Abstract

When beams from an array of afocal telescopes are presented to a beam-combining telescope, tilt errors of the beam wave fronts with respect to the combined wave front place limits on the achievable field of view (FOV); these limits, to the best of our knowledge, have not previously been correctly described in the literature. We show that if the front-end telescopes have just the right Seidel distortion coefficient, then tilt error does not limit the FOV. If this is not the case, at least small FOV’s can still be obtained, even if the FOV is not centered on the on-axis direction.

© 1999 Optical Society of America

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References

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  1. J. E. Harvey, C. Ftaclas, “Field-of-view limitations of phased telescope arrays,” Appl. Opt. 34, 5787–5798 (1995).
    [CrossRef] [PubMed]
  2. C. R. De Hainaut, D. C. Duneman, R. C. Dymale, J. P. Blea, B. D. Burton, C. E. Hines, “Wide field performance of a phased array telescope,” Opt. Eng. 34, 876–880 (1995).
    [CrossRef]
  3. L. D. Weaver, J. S. Fender, C. R. De Hainaut, “Design considerations for multiple telescope imaging arrays,” Opt. Eng. 27, 730–735 (1988).
    [CrossRef]
  4. M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), Chap. 5, see especially pp. 211–213.
  5. D. Korsch, Reflective Optics (Academic, San Diego, 1991), pp. 43, 98, 212–214, and 230.
  6. F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1957), p. 152.
  7. M. Lampton, BEAM4 Optical Ray Tracer (Stellar Software, PO Box 10183, Berkeley, Calif. 94709).

1995

C. R. De Hainaut, D. C. Duneman, R. C. Dymale, J. P. Blea, B. D. Burton, C. E. Hines, “Wide field performance of a phased array telescope,” Opt. Eng. 34, 876–880 (1995).
[CrossRef]

J. E. Harvey, C. Ftaclas, “Field-of-view limitations of phased telescope arrays,” Appl. Opt. 34, 5787–5798 (1995).
[CrossRef] [PubMed]

1988

L. D. Weaver, J. S. Fender, C. R. De Hainaut, “Design considerations for multiple telescope imaging arrays,” Opt. Eng. 27, 730–735 (1988).
[CrossRef]

Blea, J. P.

C. R. De Hainaut, D. C. Duneman, R. C. Dymale, J. P. Blea, B. D. Burton, C. E. Hines, “Wide field performance of a phased array telescope,” Opt. Eng. 34, 876–880 (1995).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), Chap. 5, see especially pp. 211–213.

Burton, B. D.

C. R. De Hainaut, D. C. Duneman, R. C. Dymale, J. P. Blea, B. D. Burton, C. E. Hines, “Wide field performance of a phased array telescope,” Opt. Eng. 34, 876–880 (1995).
[CrossRef]

De Hainaut, C. R.

C. R. De Hainaut, D. C. Duneman, R. C. Dymale, J. P. Blea, B. D. Burton, C. E. Hines, “Wide field performance of a phased array telescope,” Opt. Eng. 34, 876–880 (1995).
[CrossRef]

L. D. Weaver, J. S. Fender, C. R. De Hainaut, “Design considerations for multiple telescope imaging arrays,” Opt. Eng. 27, 730–735 (1988).
[CrossRef]

Duneman, D. C.

C. R. De Hainaut, D. C. Duneman, R. C. Dymale, J. P. Blea, B. D. Burton, C. E. Hines, “Wide field performance of a phased array telescope,” Opt. Eng. 34, 876–880 (1995).
[CrossRef]

Dymale, R. C.

C. R. De Hainaut, D. C. Duneman, R. C. Dymale, J. P. Blea, B. D. Burton, C. E. Hines, “Wide field performance of a phased array telescope,” Opt. Eng. 34, 876–880 (1995).
[CrossRef]

Fender, J. S.

L. D. Weaver, J. S. Fender, C. R. De Hainaut, “Design considerations for multiple telescope imaging arrays,” Opt. Eng. 27, 730–735 (1988).
[CrossRef]

Ftaclas, C.

