Abstract

Optical true-time-delay devices based on the White cell can be divided into two general types: polynomial cells, in which the number of delays that can be obtained is related to the number of times m that a beam bounces in the cell raised to some power, and exponential cells, in which the number of delays is proportional to some number raised to the power of m. In exponential cells, the topic to be addressed, the spatial light modulator switches between a delay element and a null path on each bounce. We describe an improved design of this switching engine, which contains a liquid-crystal switch and a White cell. We examine astigmatism and corrections for it and present a specific design.

© 2003 Optical Society of America

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References

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  1. H. Zmuda, E. N. Toughlian, “Photonic aspects of modern radar,” in The Artech House Optoelectronics Library, B. Culshaw, A. Rogers, H. Taylor, eds. (Artech House, Norwood, Mass., 1994), p. 550.
  2. K. P. Jackson, S. A. Newton, B. Moslehi, M. Tur, C. C. Cutler, J. W. Goodman, H. J. Shaw, “Optical fiber delay-line signal processing,” IEEE Trans. Microwave Theory Techn. MTT-33, 193–209 (1985).
    [CrossRef]
  3. S. Sales, J. Pampany, J. Martí, D. Pastor, “Solutions to the synthesis problem of optical delay line filters,” Opt. Lett. 20, 2438–2440 (1995).
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  6. B. L. Anderson, C. D. Liddle, “Optical true time delay for phased-array antennas: demonstration of a quadratic White cell,” Appl. Opt. 41, 4912–4921 (2002).
    [CrossRef] [PubMed]
  7. B. L. Anderson, R. Mital, “Polynomial-based optical true-time delay devices with microelectromechanical mirror arrays,” Appl. Opt. 41, 5449–5461 (2002).
    [CrossRef] [PubMed]
  8. S. A. Collins, B. L. Anderson, “Device and method for producing optically-controlled incremental time delays,” U.S. patent6,388,615 (14May2002).
  9. B. L. Anderson, S. A. Collins, C. A. Klein, E. A. Beecher, S. B. Brown, “Optically produced true-time delays for phased antenna arrays,” Appl. Opt. 36, 8493–8503 (1997).
    [CrossRef]
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    [CrossRef]

2002

1997

1995

1985

K. P. Jackson, S. A. Newton, B. Moslehi, M. Tur, C. C. Cutler, J. W. Goodman, H. J. Shaw, “Optical fiber delay-line signal processing,” IEEE Trans. Microwave Theory Techn. MTT-33, 193–209 (1985).
[CrossRef]

1976

1961

1942

Anderson, B. L.

Beecher, E. A.

Brown, S. B.

Collins, S. A.

B. L. Anderson, S. A. Collins, C. A. Klein, E. A. Beecher, S. B. Brown, “Optically produced true-time delays for phased antenna arrays,” Appl. Opt. 36, 8493–8503 (1997).
[CrossRef]

S. A. Collins, B. L. Anderson, “Device and method for producing optically-controlled incremental time delays,” U.S. patent6,388,615 (14May2002).

Cutler, C. C.

K. P. Jackson, S. A. Newton, B. Moslehi, M. Tur, C. C. Cutler, J. W. Goodman, H. J. Shaw, “Optical fiber delay-line signal processing,” IEEE Trans. Microwave Theory Techn. MTT-33, 193–209 (1985).
[CrossRef]

Edwards, T. H.

Goodman, J. W.

K. P. Jackson, S. A. Newton, B. Moslehi, M. Tur, C. C. Cutler, J. W. Goodman, H. J. Shaw, “Optical fiber delay-line signal processing,” IEEE Trans. Microwave Theory Techn. MTT-33, 193–209 (1985).
[CrossRef]

Jackson, K. P.

K. P. Jackson, S. A. Newton, B. Moslehi, M. Tur, C. C. Cutler, J. W. Goodman, H. J. Shaw, “Optical fiber delay-line signal processing,” IEEE Trans. Microwave Theory Techn. MTT-33, 193–209 (1985).
[CrossRef]

Klein, C. A.

Liddle, C. D.

Martí, J.

Mital, R.

Moslehi, B.

K. P. Jackson, S. A. Newton, B. Moslehi, M. Tur, C. C. Cutler, J. W. Goodman, H. J. Shaw, “Optical fiber delay-line signal processing,” IEEE Trans. Microwave Theory Techn. MTT-33, 193–209 (1985).
[CrossRef]

Newton, S. A.

K. P. Jackson, S. A. Newton, B. Moslehi, M. Tur, C. C. Cutler, J. W. Goodman, H. J. Shaw, “Optical fiber delay-line signal processing,” IEEE Trans. Microwave Theory Techn. MTT-33, 193–209 (1985).
[CrossRef]

Pampany, J.

Pastor, D.

Sales, S.

Shaw, H. J.

K. P. Jackson, S. A. Newton, B. Moslehi, M. Tur, C. C. Cutler, J. W. Goodman, H. J. Shaw, “Optical fiber delay-line signal processing,” IEEE Trans. Microwave Theory Techn. MTT-33, 193–209 (1985).
[CrossRef]

Toughlian, E. N.

H. Zmuda, E. N. Toughlian, “Photonic aspects of modern radar,” in The Artech House Optoelectronics Library, B. Culshaw, A. Rogers, H. Taylor, eds. (Artech House, Norwood, Mass., 1994), p. 550.

Tur, M.

K. P. Jackson, S. A. Newton, B. Moslehi, M. Tur, C. C. Cutler, J. W. Goodman, H. J. Shaw, “Optical fiber delay-line signal processing,” IEEE Trans. Microwave Theory Techn. MTT-33, 193–209 (1985).
[CrossRef]

White, J.

White, J. U.

Zmuda, H.

