Abstract

Phase-induced amplitude apodization (PIAA) is a promising technique in high contrast coronagraphs due to the characteristics of high efficiency and small inner working angle. In this letter, we present a new method for calculating the diffraction effects in PIAA coronagraphs based on boundary wave diffraction theory. We propose a numerical propagator in an azimuth boundary integral form, and then delve into its analytical propagator using stationary phase approximation. This propagator has straightforward physical meaning and obvious advantage on calculating efficiency, compared with former methods based on numerical integral or angular spectrum propagation method. Using this propagator, we can make a more direct explanation to the significant impact of pre-apodizer. This propagator can also be used to calculate the aberration propagation properties of PIAA optics. The calculating is also simplified since the decomposing procedure is not needed regardless of the form of the aberration.

© 2017 Optical Society of America

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References

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  1. O. Guyon, “Phase-induced amplitude apodization of telescope pupils for extrasolar terrestrial planet imaging,” Astron. Astrophys. 404(1), 379–387 (2003).
    [Crossref]
  2. O. Guyon, E. A. Pluzhnik, M. J. Kuchner, B. Collins, and S. T. Ridgway, “Theoretical limits on extra solar terrestrial planet detection with coronagraphs,” Astrophys. J. 167(1Suppl.), 81–99 (2006).
    [Crossref]
  3. F. Martinache, O. Guyon, E. A. Pluzhnik, R. Galicher, and S. T. Ridgway, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph.II. Performance,” Astrophys. J. 639(2), 1129–1137 (2006).
    [Crossref]
  4. R. J. Vanderbei, “Diffraction analysis of two-dimensional pupil mapping for high-contrast imaging,” Astrophys. J. 636(1), 528–543 (2006).
    [Crossref]
  5. R. Belikov, N. J. Kasdin, and R. J. Vanderbei, “Diffraction-based sensitivity analysis of apodized pupil-mapping systems,” Astrophys. J. 652(1), 833–844 (2006).
    [Crossref]
  6. E. A. Pluzhnik, O. Guyon, S. T. Ridgway, F. Martinache, R. A. Woodruff, C. Blain, and R. Galicher, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph. III. Diffraction effects and coronagraph design,” Astrophys. J. 644(2), 1246–1257 (2006).
    [Crossref]
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    [Crossref]
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    [Crossref]
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  12. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).
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    [Crossref]
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2012 (1)

2011 (1)

2010 (2)

Y. Z. Umul, “Uniform boundary diffraction wave theory of Rubinowicz,” J. Opt. Soc. Am. A 27(7), 1613–1619 (2010).
[Crossref] [PubMed]

J. E. Krist, L. Pueyo, and S. B. Shaklan, Proc. “Practical numerical propagation of arbitrary wavefronts through PIAA optics,” Proc. SPIE 7731, 77314N (2010).
[Crossref]

2009 (1)

L. Pueyo, S. B. Shaklan, A. Give’on, and J. Krist, “Numerical propagator through PIAA optics,” Proc. SPIE 7440, 74400E (2009).
[Crossref]

2006 (5)

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

F. Martinache, O. Guyon, E. A. Pluzhnik, R. Galicher, and S. T. Ridgway, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph.II. Performance,” Astrophys. J. 639(2), 1129–1137 (2006).
[Crossref]

R. J. Vanderbei, “Diffraction analysis of two-dimensional pupil mapping for high-contrast imaging,” Astrophys. J. 636(1), 528–543 (2006).
[Crossref]

R. Belikov, N. J. Kasdin, and R. J. Vanderbei, “Diffraction-based sensitivity analysis of apodized pupil-mapping systems,” Astrophys. J. 652(1), 833–844 (2006).
[Crossref]

E. A. Pluzhnik, O. Guyon, S. T. Ridgway, F. Martinache, R. A. Woodruff, C. Blain, and R. Galicher, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph. III. Diffraction effects and coronagraph design,” Astrophys. J. 644(2), 1246–1257 (2006).
[Crossref]

2005 (1)

R. J. Vanderbei and W. A. Traub, “Pupil mapping in two dimensions for high-contrast imaging,” Astrophys. J. 626(2), 1079–1090 (2005).
[Crossref]

2003 (1)

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

1977 (1)

1971 (1)

1964 (1)

Belikov, R.

