Abstract

We investigate the fundamental spatial resolution of an x-ray pore optic as a function of the pore dimensions, the photon energy, and the focal length. We achieve this by calculating the shape of the focal spot, using diffraction integrals such that the half-energy width is determined. Quantitative results are presented for the X-Ray Evolving Universe Spectroscopy (XEUS) telescope, showing that a resolution of better than 2 arc sec half-energy width is possible by use of an optic with pore sizes of approximately 0.5 mm.

© 2005 Optical Society of America

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  1. A. N. Parmar, M. Arnaud, X. Barcons, J. Bleeker, G. Hasinger, H. Inoue, G. Palumbo, M. Turner, “Science with XEUS: the X-ray Evolving Universe Spectroscopy Mission,” in UV and Gamma-Ray Space Telescope Systems, G. Hasinger, M. J. Turner, eds., Proc. SPIE5488, 388–393 (2004).
    [CrossRef]
  2. M. Bavdaz, D. Lumb, A. Peacock, “XEUS mission reference design,” in UV and Gamma-Ray Space Telescope Systems, G. Hasinger, M. J. Turner, eds., Proc. SPIE5488, 829–836 (2004).
    [CrossRef]
  3. M. Beijersbergen, M. Bavdaz, A. Peacock, E. Tomaselli, R. Fairbend, J.-P. Boutot, S. Flyckt, A. Brunton, G. Price, G. Fraser, C. Herrmann, M. Krumrey, E. Ziegler, A. Freund, “High-resolution micropore x-ray optics produced with microchannel plate technology,” in Advances in X-Ray Optics, A. Freund, T. Ishikawa, A. Khounsary, D. Mancini, A. Michette, S. Oestreich, eds., Proc. SPIE4145, 188–192 (2001).
    [CrossRef]
  4. M. Beijersbergen, S. Kraft, R. Guenther, A. Mieremet, M. Collon, M. Bavdaz, D. Lumb, A. Peacock, “Silicon pore optics: novel lightweight high-resolution X-ray optics developed for XEUS,” in UV and Gamma-Ray Space Telescope Systems, G. Hasinger, M. J. Turner, eds., Proc. SPIE5488, 868–874 (2004).
    [CrossRef]
  5. H. Wolter, “Spiegelsysteme streifenden Einfalls als abbildende Optiken für Röntgenstrahlen,” Ann. Phys. 10, 94–114 (1952).
    [CrossRef]
  6. M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, 1999), pp. 412–516.
    [CrossRef]
  7. F. Pedrotti, L. Pedrotti, Introduction to Optics (Prentice-Hall, 1987), pp. 323–348.
  8. The exact solution for the prefactor z can be found by solution of the equation ∫02(sin x/x)2dx= ½∫0∞ (sin x/x)2dx. This equation can be rewritten9 into [cos(2z) − 1]/(2z) + Si(2z) = π/4, where Si is the sine-integral function, giving z = 0.850. This solution has to be divided by π to yield the relative position of the HEW with respect to the first zero points of the central peak of the diffraction pattern.
  9. I. S. Gradsteyn, I. M. Ryzhik, Table of Integrals, Series and Products, 6th ed. (Academic, 2000).

1952 (1)

H. Wolter, “Spiegelsysteme streifenden Einfalls als abbildende Optiken für Röntgenstrahlen,” Ann. Phys. 10, 94–114 (1952).
[CrossRef]

Arnaud, M.

A. N. Parmar, M. Arnaud, X. Barcons, J. Bleeker, G. Hasinger, H. Inoue, G. Palumbo, M. Turner, “Science with XEUS: the X-ray Evolving Universe Spectroscopy Mission,” in UV and Gamma-Ray Space Telescope Systems, G. Hasinger, M. J. Turner, eds., Proc. SPIE5488, 388–393 (2004).
[CrossRef]

Barcons, X.

A. N. Parmar, M. Arnaud, X. Barcons, J. Bleeker, G. Hasinger, H. Inoue, G. Palumbo, M. Turner, “Science with XEUS: the X-ray Evolving Universe Spectroscopy Mission,” in UV and Gamma-Ray Space Telescope Systems, G. Hasinger, M. J. Turner, eds., Proc. SPIE5488, 388–393 (2004).
[CrossRef]

Bavdaz, M.

M. Bavdaz, D. Lumb, A. Peacock, “XEUS mission reference design,” in UV and Gamma-Ray Space Telescope Systems, G. Hasinger, M. J. Turner, eds., Proc. SPIE5488, 829–836 (2004).
[CrossRef]

M. Beijersbergen, M. Bavdaz, A. Peacock, E. Tomaselli, R. Fairbend, J.-P. Boutot, S. Flyckt, A. Brunton, G. Price, G. Fraser, C. Herrmann, M. Krumrey, E. Ziegler, A. Freund, “High-resolution micropore x-ray optics produced with microchannel plate technology,” in Advances in X-Ray Optics, A. Freund, T. Ishikawa, A. Khounsary, D. Mancini, A. Michette, S. Oestreich, eds., Proc. SPIE4145, 188–192 (2001).
[CrossRef]

M. Beijersbergen, S. Kraft, R. Guenther, A. Mieremet, M. Collon, M. Bavdaz, D. Lumb, A. Peacock, “Silicon pore optics: novel lightweight high-resolution X-ray optics developed for XEUS,” in UV and Gamma-Ray Space Telescope Systems, G. Hasinger, M. J. Turner, eds., Proc. SPIE5488, 868–874 (2004).
[CrossRef]

Beijersbergen, M.

M. Beijersbergen, S. Kraft, R. Guenther, A. Mieremet, M. Collon, M. Bavdaz, D. Lumb, A. Peacock, “Silicon pore optics: novel lightweight high-resolution X-ray optics developed for XEUS,” in UV and Gamma-Ray Space Telescope Systems, G. Hasinger, M. J. Turner, eds., Proc. SPIE5488, 868–874 (2004).
[CrossRef]

M. Beijersbergen, M. Bavdaz, A. Peacock, E. Tomaselli, R. Fairbend, J.-P. Boutot, S. Flyckt, A. Brunton, G. Price, G. Fraser, C. Herrmann, M. Krumrey, E. Ziegler, A. Freund, “High-resolution micropore x-ray optics produced with microchannel plate technology,” in Advances in X-Ray Optics, A. Freund, T. Ishikawa, A. Khounsary, D. Mancini, A. Michette, S. Oestreich, eds., Proc. SPIE4145, 188–192 (2001).
[CrossRef]

Bleeker, J.

A. N. Parmar, M. Arnaud, X. Barcons, J. Bleeker, G. Hasinger, H. Inoue, G. Palumbo, M. Turner, “Science with XEUS: the X-ray Evolving Universe Spectroscopy Mission,” in UV and Gamma-Ray Space Telescope Systems, G. Hasinger, M. J. Turner, eds., Proc. SPIE5488, 388–393 (2004).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, 1999), pp. 412–516.
[CrossRef]

Boutot, J.-P.

M. Beijersbergen, M. Bavdaz, A. Peacock, E. Tomaselli, R. Fairbend, J.-P. Boutot, S. Flyckt, A. Brunton, G. Price, G. Fraser, C. Herrmann, M. Krumrey, E. Ziegler, A. Freund, “High-resolution micropore x-ray optics produced with microchannel plate technology,” in Advances in X-Ray Optics, A. Freund, T. Ishikawa, A. Khounsary, D. Mancini, A. Michette, S. Oestreich, eds., Proc. SPIE4145, 188–192 (2001).
[CrossRef]

Brunton, A.

M. Beijersbergen, M. Bavdaz, A. Peacock, E. Tomaselli, R. Fairbend, J.-P. Boutot, S. Flyckt, A. Brunton, G. Price, G. Fraser, C. Herrmann, M. Krumrey, E. Ziegler, A. Freund, “High-resolution micropore x-ray optics produced with microchannel plate technology,” in Advances in X-Ray Optics, A. Freund, T. Ishikawa, A. Khounsary, D. Mancini, A. Michette, S. Oestreich, eds., Proc. SPIE4145, 188–192 (2001).
[CrossRef]

Collon, M.

M. Beijersbergen, S. Kraft, R. Guenther, A. Mieremet, M. Collon, M. Bavdaz, D. Lumb, A. Peacock, “Silicon pore optics: novel lightweight high-resolution X-ray optics developed for XEUS,” in UV and Gamma-Ray Space Telescope Systems, G. Hasinger, M. J. Turner, eds., Proc. SPIE5488, 868–874 (2004).
[CrossRef]

Fairbend, R.

