## Wide-band coronagraph with sinusoidal phase in the angular direction |

Optics Express, Vol. 20, Issue 10, pp. 10933-10943 (2012)

http://dx.doi.org/10.1364/OE.20.010933

Acrobat PDF (1487 KB)

### 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

^{10}times brighter than visible reflected light from an Earth-like planet. In the near-infrared band, the starlight is 10

^{4}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. 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]

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]

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

## 2. Sinusoidal phase mask

_{n}(.) 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]

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

^{2}+ y′′

^{2})

^{1/2}. When the radius R

_{LS}of the LS is smaller than R

_{AS}, the light field f

_{n}(r′′) is completely blocked. When n is an odd or zero, there are still values inside the LS. Therefore it is desirable that C

_{2q + 1}(q is an integer) and C

_{0}can be designed equal to zero for the designed wavelength λ

_{0}.

_{2q + 1}(λ) is always equal to zero, provided that t(θ + π, λ) = t(θ, λ), where θ is from 0 to π. That is to say, the relation C

_{2q + 1}(λ) = 0 holds for any double periodic function t(θ, λ) in the range of 0<θ≤2π.

_{0}(λ). This coefficient may be expressed as a Taylor expansion in the vicinity of λ

_{0}:

_{0}(λ)≈0 needs to be satisfied for a wide band of wavelengths. So we have the two conditions:

_{0}, where t(θ) = e

^{iG(θ)}. By taking the chromatic property into account, one can get the transmission function of the mask:where h>0. There are two periods in the range of 2π, so b must be greater than 2.This property can be easily seen from Fig. 2. As we state above, the condition C

_{2q + 1}(λ) = 0 holds because there are two periods in the range of 2π. By use of Eq. (5), Eq. (10) and the formula:one can get

_{1}(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 λ

_{0}into Eqs. (11) and (5), one can numerically calculate the coefficients C

_{0}(λ) and C

_{2m}(λ). In particular, by use of Eq. (13), the value of C

_{0}(λ) can be analytically expressed as:

## 3. Numerical tests

_{0}, we numerically calculate the Lyot-stop image which is acquired at the Lyot-stop plane using Eqs. (5) and (15). The calculation is shown in Fig. 3(a) . The infinite sum in Eq. (15) is truncated from m = −15 to m = 15. R

_{AS}is chosen to be 1 and the factor a is chosen to be 1, where a = kR

_{AS}/f. To make sure that the result is accurate, we calculate the finite sum of |C

_{2m}|

^{2}with m chosen from −15 to 15, which is 0.9996. Namely,

^{−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.

_{LS}is chosen to be smaller than R

_{AS}, then the star light is completely blocked after travelling through the Lyot stop. That is to say, the throughput of on-axis star light is zero. However, technology limits of the fabrication of phase mask will lead to phase error. The finite sizes of phase mask and of lens L2 will lead to truncation error in the spatial frequency distribution. All the above-mentioned limits will lead to the nonzero values of the field in the nearby inner area of the circle r′′ = R

_{AS}. So R

_{LS}should be chosen to be slightly less than R

_{AS}in the actual situation.

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]

_{0}and C

_{2q + 1}for the designed wavelength λ

_{0}. But when wavelength changes in the vicinity of λ

_{0}, C

_{0}(λ) is no longer equal to zero. As we state in section 2, the C

_{0}(λ) of a SPM is given by Eq. (14). It can be proven that |C

_{0}(λ)|

^{2}of the FQPM and VPM2 can be written as:

_{0}(λ)|

^{2}of SPM with the wavelength changing from 450nm to 650nm compared with those of FQPM and VPM2, where λ

_{0}= 550nm. One can see that SPM has a smaller value of |C

_{0}(λ)|

^{2}than the others for a bandwidth as large as 200nm. Especially when the wavelength lies between 500nm to 600nm, the value |C

_{0}(λ)|

^{2}of SPM is more than 10

^{2}-10

^{8}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

## Appendix A

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]

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]

_{1}from the transverse plane (x, y) to (x′, y′) in Fig.1 can be easily determined to be

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]

## Appendix B

_{2}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]

**60**(9), 1168–1177 (1970). [CrossRef]

^{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′.

## Appendix C

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

_{2}F

_{1}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

_{n}is the Pochhammer symbol, and it is defined by

## 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 |

2. | M. B. Lyot, “A study of the solar corona and prominences without eclipses,” Mon. Not. R. Astron. Soc. |

3. | E. Serabyn, D. Mawet, and R. Burruss, “An image of an exoplanet separated by two diffraction beamwidths from a star,” Nature |

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. |

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 |

6. | M. P. Cagigal, V. F. Canales, P. J. Valle, and J. E. Oti, “Coronagraphic mask design using Hermite functions,” Opt. Express |

7. | P. Baudoz, Y. Rabbia, and J. Gay, “Achromatic interfero coronagraph,” Astron. Astrophys. Suppl. Ser. |

8. | N. Baba, N. Murakami, T. Ishigaki, and N. Hashimoto, “Polarization interferometric stellar coronagraph,” Opt. Lett. |

9. | O. Guyon and M. Shao, “The pupil-swapping coronagraph,” Publ Astron. Soc. Pac. |

