Abstract

Several techniques have been developed to phase apertures in the context of astronomical telescopes with segmented mirrors. Phasing multiple apertures, however, is important in a wide range of optical applications. The application of primary interest in this paper is the phasing of multiple short pulse laser beams for fast ignition fusion experiments. In this paper analytic expressions are derived for parameters such as the far-field distribution, a line-integrated form of the far-field distribution that could be fit to measured data, enclosed energy or energy-in-a-bucket and center-of-mass that can then be used to phase two rectangular apertures. Experimental data is taken with a MEMS device to simulate the two apertures and comparisons are made between the analytic parameters and those derived from the measurements. Two methods, fitting the measured far-field distribution to the theoretical distribution and measuring the ensquared energy in the far-field, produced overall phase variance between the 100 measurements of less than 0.005 rad2 or an RMS displacement of less than 12 nm.

©2009 Optical Society of America

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References

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  1. G. Chanan, M. Troy, F. Dekens, S. Michaels, J. Nelson, T. Mast, and D. Kirkman, “Phasing the mirror segments of the keck telescopes: the broadband phasing algorithm,” Appl. Opt. 37(1), 140–155 (1998).
    [Crossref]
  2. G. Chanan, C. Ohara, and M. Troy, “Phasing the mirror segments of the Keck telescopes II: the narrow-band phasing algorithm,” Appl. Opt. 39(25), 4706–4714 (2000).
    [Crossref]
  3. A. Schumacher, N. Devaney, and L. Montoya, “Phasing segmented mirrors: a modification of the Keck narrow-band technique and its application to extremely large telescopes,” