Abstract

We report on interferometric characterization of a deep parabolic mirror with a depth of more than five times its focal length. The interferometer is of Fizeau type; its core consists of the mirror itself, a spherical null element, and a reference flat. Because of the extreme solid angle produced by the paraboloid, the alignment of the setup appears to be very critical and needs auxiliary systems for control. Aberrations caused by misalignments are removed via fitting of suitable functionals provided by means of ray tracing simulations. It turns out that the usual misalignment approximations fail under these extreme conditions.

© 2008 Optical Society of America

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  1. N. Lindlein, R. Maiwald, H. Konermann, M. Sondermann, U. Peschel, and G. Leuchs, “A new 4π-geometry optimized for focussing onto an atom with a dipole-like radiation pattern,” Laser Phys. 17, 927-934 (2007).
    [CrossRef]
  2. M. Sondermann, R. Maiwald, H. Konermann, N. Lindlein, U. Peschel, and G. Leuchs, “Design of a mode converter for efficient light-atom coupling in free space,” Appl. Phys. B 89, 489-492 (2007).
  3. S. Hell and E. H. K. Stelzer, “Properties of a 4Pi-confocal fluorescence microscope,” J. Opt. Soc. Am. A 9, 2159-2166 (1992).
  4. J. Stadler, C. Stanciu, C. Stupperich, and A. J. Meixner, “Tighter focusing with a parabolic mirror,” Opt. Lett. 33, 681-683 (2008).
    [CrossRef]
  5. W. T. Welford, “Aplanatism and Isoplanatism,” in Progress in Optics, E. Wolf, ed. (Elsivier, 1976), Vol. XIII, pp. 267-292.
  6. ARCoptix; product description: “Radial Polarization Converter”.
  7. M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge University, 1980).
  8. E. W. Young, “Optimal removal of all mislocation effects in interferometric tests,” Proc. SPIE 661, 116-124(1986).
  9. J. Schwider, R. Burow, K.-E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, “Digital wavefront measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421-3432 (1983).
  10. G. Schulz and J. Schwider, “Interferometric testing of smooth surfaces,” in Progress in Optics, E. Wolf, ed. (Elsivier, 1976), Vol. XIII, p. 93.
  11. J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf, ed. (Elsivier, 1990), Vol. XXVIII, pp. 271-359.
  12. A. E. Jensen, “Absolute calibration method for laser Twyman-Green wave front testing interferometers,” J. Opt. Soc. Am. 63, 1313A (1973).
  13. K.-E. Elßner, J. Grzanna, and G. Schulz, “Interferentielle absolutprüfung von sphärizitätsnormalen,” Opt. Acta 27, 563-580 (1980).
  14. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693-2703 (1974).
  15. K.-E. Elssner, R. Burow, J. Grzanna, and R. Spolaczyk, “Absolute sphericity measurement,” Appl. Opt. 28, 4649-4661 (1989);
  16. J. Schwider, K.-E. Elssner, J. Grzanna, and R. Spolaczyk, “Results and Error Sources in Absolute Sphericity Measurement,” in IMEKO TC Series No. 14, Proc. 1st Symposium Budapest, T. Kemény and K. Havrilla, eds. (Nova Science, 1987), pp. 93-103.
  17. G. Schulz, “Interferentielle absolutprüfung zweier flächen,” Opt. Acta 20, 699-706 (1973).
  18. C. J. Evans and R. N. Kestner, “Test optics error removal,” Appl. Opt. 35, 1015-1021 (1996).
  19. G. Seitz, “Alternatives verfahren zur absolutkalibrierung von interferometrischen anordnungen,” DGaO-meeting 1997 Kloster Banz, Poster 1, abstract p. 92, conf. Program.
  20. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358-379 (1959).
    [CrossRef]

2008 (1)

2007 (2)

N. Lindlein, R. Maiwald, H. Konermann, M. Sondermann, U. Peschel, and G. Leuchs, “A new 4π-geometry optimized for focussing onto an atom with a dipole-like radiation pattern,” Laser Phys. 17, 927-934 (2007).
[CrossRef]

M. Sondermann, R. Maiwald, H. Konermann, N. Lindlein, U. Peschel, and G. Leuchs, “Design of a mode converter for efficient light-atom coupling in free space,” Appl. Phys. B 89, 489-492 (2007).

