## Boundary diffraction wave integrals for diffraction modeling of external occulters |

Optics Express, Vol. 20, Issue 14, pp. 15196-15208 (2012)

http://dx.doi.org/10.1364/OE.20.015196

Acrobat PDF (884 KB)

### Abstract

An occulter is a large diffracting screen which may be flown in conjunction with a telescope to image extrasolar planets. The edge is shaped to minimize the diffracted light in a region beyond the occulter, and a telescope may be placed in this dark shadow to view an extrasolar system with the starlight removed. Errors in position, orientation, and shape of the occulter will diffract additional light into this region, and a challenge of modeling an occulter system is to accurately and quickly model these effects. We present a fast method for the calculation of electric fields following an occulter, based on the concept of the boundary diffraction wave: the 2D structure of the occulter is reduced to a 1D edge integral which directly incorporates the occulter shape, and which can be easily adjusted to include changes in occulter position and shape, as well as the effects of sources—such as exoplanets—which arrive off-axis to the occulter. The structure of a typical implementation of the algorithm is included.

© 2012 OSA

## 1. Introduction

*contrast*) and angular resolution. An Earth-twin which orbits a Sun-twin at 1AU emits 10

^{10}times less flux than the parent star in the visible band; if the system was located at 10 parsecs from Earth, the angular separation of the two objects would be 100mas, which for most proposed space telescopes puts the separation under 4

*λ*/

*D*. Specialized methods are required to remove this flux at these small working angles.

*occulter*, a spacecraft with a shaped edge flown in front of the telescope to block the starlight before it arrives at the telescope. The occulter size (tens of meters) and distance (tens of thousands of kilometers) are chosen so the angular extent of the occulter is smaller than some desired angle, on the order of 100 milliarcseconds, so exoplanets in orbit about the star will still be visible. The edge is shaped to control diffraction, with the form chosen to suppress the light to a factor of 10

^{10}across the telescope aperture and over a wide spectral passband.

1. N. J. Kasdin, D. N. Spergel, R. J. Vanderbei, D. Lisman, S. Shaklan, M. Thomson, P. Walkemeyer, V. Bach, E. Oakes, E. Cady, S. Martin, L. Marchen, B. Macintosh, R. E. Rudd, J. Mikula, and D. Lynch, “Advancing technology for starlight suppression via an external occulter,” Proc. SPIE **8151**, 81510J (2011). [CrossRef]

2. S. B. Shaklan, M. C. Noecker, T. Glassman, A. S. Lo, P. J. Dumont, N. J. Kasdin, E. J. Cady, R. Vanderbei, and P. R. Lawson, “Error budgeting and tolerancing of starshades for exoplanet detection,” Proc. SPIE **7731**, 77312G (2010). [CrossRef]

3. T. Glassman, A. Johnson, A. Lo, D. Dailey, H. Shelton, and J. Vogrin, “Error analysis on the NWO starshade,” Proc. SPIE **7731**, 773150 (2010). [CrossRef]

^{5}or greater in amplitude—drives the accuracy of the propagator used for the models, while the modeling of time-varying thermal shape deformations and closed-loop formation flying in broadband light drive the speed of the calculation. Some of the most in-depth simulated observations can take from hours to days to run, and the ability to quickly and accurately model propagation past an occulter is thus a major enabling factor in the characterization of occulter system performance. Section 2 gives an overview of the current techniques; the new technique presented in this work is described in Sections 3 and 3.1, and a suggested implementation scheme, along with a pseudocode overview of the algorithm and some computational results, are given in Section 4.

## 2. Current occulter modeling techniques

*N*identical tapering structures (

_{p}*petals*) around the edge, giving the whole structure the general appearance of a flower. (An example is shown in Fig. 1.) This shape is a result of their provenance as (0,1)-valued approximations to apodizers; the general method to design an occulter is to determine a smooth apodization profile

*A*(

*r*) which can provide the necessary starlight suppression, and then convert it into a binary shape with a sufficient number of petals so that performance is not undermined. The actual form of

*A*(

*r*) can be determined by optimization [4

4. R. J. Vanderbei, E. J. Cady, and N. J. Kasdin, “Optimal occulter design for finding extrasolar planets,” Astrophys. J. **665**, 794–798 (2007). [CrossRef]

5. E. Cady, L. Pueyo, R. Soummer, and N. J. Kasdin, “Performance of hybrid occulters using apodized pupil Lyot coronagraphy,” Proc. SPIE **7010**, 70101X (2008). [CrossRef]

6. C. J. Copi and G. D. Starkman, “The Big Occulting Steerable Satellite [BOSS],” Astrophys. J. **532**, 581–592 (2000). [CrossRef]

7. W. Cash, “Detection of earth-like planets around nearby stars using a petal-shaped occulter,” Nature **442**, 51–53 (2006). [CrossRef] [PubMed]

*r*,

*θ*) being polar coordinates in the plane of the occulter. Each of the [...,...] is a set of points which defines the outline of a single petal; ∪ denotes the union of these sets, which sets the boundary for the entire occulter. (We denote this boundary as

*∂*Ω, as it will be used later.)