M. Beijersbergen, M. Bavdaz, A. Peacock, E. Tomaselli, R. Fairbend, J.-P. Boutot, S. Flyckt, A. Brunton, G. Price, G. Fraser, C. Herrmann, M. Krumrey, E. Ziegler, A. Freund, “High-resolution micropore x-ray optics produced with microchannel plate technology,” in Advances in X-Ray Optics, A. Freund, T. Ishikawa, A. Khounsary, D. Mancini, A. Michette, S. Oestreich, eds., Proc. SPIE4145, 188–192 (2001).
[CrossRef]

Flyckt, S.

M. Beijersbergen, M. Bavdaz, A. Peacock, E. Tomaselli, R. Fairbend, J.-P. Boutot, S. Flyckt, A. Brunton, G. Price, G. Fraser, C. Herrmann, M. Krumrey, E. Ziegler, A. Freund, “High-resolution micropore x-ray optics produced with microchannel plate technology,” in Advances in X-Ray Optics, A. Freund, T. Ishikawa, A. Khounsary, D. Mancini, A. Michette, S. Oestreich, eds., Proc. SPIE4145, 188–192 (2001).
[CrossRef]

Fraser, G.

M. Beijersbergen, M. Bavdaz, A. Peacock, E. Tomaselli, R. Fairbend, J.-P. Boutot, S. Flyckt, A. Brunton, G. Price, G. Fraser, C. Herrmann, M. Krumrey, E. Ziegler, A. Freund, “High-resolution micropore x-ray optics produced with microchannel plate technology,” in Advances in X-Ray Optics, A. Freund, T. Ishikawa, A. Khounsary, D. Mancini, A. Michette, S. Oestreich, eds., Proc. SPIE4145, 188–192 (2001).
[CrossRef]

Freund, A.

M. Beijersbergen, M. Bavdaz, A. Peacock, E. Tomaselli, R. Fairbend, J.-P. Boutot, S. Flyckt, A. Brunton, G. Price, G. Fraser, C. Herrmann, M. Krumrey, E. Ziegler, A. Freund, “High-resolution micropore x-ray optics produced with microchannel plate technology,” in Advances in X-Ray Optics, A. Freund, T. Ishikawa, A. Khounsary, D. Mancini, A. Michette, S. Oestreich, eds., Proc. SPIE4145, 188–192 (2001).
[CrossRef]

Gradsteyn, I. S.

I. S. Gradsteyn, I. M. Ryzhik, Table of Integrals, Series and Products, 6th ed. (Academic, 2000).

Guenther, R.

M. Beijersbergen, S. Kraft, R. Guenther, A. Mieremet, M. Collon, M. Bavdaz, D. Lumb, A. Peacock, “Silicon pore optics: novel lightweight high-resolution X-ray optics developed for XEUS,” in UV and Gamma-Ray Space Telescope Systems, G. Hasinger, M. J. Turner, eds., Proc. SPIE5488, 868–874 (2004).
[CrossRef]

Hasinger, G.

A. N. Parmar, M. Arnaud, X. Barcons, J. Bleeker, G. Hasinger, H. Inoue, G. Palumbo, M. Turner, “Science with XEUS: the X-ray Evolving Universe Spectroscopy Mission,” in UV and Gamma-Ray Space Telescope Systems, G. Hasinger, M. J. Turner, eds., Proc. SPIE5488, 388–393 (2004).
[CrossRef]

Herrmann, C.

M. Beijersbergen, M. Bavdaz, A. Peacock, E. Tomaselli, R. Fairbend, J.-P. Boutot, S. Flyckt, A. Brunton, G. Price, G. Fraser, C. Herrmann, M. Krumrey, E. Ziegler, A. Freund, “High-resolution micropore x-ray optics produced with microchannel plate technology,” in Advances in X-Ray Optics, A. Freund, T. Ishikawa, A. Khounsary, D. Mancini, A. Michette, S. Oestreich, eds., Proc. SPIE4145, 188–192 (2001).
[CrossRef]

Inoue, H.

A. N. Parmar, M. Arnaud, X. Barcons, J. Bleeker, G. Hasinger, H. Inoue, G. Palumbo, M. Turner, “Science with XEUS: the X-ray Evolving Universe Spectroscopy Mission,” in UV and Gamma-Ray Space Telescope Systems, G. Hasinger, M. J. Turner, eds., Proc. SPIE5488, 388–393 (2004).
[CrossRef]

Kraft, S.