1996 (1)

1992 (1)

1989 (1)

1986 (1)

E. W. Young, “Optimal removal of all mislocation effects in interferometric tests,” Proc. SPIE 661, 116-124(1986).

1983 (1)

1980 (1)

K.-E. Elßner, J. Grzanna, and G. Schulz, “Interferentielle absolutprüfung von sphärizitätsnormalen,” Opt. Acta 27, 563-580 (1980).

1974 (1)

1973 (2)

G. Schulz, “Interferentielle absolutprüfung zweier flächen,” Opt. Acta 20, 699-706 (1973).

A. E. Jensen, “Absolute calibration method for laser Twyman-Green wave front testing interferometers,” J. Opt. Soc. Am. 63, 1313A (1973).

1959 (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358-379 (1959).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge University, 1980).

Brangaccio, D. J.

Bruning, J. H.

Burow, R.

Elssner, K.-E.

Elßner, K.-E.

K.-E. Elßner, J. Grzanna, and G. Schulz, “Interferentielle absolutprüfung von sphärizitätsnormalen,” Opt. Acta 27, 563-580 (1980).

Elssner, K.-E.

J. Schwider, K.-E. Elssner, J. Grzanna, and R. Spolaczyk, “Results and Error Sources in Absolute Sphericity Measurement,” in IMEKO TC Series No. 14, Proc. 1st Symposium Budapest, T. Kemény and K. Havrilla, eds. (Nova Science, 1987), pp. 93-103.

Evans, C. J.

Gallagher, J. E.

Grzanna, J.

K.-E. Elssner, R. Burow, J. Grzanna, and R. Spolaczyk, “Absolute sphericity measurement,” Appl. Opt. 28, 4649-4661 (1989);

J. Schwider, R. Burow, K.-E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, “Digital wavefront measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421-3432 (1983).

K.-E. Elßner, J. Grzanna, and G. Schulz, “Interferentielle absolutprüfung von sphärizitätsnormalen,” Opt. Acta 27, 563-580 (1980).

J. Schwider, K.-E. Elssner, J. Grzanna, and R. Spolaczyk, “Results and Error Sources in Absolute Sphericity Measurement,” in IMEKO TC Series No. 14, Proc. 1st Symposium Budapest, T. Kemény and K. Havrilla, eds. (Nova Science, 1987), pp. 93-103.

Hell, S.

Herriott, D. R.

Jensen, A. E.

A. E. Jensen, “Absolute calibration method for laser Twyman-Green wave front testing interferometers,” J. Opt. Soc. Am. 63, 1313A (1973).

Kestner, R. N.

Konermann, H.

M. Sondermann, R. Maiwald, H. Konermann, N. Lindlein, U. Peschel, and G. Leuchs, “Design of a mode converter for efficient light-atom coupling in free space,” Appl. Phys. B 89, 489-492 (2007).

N. Lindlein, R. Maiwald, H. Konermann, M. Sondermann, U. Peschel, and G. Leuchs, “A new 4π-geometry optimized for focussing onto an atom with a dipole-like radiation pattern,” Laser Phys. 17, 927-934 (2007).
[CrossRef]

Leuchs, G.

N. Lindlein, R. Maiwald, H. Konermann, M. Sondermann, U. Peschel, and G. Leuchs, “A new 4π-geometry optimized for focussing onto an atom with a dipole-like radiation pattern,” Laser Phys. 17, 927-934 (2007).
[CrossRef]

M. Sondermann, R. Maiwald, H. Konermann, N. Lindlein, U. Peschel, and G. Leuchs, “Design of a mode converter for efficient light-atom coupling in free space,” Appl. Phys. B 89, 489-492 (2007).

Lindlein, N.