10. | O. Guyon, “Phase-induced amplitude apodization of telescope pupils for extrasolar terrestrial planet imaging,” Astron. Astrophys. |

11. | M. J. Kuchner and W. A. Traub, “A coronagraph with a band-limited mask for finding terrestrial planets,” Astrophys. J. |

12. | M. J. Kuchner, J. Crepp, and J. Ge, “Eight-order image masks for terrestrial planet finding,” Astrophys. J. |

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. |

14. | D. Rouan, P. Riaud, A. Boccaletti, Y. Clénet, and A. Labeyrie, “The four-quadrant phase-mask coronagraph,” Publ. Astron. Soc. Pac. |

15. | L. Abe, F. Vakili, and A. Boccaletti, “The achromatic phase knife coronagraph,” Astron. Astrophys. |

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. |

17. | D. Mawet, P. Riaud, O. Absil, and J. Surdej, “Annular groove phase mask coronagraph,” Astrophys. J. |

18. | G. Foo, D. M. Palacios, and G. A. Swartzlander Jr., “Optical vortex coronagraph,” Opt. Lett. |

19. | J. H. Lee, G. Foo, E. G. Johnson, and G. A. Swartzlander Jr., “Experimental verification of an optical vortex coronagraph,” Phys. Rev. Lett. |

20. | G. A. Swartzlander Jr., “The optical vortex coronagraph,” J. Opt. A, Pure Appl. Opt. |

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. |

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. |

23. | G. A. Swartzlander Jr., “Achromatic optical vortex lens,” Opt. Lett. |

24. | Q. Cao, M. Gruber, and J. Jahns, “Generalized confocal imaging systems for free-space optical interconnections,” Appl. Opt. |

25. | S. A. Collins Jr., “Lens-system diffraction integral written in terms of Matrix Optics,” J. Opt. Soc. Am. |

26. | N. Hodgson and H. Weber, |

27. | M. Born and E. Wolf, |

28. | J. W. Goodman, |

29. | M. Abramowitz and I. A. Stegun, |

30. | A. Carlotti, G. Ricort, and C. Aime, “Phase mask coronagraphy using a Mach-Zehnder interferometer,” Astron. Astrophys. |

**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

- J. T. Trauger and W. A. Traub, “A laboratory demonstration of the capability to image an Earth-like extrasolar planet,” Nature446(7137), 771–773 (2007). [CrossRef] [PubMed]
- M. B. Lyot, “A study of the solar corona and prominences without eclipses,” Mon. Not. R. Astron. Soc.99, 580–594 (1939).
- E. Serabyn, D. Mawet, and R. Burruss, “An image of an exoplanet separated by two diffraction beamwidths from a star,” Nature464(7291), 1018–1020 (2010). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- P. Baudoz, Y. Rabbia, and J. Gay, “Achromatic interfero coronagraph,” Astron. Astrophys. Suppl. Ser.141(2), 319–329 (2000). [CrossRef]
- N. Baba, N. Murakami, T. Ishigaki, and N. Hashimoto, “Polarization interferometric stellar coronagraph,” Opt. Lett.27(16), 1373–1375 (2002). [CrossRef] [PubMed]
- O. Guyon and M. Shao, “The pupil-swapping coronagraph,” Publ Astron. Soc. Pac.118(844), 860–865 (2006). [CrossRef]
- O. Guyon, “Phase-induced amplitude apodization of telescope pupils for extrasolar terrestrial planet imaging,” Astron. Astrophys.404(1), 379–387 (2003). [CrossRef]
- 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]
- M. J. Kuchner, J. Crepp, and J. Ge, “Eight-order image masks for terrestrial planet finding,” Astrophys. J.628(1), 466–473 (2005). [CrossRef]
- 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]
- 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]
- L. Abe, F. Vakili, and A. Boccaletti, “The achromatic phase knife coronagraph,” Astron. Astrophys.374(3), 1161–1168 (2001). [CrossRef]
- 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]
- D. Mawet, P. Riaud, O. Absil, and J. Surdej, “Annular groove phase mask coronagraph,” Astrophys. J.633(2), 1191–1200 (2005). [CrossRef]
- G. Foo, D. M. Palacios, and G. A. Swartzlander., “Optical vortex coronagraph,” Opt. Lett.30(24), 3308–3310 (2005). [CrossRef] [PubMed]
- 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]
- G. A. Swartzlander., “The optical vortex coronagraph,” J. Opt. A, Pure Appl. Opt.11(9), 094022 (2009). [CrossRef]
- 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]
- 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]
- G. A. Swartzlander., “Achromatic optical vortex lens,” Opt. Lett.31(13), 2042–2044 (2006). [CrossRef] [PubMed]
- 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]
- S. A. Collins., “Lens-system diffraction integral written in terms of Matrix Optics,” J. Opt. Soc. Am.60(9), 1168–1177 (1970). [CrossRef]
- N. Hodgson and H. Weber, Optical Resonators (Springer-Verlag, 1997), Chaps. 1 and 2.
- M. Born and E. Wolf, Principles of Optics 7th ed. (Cambridge University Press, 1999), Chap. 8.
- J. W. Goodman, Introduction to Fourier Optics 3rd ed. (Roberts and Company Publishers, 2005), Chap. 2.
- M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions 9th ed. (Dover Publications, 1970), 487.
- 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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