R. Belikov, N. J. Kasdin, and R. J. Vanderbei, “Diffraction-based sensitivity analysis of apodized pupil-mapping systems,” Astrophys. J. 652(1), 833–844 (2006).
[Crossref]

Blain, C.

E. A. Pluzhnik, O. Guyon, S. T. Ridgway, F. Martinache, R. A. Woodruff, C. Blain, and R. Galicher, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph. III. Diffraction effects and coronagraph design,” Astrophys. J. 644(2), 1246–1257 (2006).
[Crossref]

Cady, E.

Collins, B.

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

Galicher, R.

F. Martinache, O. Guyon, E. A. Pluzhnik, R. Galicher, and S. T. Ridgway, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph.II. Performance,” Astrophys. J. 639(2), 1129–1137 (2006).
[Crossref]

E. A. Pluzhnik, O. Guyon, S. T. Ridgway, F. Martinache, R. A. Woodruff, C. Blain, and R. Galicher, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph. III. Diffraction effects and coronagraph design,” Astrophys. J. 644(2), 1246–1257 (2006).
[Crossref]

Give’on, A.

L. Pueyo, S. B. Shaklan, A. Give’on, and J. Krist, “Numerical propagator through PIAA optics,” Proc. SPIE 7440, 74400E (2009).
[Crossref]

Guyon, O.

E. A. Pluzhnik, O. Guyon, S. T. Ridgway, F. Martinache, R. A. Woodruff, C. Blain, and R. Galicher, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph. III. Diffraction effects and coronagraph design,” Astrophys. J. 644(2), 1246–1257 (2006).
[Crossref]

F. Martinache, O. Guyon, E. A. Pluzhnik, R. Galicher, and S. T. Ridgway, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph.II. Performance,” Astrophys. J. 639(2), 1129–1137 (2006).
[Crossref]

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

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

Kasdin, N. J.

L. Pueyo, N. J. Kasdin, and S. Shaklan, “Propagation of aberrations through phase-induced amplitude apodization coronagraph,” J. Opt. Soc. Am. A 28(2), 189–202 (2011).
[Crossref] [PubMed]

R. Belikov, N. J. Kasdin, and R. J. Vanderbei, “Diffraction-based sensitivity analysis of apodized pupil-mapping systems,” Astrophys. J. 652(1), 833–844 (2006).
[Crossref]

Krist, J.

L. Pueyo, S. B. Shaklan, A. Give’on, and J. Krist, “Numerical propagator through PIAA optics,” Proc. SPIE 7440, 74400E (2009).
[Crossref]

Krist, J. E.

J. E. Krist, L. Pueyo, and S. B. Shaklan, Proc. “Practical numerical propagation of arbitrary wavefronts through PIAA optics,” Proc. SPIE 7731, 77314N (2010).
[Crossref]

Kuchner, M. J.

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

Lachambre, J. L.

Lavigne, P.

Lit, J. W. Y.

Martinache, F.

E. A. Pluzhnik, O. Guyon, S. T. Ridgway, F. Martinache, R. A. Woodruff, C. Blain, and R. Galicher, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph. III. Diffraction effects and coronagraph design,” Astrophys. J. 644(2), 1246–1257 (2006).
[Crossref]

F. Martinache, O. Guyon, E. A. Pluzhnik, R. Galicher, and S. T. Ridgway, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph.II. Performance,” Astrophys. J. 639(2), 1129–1137 (2006).
[Crossref]

Miyamoto, K.

Otis, G.

Pluzhnik, E. A.