N. Lindlein, R. Maiwald, H. Konermann, M. Sondermann, U. Peschel, and G. Leuchs, “A new 4π-geometry optimized for focussing onto an atom with a dipole-like radiation pattern,” Laser Phys. 17, 927-934 (2007).
[CrossRef]

M. Sondermann, R. Maiwald, H. Konermann, N. Lindlein, U. Peschel, and G. Leuchs, “Design of a mode converter for efficient light-atom coupling in free space,” Appl. Phys. B 89, 489-492 (2007).

Maiwald, R.

N. Lindlein, R. Maiwald, H. Konermann, M. Sondermann, U. Peschel, and G. Leuchs, “A new 4π-geometry optimized for focussing onto an atom with a dipole-like radiation pattern,” Laser Phys. 17, 927-934 (2007).
[CrossRef]

M. Sondermann, R. Maiwald, H. Konermann, N. Lindlein, U. Peschel, and G. Leuchs, “Design of a mode converter for efficient light-atom coupling in free space,” Appl. Phys. B 89, 489-492 (2007).

Meixner, A. J.

Merkel, K.

Peschel, U.

M. Sondermann, R. Maiwald, H. Konermann, N. Lindlein, U. Peschel, and G. Leuchs, “Design of a mode converter for efficient light-atom coupling in free space,” Appl. Phys. B 89, 489-492 (2007).

N. Lindlein, R. Maiwald, H. Konermann, M. Sondermann, U. Peschel, and G. Leuchs, “A new 4π-geometry optimized for focussing onto an atom with a dipole-like radiation pattern,” Laser Phys. 17, 927-934 (2007).
[CrossRef]

Richards, B.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358-379 (1959).
[CrossRef]

Rosenfeld, D. P.

Schulz, G.

K.-E. Elßner, J. Grzanna, and G. Schulz, “Interferentielle absolutprüfung von sphärizitätsnormalen,” Opt. Acta 27, 563-580 (1980).

G. Schulz, “Interferentielle absolutprüfung zweier flächen,” Opt. Acta 20, 699-706 (1973).

G. Schulz and J. Schwider, “Interferometric testing of smooth surfaces,” in Progress in Optics, E. Wolf, ed. (Elsivier, 1976), Vol. XIII, p. 93.

Schwider, J.

J. Schwider, R. Burow, K.-E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, “Digital wavefront measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421-3432 (1983).

G. Schulz and J. Schwider, “Interferometric testing of smooth surfaces,” in Progress in Optics, E. Wolf, ed. (Elsivier, 1976), Vol. XIII, p. 93.

J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf, ed. (Elsivier, 1990), Vol. XXVIII, pp. 271-359.

J. Schwider, K.-E. Elssner, J. Grzanna, and R. Spolaczyk, “Results and Error Sources in Absolute Sphericity Measurement,” in IMEKO TC Series No. 14, Proc. 1st Symposium Budapest, T. Kemény and K. Havrilla, eds. (Nova Science, 1987), pp. 93-103.

Seitz, G.

G. Seitz, “Alternatives verfahren zur absolutkalibrierung von interferometrischen anordnungen,” DGaO-meeting 1997 Kloster Banz, Poster 1, abstract p. 92, conf. Program.

Sondermann, M.

M. Sondermann, R. Maiwald, H. Konermann, N. Lindlein, U. Peschel, and G. Leuchs, “Design of a mode converter for efficient light-atom coupling in free space,” Appl. Phys. B 89, 489-492 (2007).

N. Lindlein, R. Maiwald, H. Konermann, M. Sondermann, U. Peschel, and G. Leuchs, “A new 4π-geometry optimized for focussing onto an atom with a dipole-like radiation pattern,” Laser Phys. 17, 927-934 (2007).
[CrossRef]

Spolaczyk, R.

K.-E. Elssner, R. Burow, J. Grzanna, and R. Spolaczyk, “Absolute sphericity measurement,” Appl. Opt. 28, 4649-4661 (1989);

J. Schwider, R. Burow, K.-E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, “Digital wavefront measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421-3432 (1983).

J. Schwider, K.-E. Elssner, J. Grzanna, and R. Spolaczyk, “Results and Error Sources in Absolute Sphericity Measurement,” in IMEKO TC Series No. 14, Proc. 1st Symposium Budapest, T. Kemény and K. Havrilla, eds. (Nova Science, 1987), pp. 93-103.

