1ff10].
(B1)

According to the Collins formula [25

25. S. A. Collins Jr., “Lens-system diffraction integral written in terms of Matrix Optics,” J. Opt. Soc. Am. 60(9), 1168–1177 (1970). [CrossRef]

,26

26. N. Hodgson and H. Weber, Optical Resonators (Springer-Verlag, 1997), Chaps. 1 and 2.

], one can get the complex amplitude U(x′′, y′′)

U(x'',y'')=(iλB)1exp(i3kf)U(x',y')×exp{ik2B[A(x'2+y'2)+D(x''2+y''2)2(x'x''+y'y'')]}dx'dy'.
(B2)

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

U(x'',y'')=(iλf)1A1t(θ)J1(ar')ar'exp(ikr'2/2f)×exp{ik2f[(x'2+y'2)2(x'x''+y'y'')]}dx'dy'=A1(iλf)1t(θ)J1(ar')ar'exp[ikf(x'x''+y'y'')]dx'dy'.
(B3)

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

27. M. Born and E. Wolf, Principles of Optics 7th ed. (Cambridge University Press, 1999), Chap. 8.

,28

28. J. W. Goodman, Introduction to Fourier Optics 3rd ed. (Roberts and Company Publishers, 2005), Chap. 2.

]

U(x'',y'')=A1(iλf)10r'J1(ar')ar'{02πt(θ)exp[ikfr'r''cos(θϕ)]dθ}dr'=A1(iλf)1n=0r'J1(ar')ar'Jn(kfr'r'')×{02π(i)nt(θ)exp[ik(θϕ)]dθ}dr'=A1(iλf)1n=0r'J1(ar')ar'Jn(kfr'r'')×(i)nexp(inϕ){02πt(θ)exp(inθ)dθ}dr'=A2n=(i)nCnHn{J1(ar')ar'}exp(inϕ),
(B4)

where Hn{.} is the n-th order Hankel transform, and Cn 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

29. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions 9th ed. (Dover Publications, 1970), 487.

,30

30. 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'}=0J2m(r''r')J1(r')dr'=Γ(m+1)r''2Γ(2)Γ(m)×F21(1+m,1m;2;1r''2),r''1
(C1)

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

H2m{J1(r')r'}=mr''2n=0m1(1+m)n(1m)n2nn!1r''2n,
(C2)

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

(x)n=x(x+1)(x+2)(x+n1).
(C3)

So we 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,
(C4)

where the property of Г(m)=(m-1)! 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.
(C5)

References and links

1.

J. T. Trauger and W. A. Traub, “A laboratory demonstration of the capability to image an Earth-like extrasolar planet,” Nature 446(7137), 771–773 (2007). [CrossRef] [PubMed]

2.

M. B. Lyot, “A study of the solar corona and prominences without eclipses,” Mon. Not. R. Astron. Soc. 99, 580–594 (1939).

3.

E. Serabyn, D. Mawet, and 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, 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.

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

8.

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

9.

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

10.

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

11.

M. J. Kuchner and W. A. Traub, “A coronagraph with a band-limited mask for finding terrestrial planets,” Astrophys. J. 570(2), 900–908 (2002). [CrossRef]

12.

M. J. Kuchner, J. Crepp, and J. Ge, “Eight-order image masks for terrestrial planet finding,” Astrophys. J. 628(1), 466–473 (2005). [CrossRef]

13.

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]

14.

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]

15.

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

16.

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]

17.

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

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]

25.

S. A. Collins Jr., “Lens-system diffraction integral written in terms of Matrix Optics,” J. Opt. Soc. Am. 60(9), 1168–1177 (1970). [CrossRef]

26.

N. Hodgson and H. Weber, Optical Resonators (Springer-Verlag, 1997), Chaps. 1 and 2.

27.

M. Born and E. Wolf, Principles of Optics 7th ed. (Cambridge University Press, 1999), Chap. 8.

28.

J. W. Goodman, Introduction to Fourier Optics 3rd ed. (Roberts and Company Publishers, 2005), Chap. 2.

29.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions 9th ed. (Dover Publications, 1970), 487.

30.

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: February 21, 2012
Revised Manuscript: April 5, 2012
Manuscript Accepted: April 5, 2012
Published: April 26, 2012

Citation
Ourui Ma, Qing Cao, and Fanzhen Hou, "Wide-band coronagraph with sinusoidal phase in the angular direction," Opt. Express 20, 10933-10943 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-10-10933


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References

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  4. 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, 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. Express16(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. Express17(22), 20515–20520 (2009). [CrossRef] [PubMed]
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  8. N. Baba, N. Murakami, T. Ishigaki, and N. Hashimoto, “Polarization interferometric stellar coronagraph,” Opt. Lett.27(16), 1373–1375 (2002). [CrossRef] [PubMed]
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  10. O. Guyon, “Phase-induced amplitude apodization of telescope pupils for extrasolar terrestrial planet imaging,” Astron. Astrophys.404(1), 379–387 (2003). [CrossRef]
  11. M. J. Kuchner and W. A. Traub, “A coronagraph with a band-limited mask for finding terrestrial planets,” Astrophys. J.570(2), 900–908 (2002). [CrossRef]
  12. M. J. Kuchner, J. Crepp, and J. Ge, “Eight-order image masks for terrestrial planet finding,” Astrophys. J.628(1), 466–473 (2005). [CrossRef]
  13. 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]
  14. 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]
  15. L. Abe, F. Vakili, and A. Boccaletti, “The achromatic phase knife coronagraph,” Astron. Astrophys.374(3), 1161–1168 (2001). [CrossRef]
  16. 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]
  17. D. Mawet, P. Riaud, O. Absil, and J. Surdej, “Annular groove phase mask coronagraph,” Astrophys. J.633(2), 1191–1200 (2005). [CrossRef]
  18. G. Foo, D. M. Palacios, and G. A. Swartzlander., “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., “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, 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., “Achromatic optical vortex lens,” Opt. Lett.31(13), 2042–2044 (2006). [CrossRef] [PubMed]
  24. 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]
  25. S. A. Collins., “Lens-system diffraction integral written in terms of Matrix Optics,” J. Opt. Soc. Am.60(9), 1168–1177 (1970). [CrossRef]
  26. N. Hodgson and H. Weber, Optical Resonators (Springer-Verlag, 1997), Chaps. 1 and 2.
  27. M. Born and E. Wolf, Principles of Optics 7th ed. (Cambridge University Press, 1999), Chap. 8.
  28. J. W. Goodman, Introduction to Fourier Optics 3rd ed. (Roberts and Company Publishers, 2005), Chap. 2.
  29. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions 9th ed. (Dover Publications, 1970), 487.
  30. A. Carlotti, G. Ricort, and C. Aime, “Phase mask coronagraphy using a Mach-Zehnder interferometer,” Astron. Astrophys.504(2), 663–671 (2009). [CrossRef]

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