M. Beijersbergen, S. Kraft, R. Guenther, A. Mieremet, M. Collon, M. Bavdaz, D. Lumb, A. Peacock, “Silicon pore optics: novel lightweight high-resolution X-ray optics developed for XEUS,” in UV and Gamma-Ray Space Telescope Systems, G. Hasinger, M. J. Turner, eds., Proc. SPIE5488, 868–874 (2004).
[CrossRef]

Krumrey, M.

M. Beijersbergen, M. Bavdaz, A. Peacock, E. Tomaselli, R. Fairbend, J.-P. Boutot, S. Flyckt, A. Brunton, G. Price, G. Fraser, C. Herrmann, M. Krumrey, E. Ziegler, A. Freund, “High-resolution micropore x-ray optics produced with microchannel plate technology,” in Advances in X-Ray Optics, A. Freund, T. Ishikawa, A. Khounsary, D. Mancini, A. Michette, S. Oestreich, eds., Proc. SPIE4145, 188–192 (2001).
[CrossRef]

Lumb, D.

M. Bavdaz, D. Lumb, A. Peacock, “XEUS mission reference design,” in UV and Gamma-Ray Space Telescope Systems, G. Hasinger, M. J. Turner, eds., Proc. SPIE5488, 829–836 (2004).
[CrossRef]

M. Beijersbergen, S. Kraft, R. Guenther, A. Mieremet, M. Collon, M. Bavdaz, D. Lumb, A. Peacock, “Silicon pore optics: novel lightweight high-resolution X-ray optics developed for XEUS,” in UV and Gamma-Ray Space Telescope Systems, G. Hasinger, M. J. Turner, eds., Proc. SPIE5488, 868–874 (2004).
[CrossRef]

Mieremet, A.

M. Beijersbergen, S. Kraft, R. Guenther, A. Mieremet, M. Collon, M. Bavdaz, D. Lumb, A. Peacock, “Silicon pore optics: novel lightweight high-resolution X-ray optics developed for XEUS,” in UV and Gamma-Ray Space Telescope Systems, G. Hasinger, M. J. Turner, eds., Proc. SPIE5488, 868–874 (2004).
[CrossRef]

Palumbo, G.

A. N. Parmar, M. Arnaud, X. Barcons, J. Bleeker, G. Hasinger, H. Inoue, G. Palumbo, M. Turner, “Science with XEUS: the X-ray Evolving Universe Spectroscopy Mission,” in UV and Gamma-Ray Space Telescope Systems, G. Hasinger, M. J. Turner, eds., Proc. SPIE5488, 388–393 (2004).
[CrossRef]

Parmar, A. N.

A. N. Parmar, M. Arnaud, X. Barcons, J. Bleeker, G. Hasinger, H. Inoue, G. Palumbo, M. Turner, “Science with XEUS: the X-ray Evolving Universe Spectroscopy Mission,” in UV and Gamma-Ray Space Telescope Systems, G. Hasinger, M. J. Turner, eds., Proc. SPIE5488, 388–393 (2004).
[CrossRef]

Peacock, A.

M. Bavdaz, D. Lumb, A. Peacock, “XEUS mission reference design,” in UV and Gamma-Ray Space Telescope Systems, G. Hasinger, M. J. Turner, eds., Proc. SPIE5488, 829–836 (2004).
[CrossRef]

M. Beijersbergen, M. Bavdaz, A. Peacock, E. Tomaselli, R. Fairbend, J.-P. Boutot, S. Flyckt, A. Brunton, G. Price, G. Fraser, C. Herrmann, M. Krumrey, E. Ziegler, A. Freund, “High-resolution micropore x-ray optics produced with microchannel plate technology,” in Advances in X-Ray Optics, A. Freund, T. Ishikawa, A. Khounsary, D. Mancini, A. Michette, S. Oestreich, eds., Proc. SPIE4145, 188–192 (2001).
[CrossRef]

M. Beijersbergen, S. Kraft, R. Guenther, A. Mieremet, M. Collon, M. Bavdaz, D. Lumb, A. Peacock, “Silicon pore optics: novel lightweight high-resolution X-ray optics developed for XEUS,” in UV and Gamma-Ray Space Telescope Systems, G. Hasinger, M. J. Turner, eds., Proc. SPIE5488, 868–874 (2004).
[CrossRef]

Pedrotti, F.