Stadler, J.

Stanciu, C.

Stelzer, E. H. K.

Stupperich, C.

Welford, W. T.

W. T. Welford, “Aplanatism and Isoplanatism,” in Progress in Optics, E. Wolf, ed. (Elsivier, 1976), Vol. XIII, pp. 267-292.

White, A. D.

Wolf, E.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358-379 (1959).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge University, 1980).

Young, E. W.

E. W. Young, “Optimal removal of all mislocation effects in interferometric tests,” Proc. SPIE 661, 116-124(1986).

Appl. Opt. (4)

Appl. Phys. B (1)

M. Sondermann, R. Maiwald, H. Konermann, N. Lindlein, U. Peschel, and G. Leuchs, “Design of a mode converter for efficient light-atom coupling in free space,” Appl. Phys. B 89, 489-492 (2007).

J. Opt. Soc. Am. (1)

A. E. Jensen, “Absolute calibration method for laser Twyman-Green wave front testing interferometers,” J. Opt. Soc. Am. 63, 1313A (1973).

J. Opt. Soc. Am. A (1)

Laser Phys. (1)

N. Lindlein, R. Maiwald, H. Konermann, M. Sondermann, U. Peschel, and G. Leuchs, “A new 4π-geometry optimized for focussing onto an atom with a dipole-like radiation pattern,” Laser Phys. 17, 927-934 (2007).
[CrossRef]

Opt. Acta (2)

G. Schulz, “Interferentielle absolutprüfung zweier flächen,” Opt. Acta 20, 699-706 (1973).

K.-E. Elßner, J. Grzanna, and G. Schulz, “Interferentielle absolutprüfung von sphärizitätsnormalen,” Opt. Acta 27, 563-580 (1980).

Opt. Lett. (1)

Proc. R. Soc. A (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358-379 (1959).
[CrossRef]

Proc. SPIE (1)

E. W. Young, “Optimal removal of all mislocation effects in interferometric tests,” Proc. SPIE 661, 116-124(1986).

Other (7)

G. Schulz and J. Schwider, “Interferometric testing of smooth surfaces,” in Progress in Optics, E. Wolf, ed. (Elsivier, 1976), Vol. XIII, p. 93.

J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf, ed. (Elsivier, 1990), Vol. XXVIII, pp. 271-359.

J. Schwider, K.-E. Elssner, J. Grzanna, and R. Spolaczyk, “Results and Error Sources in Absolute Sphericity Measurement,” in IMEKO TC Series No. 14, Proc. 1st Symposium Budapest, T. Kemény and K. Havrilla, eds. (Nova Science, 1987), pp. 93-103.

W. T. Welford, “Aplanatism and Isoplanatism,” in Progress in Optics, E. Wolf, ed. (Elsivier, 1976), Vol. XIII, pp. 267-292.

ARCoptix; product description: “Radial Polarization Converter”.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge University, 1980).

G. Seitz, “Alternatives verfahren zur absolutkalibrierung von interferometrischen anordnungen,” DGaO-meeting 1997 Kloster Banz, Poster 1, abstract p. 92, conf. Program.

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Figures (29)

Fig. 1
Fig. 1

Fizeau interferometer for testing a parabolic mirror.

Fig. 2
Fig. 2

Core of the Fizeau test setup: Comprised of the paraboloid, the plane reference surface, and the spherical steel sphere as a null element for the spherical wave formed in the focal plane of the parabolic mirror.

Fig. 3
Fig. 3

Superposition of the plane waves coming from the cat’s eye reflector with the nearly plane wave reflected from the outer rim of the paraboloid. The aberrations in the central part are due to interference of the test wave with the cat’s eye reference. The remaining aberrations are due to an imperfect cat’s eye reflector. But a simulation of the wave aberrations (on the right) of an off-axis angle of 1 arc min for the illuminating beam shows that the achievable accuracy of this adjustment is satisfactory (P/V: 0.02 wavelengths) since, in the left hand fringe pattern, only a misalignment of a few arc seconds is indicated.