*z*= 5 × 10

^{7}m; the wavelength

*λ*= 5 × 10

^{−7}m; the occulter radius

*R*= 25m; and the maximum excursions in the plane of the telescope |

*ξ*| and |

*η*| ≤ 3m. These values place the occulter well within the Fresnel regime; that is, the region in which it is valid to approximate the exponent in the propagation integral with the first two terms of a power series (the Fresnel approximation). This approximation is suitable when [8].

*A*and wavelength

*λ*is normally-incident on the occulter, as would be the case for an occulter properly aligned with a target star, the electric field at a distance

*z*past the occulter would be written with the Fresnel approximation and Babinet’s principle as

*ξ*and

*η*are Cartesian coordinates at the downstream location. The “1–” term before the integral results from the use of Babinet’s principle. Errors in (for example) occulter shape will modify Ω, but Eq. (4) will hold regardless. Computing the integral for points in the plane of a telescope aperture is the key step in modeling the performance of an occulter.

10. R. Soummer, L. Pueyo, A. Sivaramakrishnan, and R. J. Vanderbei, “Fast computation of Lyot-style coronagraph propagation,” Opt. Express **15**(24), 15935–15951, (2007). [CrossRef] [PubMed]

2. S. B. Shaklan, M. C. Noecker, T. Glassman, A. S. Lo, P. J. Dumont, N. J. Kasdin, E. J. Cady, R. Vanderbei, and P. R. Lawson, “Error budgeting and tolerancing of starshades for exoplanet detection,” Proc. SPIE **7731**, 77312G (2010). [CrossRef]

^{6}× 10

^{6}or more to capture the full shape of the profile, not including array padding.

*N*-fold symmetry in the azimuthal direction, and most errors introduce small perturbations to that. Taking advantage of the symmetry leads to three main approaches to modeling propagation past an external occulter:

_{p}- Integrals in
*r*: The Bessel function expansion [4] does the integral over4. R. J. Vanderbei, E. J. Cady, and N. J. Kasdin, “Optimal occulter design for finding extrasolar planets,” Astrophys. J.

**665**, 794–798 (2007). [CrossRef]*θ*explicitly, reducing the 2D integral to a series of single integrals in*r*whose contributions fall off exponentially fast near the optical axis, which makes computation extremely quick in this region. The series can be modified to incorporate some shape errors [11] at the expense of slower convergence. Unfortunately, it is slower at locations far from the optical axis, and requires a good deal of effort to incorporate many types of errors, such as occulter tilt, into the model. Aside from the propagation modeling, the first, radially-independent term in this series is used in optimization approaches [4**665**, 794–798 (2007). [CrossRef]] to design occulter shapes.5. E. Cady, L. Pueyo, R. Soummer, and N. J. Kasdin, “Performance of hybrid occulters using apodized pupil Lyot coronagraphy,” Proc. SPIE

**7010**, 70101X (2008). [CrossRef] - Integrals in
*θ*: The Dubra-Ferrari integral [12] starts one step back from the Fresnel form of Eq. (4), at the Raleigh-Sommerfeld diffraction integral, and evaluates the integral over12. A. Dubra and J. A. Ferrari, “Diffracted field by an arbitrary aperture,” Am. J. of Phys.

**67**, 87–92 (1999). [CrossRef]*r*to produce a pair of single integrals in angular coordinates, one of which holds for points inside the geometric extent of the starshade, and one which holds outside. This approach can incorporate most errors straightforwardly and spends the same amount of time to determine the field any point in the telescope aperture plane. Because of the manner in which the integrals are segmented, however, the integrands are more difficult to determine correctly for points physically outside the extent of the starshade. - Perturbations to a nominal shape: The slit approach [13] assumes the electric field for an occulter with no errors has been calculated by one of the above methods, and approximates the difference between the shape with and without errors as a series of long, thin boxes whose Fresnel transforms can be calculated quickly and exactly. This method is very fast but necessarily approximate.
13. P. Dumont, S. Shaklan, E. Cady, J. Kasdin, and R. Vanderbei, “Analysis of external occulters in the presence of defects,” Proc. SPIE