F. Pedrotti, L. Pedrotti, Introduction to Optics (Prentice-Hall, 1987), pp. 323–348.

Pedrotti, L.

F. Pedrotti, L. Pedrotti, Introduction to Optics (Prentice-Hall, 1987), pp. 323–348.

Price, G.

M. Beijersbergen, M. Bavdaz, A. Peacock, E. Tomaselli, R. Fairbend, J.-P. Boutot, S. Flyckt, A. Brunton, G. Price, G. Fraser, C. Herrmann, M. Krumrey, E. Ziegler, A. Freund, “High-resolution micropore x-ray optics produced with microchannel plate technology,” in Advances in X-Ray Optics, A. Freund, T. Ishikawa, A. Khounsary, D. Mancini, A. Michette, S. Oestreich, eds., Proc. SPIE4145, 188–192 (2001).
[CrossRef]

Ryzhik, I. M.

I. S. Gradsteyn, I. M. Ryzhik, Table of Integrals, Series and Products, 6th ed. (Academic, 2000).

Tomaselli, E.

M. Beijersbergen, M. Bavdaz, A. Peacock, E. Tomaselli, R. Fairbend, J.-P. Boutot, S. Flyckt, A. Brunton, G. Price, G. Fraser, C. Herrmann, M. Krumrey, E. Ziegler, A. Freund, “High-resolution micropore x-ray optics produced with microchannel plate technology,” in Advances in X-Ray Optics, A. Freund, T. Ishikawa, A. Khounsary, D. Mancini, A. Michette, S. Oestreich, eds., Proc. SPIE4145, 188–192 (2001).
[CrossRef]

Turner, M.

A. N. Parmar, M. Arnaud, X. Barcons, J. Bleeker, G. Hasinger, H. Inoue, G. Palumbo, M. Turner, “Science with XEUS: the X-ray Evolving Universe Spectroscopy Mission,” in UV and Gamma-Ray Space Telescope Systems, G. Hasinger, M. J. Turner, eds., Proc. SPIE5488, 388–393 (2004).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, 1999), pp. 412–516.
[CrossRef]

Wolter, H.

H. Wolter, “Spiegelsysteme streifenden Einfalls als abbildende Optiken für Röntgenstrahlen,” Ann. Phys. 10, 94–114 (1952).
[CrossRef]

Ziegler, E.

M. Beijersbergen, M. Bavdaz, A. Peacock, E. Tomaselli, R. Fairbend, J.-P. Boutot, S. Flyckt, A. Brunton, G. Price, G. Fraser, C. Herrmann, M. Krumrey, E. Ziegler, A. Freund, “High-resolution micropore x-ray optics produced with microchannel plate technology,” in Advances in X-Ray Optics, A. Freund, T. Ishikawa, A. Khounsary, D. Mancini, A. Michette, S. Oestreich, eds., Proc. SPIE4145, 188–192 (2001).
[CrossRef]

Ann. Phys. (1)

H. Wolter, “Spiegelsysteme streifenden Einfalls als abbildende Optiken für Röntgenstrahlen,” Ann. Phys. 10, 94–114 (1952).
[CrossRef]

Other (8)

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, 1999), pp. 412–516.
[CrossRef]

F. Pedrotti, L. Pedrotti, Introduction to Optics (Prentice-Hall, 1987), pp. 323–348.

The exact solution for the prefactor z can be found by solution of the equation ∫02(sin x/x)2dx= ½∫0∞ (sin x/x)2dx. This equation can be rewritten9 into [cos(2z) − 1]/(2z) + Si(2z) = π/4, where Si is the sine-integral function, giving z = 0.850. This solution has to be divided by π to yield the relative position of the HEW with respect to the first zero points of the central peak of the diffraction pattern.

I. S. Gradsteyn, I. M. Ryzhik, Table of Integrals, Series and Products, 6th ed. (Academic, 2000).

A. N. Parmar, M. Arnaud, X. Barcons, J. Bleeker, G. Hasinger, H. Inoue, G. Palumbo, M. Turner, “Science with XEUS: the X-ray Evolving Universe Spectroscopy Mission,” in UV and Gamma-Ray Space Telescope Systems, G. Hasinger, M. J. Turner, eds., Proc. SPIE5488, 388–393 (2004).
[CrossRef]

M. Bavdaz, D. Lumb, A. Peacock, “XEUS mission reference design,” in UV and Gamma-Ray Space Telescope Systems, G. Hasinger, M. J. Turner, eds., Proc. SPIE5488, 829–836 (2004).
[CrossRef]