Fig. 4
Fig. 4

Simulated interferograms due to alignment aberrations obtained by the program RAYTRACE: On the left, axial shift of 1 μm of the paraboloid (or steel sphere) against the interferometer frame; in the middle, lateral shift of 1 μm of the paraboloid (or steel sphere); on the right, collimation error ( R = 100 m ) of the impinging wavefront.

Fig. 5
Fig. 5

Different experimental adjustment states of the steel sphere relative to the parabolic mirror: On the left, two z adjustments; on the right, lateral displacements in x and in x, y direction.

Fig. 6
Fig. 6

Simulated misalignment in axial direction: on the left, 1 μm axial shift of the paraboloid; on the right, residuum due to the approximation of Eq. (1) (all values are in waves).

Fig. 7
Fig. 7

Simulation result showing the remaining residuum after removal of an axial shift of 1 μm , as in Fig. 6 before, but using the method of virtual shifts due to Eq. (2) and a Zernike fit of the degree 12 to the aberrations obtained by ray tracing (all values are in waves). Now, the P/V is only 0.0039 waves.

Fig. 8
Fig. 8

On the left: interference fringes shown together with the selection mask which has to be centered and the outer and inner diameter matched to the dimension of the valid data field. On the right: wrapped data giving an indication of the quality of the data.

Fig. 9
Fig. 9

Relevant phase aberration data introduced by the paraboloid in the null test configuration of Fig. 1: On the left, grayscale plot of the deviations; on the right, pseudo 3D-plot of the deviations (all values are in waves).

Fig. 10
Fig. 10

Zernike fit of the relevant phase aberration data: On the left, contour line plot; on the right, pseudo 3D-plot (all values are in waves).

Fig. 11
Fig. 11

Demonstration example for the occurrence of three-beam interference (waves proportional to R 1 = R gl , ( 1 R gl ) 2 R h , and R 2 ) in a plane surface Fizeau interferometer with a highly reflective test sample ( R gl is the reflectivity of glass, R h is the reflectivity of the coated surface). In addition, an even weaker wave R 3 is generated that can be ignored in this context. Reflections from the back surface of the reference plate are not shown.

Fig. 12
Fig. 12

Intensities detected in the focal plane of the imaging telescope. The central spot stems from the parabolic mirror in the null test arrangement. The spot in the upper half of the image belongs to the reference wave and the spot below to the parasitic intensity of the doubly-reflected beam at the null test configuration (parasitic beam). The images on the left and in the middle differ by the amount of tilt of the reference plate. The situation in the middle is chosen to be able to block the parasitic wave that is realized on the right. The parasitic intensity has been suppressed by an opaque screen (on the right).

Fig. 13
Fig. 13

Measurement with NG-filter with 10% transparency. It can be inferred from the video picture (upper part) that the NG-filter introduced strong aberrations which screen the phase aberrations introduced by the parabolic mirror (all values are in waves).

Fig. 14
Fig. 14

Evaluation result of test-beam with parasitic beam: upper part, video image; lower left, wrapped phase; lower right, unwrapped phase after removal of linear aberration term. Please note that, in the lower region, the contrast of the resulting interference pattern was below a threshold used in the software to avoid false phase values. In the lower part of the unwrapped deviation picture, a failure of the unwrapping process can be seen which is a direct result of the poor contrast of the parasitic fringes.

Fig. 15
Fig. 15

Evaluation result of the phase aberrations introduced by the paraboloid with tilted reference surface and suppressed parasitic wave: Above, fringe pattern on the CCD-chip (left) and wrapped phase after phase shifting evaluation (right); below, Zernike fit of the wave aberrations of the test beam (all values are in waves).

Fig. 16
Fig. 16

Difference of the Zernike evaluations resulting from two measurements—one with and the other without tilt fringes (all values are in waves).

Fig. 17
Fig. 17

Interferometric evaluation of the deviations of the used plane Fizeau reference surface against a supersmooth Fabry–Perot plate with λ t / 50 (all values are in waves).