E. A. Pluzhnik, O. Guyon, S. T. Ridgway, F. Martinache, R. A. Woodruff, C. Blain, and R. Galicher, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph. III. Diffraction effects and coronagraph design,” Astrophys. J. 644(2), 1246–1257 (2006).
[Crossref]

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

F. Martinache, O. Guyon, E. A. Pluzhnik, R. Galicher, and S. T. Ridgway, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph.II. Performance,” Astrophys. J. 639(2), 1129–1137 (2006).
[Crossref]

Pueyo, L.

L. Pueyo, N. J. Kasdin, and S. Shaklan, “Propagation of aberrations through phase-induced amplitude apodization coronagraph,” J. Opt. Soc. Am. A 28(2), 189–202 (2011).
[Crossref] [PubMed]

J. E. Krist, L. Pueyo, and S. B. Shaklan, Proc. “Practical numerical propagation of arbitrary wavefronts through PIAA optics,” Proc. SPIE 7731, 77314N (2010).
[Crossref]

L. Pueyo, S. B. Shaklan, A. Give’on, and J. Krist, “Numerical propagator through PIAA optics,” Proc. SPIE 7440, 74400E (2009).
[Crossref]

Ridgway, S. T.

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

E. A. Pluzhnik, O. Guyon, S. T. Ridgway, F. Martinache, R. A. Woodruff, C. Blain, and R. Galicher, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph. III. Diffraction effects and coronagraph design,” Astrophys. J. 644(2), 1246–1257 (2006).
[Crossref]

F. Martinache, O. Guyon, E. A. Pluzhnik, R. Galicher, and S. T. Ridgway, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph.II. Performance,” Astrophys. J. 639(2), 1129–1137 (2006).
[Crossref]

Shaklan, S.

Shaklan, S. B.

J. E. Krist, L. Pueyo, and S. B. Shaklan, Proc. “Practical numerical propagation of arbitrary wavefronts through PIAA optics,” Proc. SPIE 7731, 77314N (2010).
[Crossref]

L. Pueyo, S. B. Shaklan, A. Give’on, and J. Krist, “Numerical propagator through PIAA optics,” Proc. SPIE 7440, 74400E (2009).
[Crossref]

Suzuki, T.

Traub, W. A.

R. J. Vanderbei and W. A. Traub, “Pupil mapping in two dimensions for high-contrast imaging,” Astrophys. J. 626(2), 1079–1090 (2005).
[Crossref]

Umul, Y. Z.

Vanderbei, R. J.

R. J. Vanderbei, “Diffraction analysis of two-dimensional pupil mapping for high-contrast imaging,” Astrophys. J. 636(1), 528–543 (2006).
[Crossref]

R. Belikov, N. J. Kasdin, and R. J. Vanderbei, “Diffraction-based sensitivity analysis of apodized pupil-mapping systems,” Astrophys. J. 652(1), 833–844 (2006).
[Crossref]

R. J. Vanderbei and W. A. Traub, “Pupil mapping in two dimensions for high-contrast imaging,” Astrophys. J. 626(2), 1079–1090 (2005).
[Crossref]

Woodruff, R. A.

E. A. Pluzhnik, O. Guyon, S. T. Ridgway, F. Martinache, R. A. Woodruff, C. Blain, and R. Galicher, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph. III. Diffraction effects and coronagraph design,” Astrophys. J. 644(2), 1246–1257 (2006).
[Crossref]

Astron. Astrophys. (1)

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

Astrophys. J. (6)

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

F. Martinache, O. Guyon, E. A. Pluzhnik, R. Galicher, and S. T. Ridgway, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph.II. Performance,” Astrophys. J. 639(2), 1129–1137 (2006).
[Crossref]

R. J. Vanderbei, “Diffraction analysis of two-dimensional pupil mapping for high-contrast imaging,” Astrophys. J. 636(1), 528–543 (2006).
[Crossref]

R. Belikov, N. J. Kasdin, and R. J. Vanderbei, “Diffraction-based sensitivity analysis of apodized pupil-mapping systems,” Astrophys. J. 652(1), 833–844 (2006).
[Crossref]