H. Zmuda, E. N. Toughlian, “Photonic aspects of modern radar,” in The Artech House Optoelectronics Library, B. Culshaw, A. Rogers, H. Taylor, eds. (Artech House, Norwood, Mass., 1994), p. 550.

Appl. Opt.

IEEE Trans. Microwave Theory Techn.

K. P. Jackson, S. A. Newton, B. Moslehi, M. Tur, C. C. Cutler, J. W. Goodman, H. J. Shaw, “Optical fiber delay-line signal processing,” IEEE Trans. Microwave Theory Techn. MTT-33, 193–209 (1985).
[CrossRef]

J. Opt. Soc. Am.

Opt. Lett.

Other

H. Zmuda, E. N. Toughlian, “Photonic aspects of modern radar,” in The Artech House Optoelectronics Library, B. Culshaw, A. Rogers, H. Taylor, eds. (Artech House, Norwood, Mass., 1994), p. 550.

S. A. Collins, B. L. Anderson, “Device and method for producing optically-controlled incremental time delays,” U.S. patent6,388,615 (14May2002).

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Figures (15)

Fig. 1
Fig. 1

Original White cell consists of three spherical mirrors, of identical radii of curvature R, separated by R. The centers of curvature (C.C.) for each mirror are shown. A spot introduced into the White cell via the input turning mirror produces a series of spots on mirror A.

Fig. 2
Fig. 2

Spot pattern produced on mirror A by six input beams undergoing 16 bounces apiece.

Fig. 3
Fig. 3

Three-dimensional view of the TTD White cell with a binary architecture that uses glass blocks.

Fig. 4
Fig. 4

Three-dimensional view of the new binary cell configuration. PBS, polarizing beam splitter.

Fig. 5
Fig. 5

Astigmatic focus at the plane of the SLM surface. The astigmatic image creates tangential and sagittal focal planes. The surface of the SLM is located in between these two focal planes at the circle of least confusion for best imaging.

Fig. 6
Fig. 6

Normalized astigmatism versus angle of incidence θ in the binary cell according to expression (14). The astigmatic image distance is normalized to R WC/2f 1 2.

Fig. 7
Fig. 7

Depth of focus due to beam divergence at the SLM surface. The depth of focus depends on the size of the beam waist as well as the wavelength of light. The astigmatic image distance Δd sm after one, three, and m max passes is also shown.

Fig. 8
Fig. 8

Top view of the binary cell with a flat mirror to fold the path of the SLM branch. Folding the light path will allow us to reduce the angle of incidence θ on the White cell mirrors B and C.

Fig. 9
Fig. 9

Three-dimensional view of a binary White cell (a) with a folding mirror and (b) vertically oriented mirrors B and C.

Fig. 10
Fig. 10

Three-dimensional view of a binary White cell with a cylindrical mirror to compensate for non-Seidel astigmatism.

Fig. 11
Fig. 11

Normalized focal length of the cylindrical lens needed to correct for the astigmatism as a function of the angle of incidence on the White cell (spherical) mirrors, θ. The focal length in this plot is normalized by dividing by the radius of curvature R WC of the White cell mirrors.

Fig. 12
Fig. 12

Design configuration of binary cell TTD controller of the design of Table 1. Note that the SLM surface is embedded deep within the SLM housing (heavy dashed line), making it physically impossible to place the turning mirrors in the same plane as the SLM surface.

Fig. 13
Fig. 13

Astigmatism versus angle of incidence θ in the binary bell of the design of Table 1. The circled data points on the graph denote Δd sm for the angles 7.5°, 3°, and 1°.

Fig. 14
Fig. 14

Simulations in oslo showing the reduction in astigmatism as the angle is decreased. The optical system parameters are those of Table 1. On the left side of each figure is a plot showing the sagittal (×) and tangential (+) foci. The vertical axis is the position of the original object with respect to the optical axis, as a fraction of the object plane size (maximum is 1). The horizontal axis is the distance (in millimeters) along the optical axis in the vicinity of the image (this axis is tipped at 2θ). The bars indicate the scale in millimeters. (a) θ = 7.5°, (b) θ = 3°, (c) θ = 1°. The astigmatism is the difference between the two curves on the horizontal axis. Note the differences in the scale on the horizontal axes of the plots.

Fig. 15
Fig. 15

Cylindrical lens is placed in front of the spherical mirror to correct the astigmatism. The focal length is f CYL = 24 m for θ = 7.5°. The astigmatism for an object on axis is now 0.00144 mm.

Tables (2)

Tables Icon

Table 1 Design Parameters of the Binary White Cell TTD Switching Engine

Tables Icon

Table 2 Sagittal and Tangential Foci and Astigmatism Δdsm as a Function of the Angle in the White Cell Whose Parameters Are Listed in Table 1 a

Equations (22)

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

Tmax=2m/2.
dsm=f1f22RWCf1f2RWC-f1dauxRWC-2f1f22+f22RWC,
d1=f1,
d2=f2.
M=f2/f1.
1do+1diT=2R cos θ,
1do+1diS=2 cos θR,
1do+1di=1f,
fT=R cos θ2,
fS=R2 cos θ.
dsm_t=f1f22f1f2-f1daux+f22-2f12RWC cos θ,
dsm_s=f1f22f1f2-f1daux+f22-2f12cos θRWC.
Δdsm=dsm_t-dsm_s=2f12RWC1cos θ-cos θ,
Δdsm2f12/RWC=1+θ22-1-θ22θ2.
depth of focus=2π w02λ,
mmax=ΔΔdsm.
mmax=2π w02Δdsmλ.
RWC_S=RWC_T cos2θ,
1fT=1fS+2fCYL,
fCYL=R cosθsin2θ.
kw022z
kw022=2π633 nm25 μm22=3.1 mmz.

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