**7440**, 744008 (2009). [CrossRef]

## 3. The boundary diffraction wave formulation

14. K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz theory of the boundary diffraction wave part I,” J. Opt. Soc. Am **52**, 615–622 (1962). [CrossRef]

15. K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz theory of the boundary diffraction wave part II,” J. Opt. Soc. Am **52**, 626–636 (1962). [CrossRef]

14. K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz theory of the boundary diffraction wave part I,” J. Opt. Soc. Am **52**, 615–622 (1962). [CrossRef]

*e.g.*[9]) for the diffraction integral from the aperture. Consider a volume of space, bounded by a surface

*S*; the field at any point

*P*within the volume may be expressed as an integral over all surface points

*Q*on

*S*: Here we let

*s*be the distance between

*P*and

*Q*,

*n*be a vector normal to

*S*at point

*Q*, and ∇

*is the gradient evaluated at point*

_{Q}*Q*. In the case of the occulter, the volume is the space

*z*≥ 0, and

*S*stretches over the

*z*= 0 plane, as well as having a component at infinity which is assumed to vanish. (Showing that this term vanishes requires some minor additional assumptions; see Sec. 8.3.2 of [9]).

14. K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz theory of the boundary diffraction wave part I,” J. Opt. Soc. Am **52**, 615–622 (1962). [CrossRef]

**V**of this form, there exists a vector potential

**W**, such that

*V*= ∇ ×

**W**, and for plane waves incident on an aperture, that associated vector potential is Here

**r**is the vector from the origin to

*P*;

*Q*is a point within the aperture on

*S*, and

**r**′ is the vector from the origin to

*Q. k*= 2

*π*/

*λ*is the wavenumber and

*λ*is the wavelength under consideration. The various

*s*-variables become and we assume the form of the plane wave in the vector direction

**p**to be with amplitude

*A*. A coordinate diagram is shown in Fig. 2.

*S*with boundary

*∂*Ω. We recall here that

*S*is the entire bounding surface of the half-plane

*z*≥ 0, including the

*z*= 0 plane; Ω is the section of it on which the occulter lies. (We are not considering the occulter case at the moment, but its complement; we will return to the occulter shortly.) In this case, the electric field from Eq. (7) across the aperture is: Our approach is to reduce this integral to a line integral, using Stokes’ Theorem. To do this, we consider singularities of the vector potential.

**W**will have singularities somewhere on

*S*for a given

*P*, as otherwise the field in the half-space past the aperture will be zero. In the specific case of the plane wave, the lone singularity occurs at 1 +

*ŝ·*

**p**= 0, which occurs only when

*ŝ*is parallel to the propagation direction of the plane wave; thus, the singularity will fall in the aperture

*only*for points

*P*which fall in the beam as dictated by geometric optics. (Fig. 3 shows an example of this.) Miyamoto and Wolf [15

15. K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz theory of the boundary diffraction wave part II,” J. Opt. Soc. Am **52**, 626–636 (1962). [CrossRef]

*P*, depending on the singularity location: where

*U*

^{(B)}(

*P*) is the counterclockwise line integral about the edge of Ω: Here,

*ℓ*is a unit vector in the direction tangent to the edge at any point, and d

*ℓ*is a differential element of the boundary.

*U*(

*P*) satisfies or

*U*

^{(G)}(

*P*): This equation is the general form of the occulter field, in the boundary-diffraction-wave formulation.

### 3.1. Vector potentials for occulter propagation

*P*as

**r**= (

*ξ*,

*η*,

*z*);

**r**′, the vector from the origin to

*Q*, is defined as

**r**′ = (

*x*,

*y*, 0). (See again Fig. 2.) We can write the

*s*-variables explicitly as

*ψ*

_{1}off-axis and

*ψ*

_{2}from the x-axis will have Before the expression for the vector potential becomes too complicated, we will define three intermediate variables (

*f*,

*g*, and

*h*) which we can substitute into Eq. (19); these choices will greatly simplify subsequent computations. Substituting these variables gives:

*ψ*

_{1}≪ 1, the usual case for objects in an exoplanetary system, we can note that and so the potential becomes This form proves to be particularly useful, as (

*f*

^{2}+

*g*

^{2}) and

**v**·

*ℓ*can be calculated without loss of precision, as neither will be of order

*z*for the small

*ψ*

_{1}case. A typical occulter might have

*z*= 5×10

^{7}m,

*R*= 25m, and |

*ξ*| and |

*η*| ≤ 3m, and

*k*≈ 10

^{7}m

^{−1}; a representative exoplanet target for this occulter might have

*ψ*

_{1}= 5 × 10

^{−7}rad. We note that exponential terms of order

*kz*have been separated from smaller terms; these should be calculated independently, as

*kz*∼ 10

^{14}, and combining terms will lose precision in evaluating the exponent.

## 4. Efficient implementation

*N*points representing the edge of the occulter which form a simple closed curve, as well as a list of

*M*downstream points and a list of

*L*wavelengths at which the field is to be determined. We also assume that the edge is sufficiently well-sampled that the section of the edge between explicitly-specified points is well-modeled by a linear segment, as this allows us to use the midpoint rule for numerically approximating the integral. Given the slowly-changing shape over the majority of the petal, this is a good assumption; small regions such as petal tips may be specified more finely.

*ξ*

_{0},

*η*

_{0}); this can be done efficiently with an

*O*(

*N*) routine such as polywind in Numerical Recipes [16], and holds regardless of the complexity of the occulter shape. This approach also speeds up multiband calculations, as the geometric extent of the shadow determined this way is wavelength-independent. Lateral errors in the occulter-telescope position may be included by adding a Δ

*ξ*and Δ

*η*to all points at the telescope plane, and that should be done first. Note that in the case of an off-axis source, the geometric shadow is shifted laterally as well; in this case, the winding number should be calculated around (

*ξ*

_{0}−

*z*sin

*ψ*

_{1}cos

*ψ*

_{2},

*η*

_{0}−

*z*sin

*ψ*

_{1}sin

*ψ*

_{2}). We then have: In some cases—for example, when the telescope aperture remains close to the center of the occulter and the perturbations to the ideal shape of the occulter are small—the winding number may be able to be determined independently. This determination should be done if possible, as even an efficient algorithm can increase runtime significantly for boundaries containing a large number of points.

*N*line segments running counterclockwise around the occulter edge, we derive the midpoint of line segment

*j*going from (

*x*,

_{j}*y*, 0) to (

_{j}*x*

_{j}_{+1},

*y*

_{j}_{+1}, 0): and the vector pointing along the segment with (

*x*

_{j+1},

*y*

_{j+1}) equal to (

*x*

_{1},

*y*

_{1}) when

*j*=

*N*. This derivation need only be done once at the beginning, as it holds for all

*M*points and

*L*wavelengths. (We note that, while the midpoint rule has been used for the quadrature, this is certainly not a requirement, and higher order methods could be used; Eq. (39) will still hold. Using a higher-order method may come at the cost of runtime.)

*f*,

*g*, and

*h*for all

*N*segments following Eq. (29): and we then need only two intermediate terms: to give us: Note that all of the steps prior to Eq. (49) are wavelength-independent; different values of

*k*may be iterated over in the final step. It is advisable to keep terms of order

*kz*(e.g.

*kz*cos

*ψ*

_{1}or

*kh*) separate when evaluating to maintain numerical precision. A typical implementation might take the form shown in Algorithm 1. The for-loops over

*j*in particular are well-suited to vectorization where this is supported.

### 4.1. Computational results

*θ*centered about (

*ξ*

_{0},

*η*

_{0}), and so the angular distance between occulter edge points must be rederived for each point at the telescope aperture. This calculation turns out to be one of the major sources of overhead during the DF calculation.

^{−15}.

## 5. Conclusion

## Acknowledgments

## References and links

1. | N. J. Kasdin, D. N. Spergel, R. J. Vanderbei, D. Lisman, S. Shaklan, M. Thomson, P. Walkemeyer, V. Bach, E. Oakes, E. Cady, S. Martin, L. Marchen, B. Macintosh, R. E. Rudd, J. Mikula, and D. Lynch, “Advancing technology for starlight suppression via an external occulter,” Proc. SPIE |

2. | S. B. Shaklan, M. C. Noecker, T. Glassman, A. S. Lo, P. J. Dumont, N. J. Kasdin, E. J. Cady, R. Vanderbei, and P. R. Lawson, “Error budgeting and tolerancing of starshades for exoplanet detection,” Proc. SPIE |

3. | T. Glassman, A. Johnson, A. Lo, D. Dailey, H. Shelton, and J. Vogrin, “Error analysis on the NWO starshade,” Proc. SPIE |

4. | R. J. Vanderbei, E. J. Cady, and N. J. Kasdin, “Optimal occulter design for finding extrasolar planets,” Astrophys. J. |

5. | E. Cady, L. Pueyo, R. Soummer, and N. J. Kasdin, “Performance of hybrid occulters using apodized pupil Lyot coronagraphy,” Proc. SPIE |

6. | C. J. Copi and G. D. Starkman, “The Big Occulting Steerable Satellite [BOSS],” Astrophys. J. |