E. A. Pluzhnik, O. Guyon, S. T. Ridgway, F. Martinache, R. A. Woodruff, C. Blain, and R. Galicher, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph. III. Diffraction effects and coronagraph design,” Astrophys. J. 644(2), 1246–1257 (2006).
[Crossref]

R. J. Vanderbei and W. A. Traub, “Pupil mapping in two dimensions for high-contrast imaging,” Astrophys. J. 626(2), 1079–1090 (2005).
[Crossref]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (2)

Opt. Express (1)

Proc. SPIE (2)

L. Pueyo, S. B. Shaklan, A. Give’on, and J. Krist, “Numerical propagator through PIAA optics,” Proc. SPIE 7440, 74400E (2009).
[Crossref]

J. E. Krist, L. Pueyo, and S. B. Shaklan, Proc. “Practical numerical propagation of arbitrary wavefronts through PIAA optics,” Proc. SPIE 7731, 77314N (2010).
[Crossref]

Other (1)

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

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Figures (7)

Fig. 1
Fig. 1

PIAA system composed by a pair of specially figured lenses. Light incomings from the top. The maximum size of L1 is noted by r1. L1 is a little oversized in order to mitigate the diffraction effects.

Fig. 2
Fig. 2

The output fields calculated by four different methods: boundary wave integral (a), stationary phase approximation of boundary wave (b), S-Huygens approximation of 2D integral (c), and hybrid PASP method (d) proposed by Krist [8]. The radius of the second mirror is 50mm, and the first mirror is 2% oversized. The mirror separation is 500mm. The incoming light wavelength is 800nm.

Fig. 3
Fig. 3

PSFs at the focus of PIAA system, corresponding to the three output fields (b), (c), and (d) obtained above. The PSFs were computed by 10,000 points discretization.

Fig. 4
Fig. 4

Output fields and PSFs of a “diffraction-free” PIAA system with cosine-tapered pre-apodizer.

Fig. 5
Fig. 5

(a) Output field of the second term of Eq. (19) and (b) its contribution to final PSF. In (a), the upper line is the profile of the apodization function, representing the geometrical propagation properties of PIAA systems. The lower line is the contribution of the second term of Eq. (19), representing the diffraction part properties.

Fig. 6
Fig. 6

The output fields calculated by three different methods: stationary phase approximation of boundary wave(a), S-Huygens approximation of 2D integral(b), and hybrid PASP method (c) proposed by Krist [8].The relative difference in amplitude between (b) and (c) (d) is less than 1.5% at most positions, while the maximum difference is no more than 21% (occurred at the very near edge). The discrepancy in the phase near the edge is about 0.25 rad. However, all of the corresponding PSFs show good accordance while the main differences occur at the “valley” points.

Fig. 7
Fig. 7

Up: The phase of output fields calculated by three different methods, corresponding to the results in Fig. 6 (a) ~(c). From the 2D images, the discrepancy at the very near can be seen directly. Bottom: The PSFs at the focal plane of the corresponding electric fields (a) ~(c), stationary phase approximation (d), S-Huygens approximation (e), and hybrid PASP method (f) respectively.

Equations (22)