Harvey, J. E.

Hines, C. E.

C. R. De Hainaut, D. C. Duneman, R. C. Dymale, J. P. Blea, B. D. Burton, C. E. Hines, “Wide field performance of a phased array telescope,” Opt. Eng. 34, 876–880 (1995).
[CrossRef]

Jenkins, F. A.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1957), p. 152.

Korsch, D.

D. Korsch, Reflective Optics (Academic, San Diego, 1991), pp. 43, 98, 212–214, and 230.

Weaver, L. D.

L. D. Weaver, J. S. Fender, C. R. De Hainaut, “Design considerations for multiple telescope imaging arrays,” Opt. Eng. 27, 730–735 (1988).
[CrossRef]

White, H. E.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1957), p. 152.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), Chap. 5, see especially pp. 211–213.

Appl. Opt.

Opt. Eng.

C. R. De Hainaut, D. C. Duneman, R. C. Dymale, J. P. Blea, B. D. Burton, C. E. Hines, “Wide field performance of a phased array telescope,” Opt. Eng. 34, 876–880 (1995).
[CrossRef]

L. D. Weaver, J. S. Fender, C. R. De Hainaut, “Design considerations for multiple telescope imaging arrays,” Opt. Eng. 27, 730–735 (1988).
[CrossRef]

Other

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), Chap. 5, see especially pp. 211–213.

D. Korsch, Reflective Optics (Academic, San Diego, 1991), pp. 43, 98, 212–214, and 230.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1957), p. 152.

M. Lampton, BEAM4 Optical Ray Tracer (Stellar Software, PO Box 10183, Berkeley, Calif. 94709).

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Figures (1)

Fig. 1
Fig. 1

Light path through two telescopes of a phased array. y 1 and y 2 specify the locations of the telescopes and are the locations at which principal rays from wave-front segments A and B strike plane P 1. P 1, telescope entrance plane; P 2, telescope exit plane; and P 3, beam combiner entrance plane. These planes may be thought of as, but need not actually be, pupil planes. Flat mirrors a i , b i , c i , and d i (i = 1, 2) constitute the relay optics. They can be moved to advance or delay the light in either path or to present the light to the beam combiner at any vertical position in plane P 3, or they can be tilted to render light from a chosen direction parallel to the beam combiner’s optic axis. As shown, the relay mirrors are set in their zero positions so that light from direction θ = 0 goes straight into the beam combiner. Thus the angle at which wave-front segments A′ and B′ enter the beam combiner is θ′, the same angle at which they left the afocal front-end telescopes. The principal rays of segments A′ and B′ (the rays at the centers of the segments) strike P 3 at positions y 1′ and y 2′. When the relay mirrors are tilted to send light from some direction θ0′ (not shown) straight into the beam combiner, angle θ′ is replaced at P 3 by θ′ - θ0′. Inset: Seemingly severe piston errors of two segments with respect to a parallel wave front become much less severe tilt errors with respect to a wave front that passes through their centers. See text for further description.

Equations (54)