7. | W. Cash, “Detection of earth-like planets around nearby stars using a petal-shaped occulter,” Nature |

8. | J. W. Goodman, |

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

10. | R. Soummer, L. Pueyo, A. Sivaramakrishnan, and R. J. Vanderbei, “Fast computation of Lyot-style coronagraph propagation,” Opt. Express |

11. | E. Cady, “Design, tolerancing, and experimental verification of occulters for finding extrasolar planets,” PhD thesis, Princeton University, 2010. |

12. | A. Dubra and J. A. Ferrari, “Diffracted field by an arbitrary aperture,” Am. J. of Phys. |

13. | P. Dumont, S. Shaklan, E. Cady, J. Kasdin, and R. Vanderbei, “Analysis of external occulters in the presence of defects,” Proc. SPIE |

14. | K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz theory of the boundary diffraction wave part I,” J. Opt. Soc. Am |

15. | K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz theory of the boundary diffraction wave part II,” J. Opt. Soc. Am |

16. | W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, |

**OCIS Codes**

(050.1940) Diffraction and gratings : Diffraction

(050.1970) Diffraction and gratings : Diffractive optics

(350.6090) Other areas of optics : Space optics

(120.6085) Instrumentation, measurement, and metrology : Space instrumentation

(070.7345) Fourier optics and signal processing : Wave propagation

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: May 9, 2012

Revised Manuscript: June 1, 2012

Manuscript Accepted: June 5, 2012

Published: June 21, 2012

**Citation**

Eric Cady, "Boundary diffraction wave integrals for diffraction modeling of external occulters," Opt. Express **20**, 15196-15208 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-14-15196

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### References

- N. J. Kasdin, D. N. Spergel, R. J. Vanderbei, D. Lisman, S. Shaklan, M. Thomson, P. Walkemeyer, V. Bach, E. Oakes, E. Cady, S. Martin, L. Marchen, B. Macintosh, R. E. Rudd, J. Mikula, and D. Lynch, “Advancing technology for starlight suppression via an external occulter,” Proc. SPIE8151, 81510J (2011). [CrossRef]
- S. B. Shaklan, M. C. Noecker, T. Glassman, A. S. Lo, P. J. Dumont, N. J. Kasdin, E. J. Cady, R. Vanderbei, and P. R. Lawson, “Error budgeting and tolerancing of starshades for exoplanet detection,” Proc. SPIE7731, 77312G (2010). [CrossRef]
- T. Glassman, A. Johnson, A. Lo, D. Dailey, H. Shelton, and J. Vogrin, “Error analysis on the NWO starshade,” Proc. SPIE7731, 773150 (2010). [CrossRef]
- R. J. Vanderbei, E. J. Cady, and N. J. Kasdin, “Optimal occulter design for finding extrasolar planets,” Astrophys. J.665, 794–798 (2007). [CrossRef]
- E. Cady, L. Pueyo, R. Soummer, and N. J. Kasdin, “Performance of hybrid occulters using apodized pupil Lyot coronagraphy,” Proc. SPIE7010, 70101X (2008). [CrossRef]
- C. J. Copi and G. D. Starkman, “The Big Occulting Steerable Satellite [BOSS],” Astrophys. J.532, 581–592 (2000). [CrossRef]
- W. Cash, “Detection of earth-like planets around nearby stars using a petal-shaped occulter,” Nature442, 51–53 (2006). [CrossRef] [PubMed]
- J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
- M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).
- R. Soummer, L. Pueyo, A. Sivaramakrishnan, and R. J. Vanderbei, “Fast computation of Lyot-style coronagraph propagation,” Opt. Express15(24), 15935–15951, (2007). [CrossRef] [PubMed]
- E. Cady, “Design, tolerancing, and experimental verification of occulters for finding extrasolar planets,” PhD thesis, Princeton University, 2010.
- A. Dubra and J. A. Ferrari, “Diffracted field by an arbitrary aperture,” Am. J. of Phys.67, 87–92 (1999). [CrossRef]
- P. Dumont, S. Shaklan, E. Cady, J. Kasdin, and R. Vanderbei, “Analysis of external occulters in the presence of defects,” Proc. SPIE7440, 744008 (2009). [CrossRef]
- K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz theory of the boundary diffraction wave part I,” J. Opt. Soc. Am52, 615–622 (1962). [CrossRef]
- K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz theory of the boundary diffraction wave part II,” J. Opt. Soc. Am52, 626–636 (1962). [CrossRef]
- W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes. The Art of Scientific Computing (Cambridge University Press, 2007).

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