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{ U(P)= U G (P)+ U B (P), P in illuminated region U(P)= U B (P), P in shadow region ,
U G ( r ˜ , θ ˜ )=A( r ˜ ) e jk P 0 ,
U B ( r ˜ , θ ˜ )= 1 4π τ e jk[ S+| n |Z| n |h(r)+| n | h ˜ ( r ˜ ) ] S p × s 1 p s d l .
S= r 1 2 + r ˜ 2 2 r 1 r ˜ cos(θ θ ˜ )+ [h( r 1 ) h ˜ ( r ˜ )] 2 , s = 1 S ( r ˜ cos θ ˜ r 1 cosθ r ˜ sin θ ˜ r 1 sinθ h ˜ ( r ˜ )h( r 1 ) )
d l = r 1 dθ ( sinθ, cosθ, 0 ) T .
U B ( r ˜ , θ ˜ )= 1 4π e jk| n |[ Zh( r 1 )+ h ˜ ( r ˜ ) ] 0 2π e jkS S r 1 [ r ˜ cos( θ θ ˜ ) r 1 ] S+h( r 1 ) h ˜ ( r ˜ ) dθ ,
U( r ˜ , θ ˜ )=A( r ˜ )+ 1 4π e jk{ | n |[ Zh( r 1 )+ h ˜ ( r ˜ ) ] P 0 } 0 2π e jkS S r 1 [ r ˜ cos( θ θ ˜ ) r 1 ] S+h( r 1 ) h ˜ ( r ˜ ) dθ .
I( k )= + g( x ) e jkf( x ) dx =sgn[ f ( x 0 ) ] 2π k f ( x 0 ) g( x 0 ) e jkf( x 0 ) e j π 4 ,
0 2π e jkS S r 1 [ r ˜ cos(θ θ ˜ ) r 1 ] S+h( r 1 ) h ˜ ( r ˜ ) dθ = lim N 1 2N 2Nπ 2Nπ e jkS S r 1 [ r ˜ cos(θ θ ˜ ) r 1 ] S+h( r 1 ) h ˜ ( r ˜ ) dθ 1 2N + e jkf(θ) g(θ)dθ ,
f( θ )=S( θ )= r 1 2 + r ˜ 2 2 r 1 r ˜ cos( θ θ ˜ )+ [ h( r 1 ) h ˜ ( r ˜ ) ] 2 ,
g(θ)= r 1 [ r ˜ cos(θ θ ˜ ) r 1 ] S[S+h( r 1 ) h ˜ ( r ˜ )] ,
{ sin(θ- θ ˜ )=0 θ= θ ˜ +mπ, m=-2N, -2N+1,, 2N-1 .
{ cos(θ θ ˜ ) 2 f θ 2 = r 1 r ˜ S 1 , S 1 = ( r 1 r ˜ ) 2 + [h( r 1 ) h ˜ ( r ˜ )] 2
I 1 = 2π S 1 k r 1 r ˜ r 1 r ˜ r 1 2 S 1 [ S 1 +h( r 1 ) h ˜ ( r ˜ ) ] e jk S 1 e j π 4 ,
I 1 = 2π S 1 k r 1 r ˜ r 1 r ˜ + r 1 2 S 1 [ S 1 +h( r 1 ) h ˜ ( r ˜ ) ] e jk S 1 e j π 4 , S 1 = ( r 1 + r ˜ ) 2 + [ h( r 1 ) h ˜ ( r ˜ ) ] 2
0 2π e jkS S r 1 [ r ˜ cos(θ θ ˜ ) r 1 ] S+h( r 1 ) h ˜ ( r ˜ ) dθ =(2N I 1 +2N I 1 )/2N= I 1 + I 1 .
U( r ˜ , θ ˜ )=A( r ˜ )+(1/ 4π ) e jk{ | n |[ Zh( r 1 )+ h ˜ ( r ˜ ) ] P 0 } ( I 1 + I 1 ).
A( r ˜ )=a e 10 ( r ˜ /R ) 2
U B ( r ˜ , θ ˜ )= 1 4π τ T( r 1 ) W d l 1 4π ( T(r,θ)× W ) n dS .
E(r,θ)= E A (r,θ) e jk E p (r,θ) .
g(θ)= r 1 [ r ˜ cos(θ θ ˜ ) r 1 ] S[S+h( r 1 ) h ˜ ( r ˜ )] E( r 1 ,θ),
U( r ˜ , θ ˜ )=A( r ˜ )E( R ˜ (r), θ ˜ )+ 1 4π e jk{| n |[Zh( r 1 )+ h ˜ ( r ˜ )]P} [ I 1 E( r 1 , θ ˜ )+ I 1 E( r 1 , θ ˜ +π)]

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