Fig. 18
Fig. 18

Repeatability test: Difference of the Zernike evaluations of two measurements performed immediately one after the other (all values are in waves).

Fig. 19
Fig. 19

Difference of the Zernike evaluations of two measurements with readjustment of the field selecting mask (all values are in waves).

Fig. 20
Fig. 20

Interference pattern of the doubly-reflected plane wave and corresponding phase shift evaluations: Upper part, intensity distribution seen by the CCD-camera; lower part/left, wrapped phase; lower part/right, phase distribution showing, in a qualitative manner, the threefold symmetry of the surface under test (all values are in waves).

Fig. 21
Fig. 21

Difference of the Zernike evaluated wave aberrations measured with two different reference steel spheres (all values are in waves).

Fig. 22
Fig. 22

Difference of the Zernike evaluated wave aberrations of two consecutive runs, where the second run was carried out after a 180 ° rotation of the steel sphere around the optical axis of the paraboloid (all values are in waves).

Fig. 23
Fig. 23

Difference of the Zernike evaluated wave aberrations of two consecutive runs, where the second run was carried out after a 90 ° rotation of the steel sphere around the optical axis of the paraboloid (all values are in waves).

Fig. 24
Fig. 24

Reproducibility test with runs taken from measurements done with a time lapse of about 4 weeks, two different adjustments. Please note the systematic contribution of antisymmetric aberration, whose cause has to be attributed to a lateral shift of the two data sets measured with 4 weeks time lapse. It is obvious from the peak/valley values that a bigger P/V value is connected with the dominant aberration features of a lateral displacement (compare the left and right picture) (all values are in waves).

Fig. 25
Fig. 25

Difference of wave aberrations of the paraboloid after a shift of 4 pixels along x and 2 pixels along y direction, see Fig. 24, left as well as Fig. 27, left (all values are in waves).

Fig. 26
Fig. 26

Reproducibility tests of Fig. 25, data fields laterally shifted with respect to each other to demonstrate lateral displacement sensitivity: Left, difference of data fields of Fig. 24, left (one of them shifted by 4 pixels in x and 2 pixels in y direction); right, difference of data fields of Fig. 25, right (one of them shifted by 7 pixels in x and 4 pixels in y direction). In both cases the aberration features indicating lateral displacement of the fields have vanished; the peak/valley values are almost halved (all values are in waves).

Fig. 27
Fig. 27

Surface deviations of the paraboloid normalized via software from the measured wave aberrations (all values are in waves).

Fig. 28
Fig. 28

Square of the total electric field in the focal plane of the parabolic mirror by assuming, in both cases, a total incident light power of 1 W : Left, ideal case without aberrations; right, actual case for our mirror having the measured aberrations of Fig. 11. The maximum value of the square of the electric field is smaller by a factor of about 27.

Fig. 29
Fig. 29

Square of the total electric field in the focal plane of the parabolic mirror by assuming aberrations with measured symmetry but a P/V value of only λ e / 4 . Again, the total incident light power is 1 W .

Tables (1)

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Table 1 Summary of Test Results

Equations (6)

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Φ mis : Φ mis = 4 k ( d · n ) ( e · n ) ,
Φ mis = Z ξ ( x , y ) δ ξ + Z η ( x , y ) δ η + Z ζ ( x , y ) δ ζ + Z Ω x ( x , y ) Ω x + Z Ω y ( x , y ) Ω y + L ( x , y ) , L ( x , y ) = a + b x + c y .
r r = 4 f 2 r = 4 f 2 r .
I | 2 π r d r | = I | 2 π r d r | I ( r = 4 f 2 r ) = ( r 2 f ) 4 I ( r ) .
W total ( r , φ ) = W ( r , φ ) + W ( r , φ ) = W ( r , φ ) + W ( 4 f 2 r , φ + π ) = W ( 4 f 2 r , φ + π ) + W ( r , φ ) .
r = 2 f tan θ cos θ = cos ( arctan ( r 2 f ) ) = 1 1 + ( r 2 / 4 f 2 ) g ( r ) = 1 + ( r 2 / 4 f 2 ) 4 .

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