M. Beijersbergen, M. Bavdaz, A. Peacock, E. Tomaselli, R. Fairbend, J.-P. Boutot, S. Flyckt, A. Brunton, G. Price, G. Fraser, C. Herrmann, M. Krumrey, E. Ziegler, A. Freund, “High-resolution micropore x-ray optics produced with microchannel plate technology,” in Advances in X-Ray Optics, A. Freund, T. Ishikawa, A. Khounsary, D. Mancini, A. Michette, S. Oestreich, eds., Proc. SPIE4145, 188–192 (2001).
[CrossRef]

M. Beijersbergen, S. Kraft, R. Guenther, A. Mieremet, M. Collon, M. Bavdaz, D. Lumb, A. Peacock, “Silicon pore optics: novel lightweight high-resolution X-ray optics developed for XEUS,” in UV and Gamma-Ray Space Telescope Systems, G. Hasinger, M. J. Turner, eds., Proc. SPIE5488, 868–874 (2004).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Top, a Wolter-I optic: incoming x-ray photons are reflected under grazing incidence from a parabolic and then a hyperbolic surface of revolution, traditionally made from a number of nested shells: Bottom, a Wolter-I optic, or an approximation of it with conical surfaces, can be implemented by use of a large number of rectangular pores in concentric circles.

Fig. 2
Fig. 2

(a) The XEUS pore resembles (b) a rectangle when the dimensions of the pore are small compared with the radial position of the pore.

Fig. 3
Fig. 3

HEW size in the radial direction of a beam of a pore with a height of 500 µm for an energy of 0.5 keVas a function of distance behind the pore. The lines are drawn to guide the eye.

Fig. 4
Fig. 4

Shape of beam at the optic and at the focal plane (detector). Although the HEW in the focal plane is smaller, the beam itself is not.

Fig. 5
Fig. 5

HEW in the radial direction in the focal plane 50 m behind the optic according to near-field, far-field, and Fresnel–Kirchhoff diffraction for three energies E (0.2, 0.5, and 1.0 keV), as a function of pore height. The lines are drawn to guide the eye.

Fig. 6
Fig. 6

Similar to the middle of Fig. 5, except that the focal plane is located 25 m behind the optic. The lines are drawn to guide the eye.

Fig. 7
Fig. 7

Normalized diffraction pattern for Nϕ = 1, 5, 60 are shown in the upper graph. The corresponding integrated intensities (normalized) are in the lower graph, which shows that the HEW decreases with Nϕ.

Fig. 8
Fig. 8

Two possible optic configurations: (a) annular optic, (b) sector optic.

Fig. 9
Fig. 9

Top, contour plots and bottom, horizontal and vertical cross sections of the focal spot for a single pore (left), a sector optic (center), and an annular optic (right).

Tables (2)

Tables Icon

Table 1 Minimum Pore Width To Ensure That Diffraction in the Azimuthal Direction Leads to a HEW of <2 arc sec

Tables Icon

Table 2 Optimum HEW and Pore Height for Different Focal Lengths and Energies As Well As the Allowed Range in Pore Height to Achieve a Resolution Better Than 2 arc sec

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

E P F ( θ ) exp [ i k ( s + s ) ] s s d A ,
F ( θ ) = 1 + cos ( θ ) 2 ,
E P F ( θ ) exp ( i k s ) s d A .
s ( x , y ) = [ f 2 + ( x x P ) 2 + ( y y P ) 2 ] 1 / 2 f + ( x x P ) 2 2 f + ( y y P ) 2 2 f f + s x + s y ,
E P exp [ i k ( s x + s y ) ] d x d y = exp ( i k s x ) d x exp ( i k s y ) d y .
w = 2 λ f a ,
E 50 = 0.270 w f = 0.541 λ a .
L b b λ ,
| E P | 2 | F ( θ ) exp ( i k s ) s d y | 2 = [ F ( θ ) cos k s s d y ] 2 + [ F ( θ ) sin k s s d y ] 2 .
Δ P = ( r + Δ r ) 2 + f 2 r 2 + f 2 r Δ r r 2 + f 2 .
Δ P < λ 2 Δ λ .
I ( y ) = I 0 ( sin α α ) 2 ( sin N ϕ α sin α ) 2 ,
α = π y λ f a , α = π y λ f a ,

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