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tan θ=M0tan θ-Eθ3,
θ=M0θ-M0E+M02-13θ3=M01-Eθ2 θ=Mθθ,
E=E+M02-1/3
Mθ=M01-Eθ2.
p1θ=y1 sin θ+T1θ+s1cosθ-θ0-y1×sinθ-θ0,
δpθ=p2θ-p1θ=L sin θ+Δscosθ-θ0-y2θ-y1θ×sinθ-θ0,
yiθ=yiθ0+ΔRi+si tanθ-θ0,  i=1, 2,
l=y2θ0-y1θ0,
y2θ-y1θ=l+Δs tanθ-θ0.
δpθ=L sin θ+Δs cosθ-θ0-l sinθ-θ0.
δpθ=L sin θ+Δz cos θ+Δs cosθ-θ0-l sinθ-θ0,
l sin θ=L sin θ,
sin θ=constant×sin θ=M0 sin θ,
E=-M02-1/2.
EMer=-M02+2M0-3/4,
δpθ=δpθ0+dδpθdθθ0 Δθ+12d2δpθdθ2θ0 Δθ2+16d3δpθdθ3θ0 Δθ3+h.t.,
Δs=-L sin θ0-Δz cos θ0
l=L cos θ0-Δz sin θ0M01-3E+M02-1θ02.
12d2δpθdθ2θ0=3Lθ0E+M02-12+Δz M02-12+Oθ02
16d3δpθdθ3θ0=LE+M02-12+Oθ0.
Wrms=|δθ| d4=δpld4|δp| D4L,
|δpθ0|4LD rλ,
dδpθdθθ04LDrλ|Δθ|,
12d2δpθdθ2θ04LDrλ|Δθ2|,
16d3δpθdθ3θ04LDrλ|Δθ3|.
W11=-M0EMerθmax3d/2
|Δθθ0=0|4r λDE+M02-121/3.
Wrms=|M0EMerθ3|d/4=|EMerθ3|D/4.
|θ|=4rλ/|EMer|D1/3,
M02-1/2M02/2
EMer-M02/4
|EMer+M02-1/2|M02/4.
|EMer||Emer+M02-1/2|M02/4,
|Δθθ0|4r λD3θ0E+M02-12+ΔzL M02-121/2|Δθ0|33|θ0|1/2  for Δz=0, |Δθ0|3α  for Δz=0 and |θ0|=α|Δθ0|,
E3=-M02-2M0+1/Ω1-1/4.
Δθ01.5×10-3 rad=0.09°=5.2 arc min.
dθdθ=M01-3Eθ2,
d2θdθ2=-6M0Eθ,
d3θdθ3=-6M0E.
ddθsinθ-θ0=cosθ-θ0dθdθ=dθdθ  at θ=θ0,
d2dθ2sinθ-θ0=-sinθ-θ0dθdθ2+cosθ-θ0d2θdθ2=d2θdθ2  at θ=θ0,
d3dθ3sinθ-θ0=-cosθ-θ0dθdθ3-3 sinθ-θ0dθdθd2θdθ2+cosθ-θ0d3θdθ3=-dθdθ3+d3θdθ3  at θ=θ0.
ddθcosθ-θ0=0  at θ=θ0,
d2dθ2cosθ-θ0=-dθdθ2  at θ=θ0,
d3dθ3cosθ-θ0=-3 dθdθd2θdθ2  at θ=θ0.
δpθ=L sin θ+Δz cos θ+Δs cosθ-θ0-l sinθ-θ0=L sin θ0+Δz cos θ0+Δs  at θ=θ0,
dδpθdθ=L cos θ-Δz sin θ+Δs ddθcosθ-θ0-l ddθsinθ-θ0=L cos θ0-Δz sin θ0-l dθdθ  at θ=θ0,
d2δpθdθ2=-L sin θ-Δz cos θ+Δs d2dθ2cosθ-θ0-l d2dθ2sinθ-θ0=-L sin θ0-Δz cos θ0-Δsdθdθ2-l d2θdθ2  at θ=θ0,
d3δpθdθ3=-L cos θ+Δz sin θ+Δs d3dθ3cosθ-θ0-l d3dθ3sinθ-θ0=-L cos θ0+Δz sin θ0-3Δs dθdθd2θdθ2-l-dθdθ3+d3θdθ3  at θ=θ0.
Δs=-L sin θ0-Δz cos θ0.
l=L cos θ0-Δz sin θ0M01-3Eθ02=L cos α0-Δz sin θ0M01-3E+M02-1θ02.
d2δpθdθ2θ03Lθ02E+M02-1+ΔzM02-1.
d3δpθdθ3θ03L2E+M02-1.
Wrms2=4πd20d/202πδθρ cos α2ρdρdα=δθ d42,

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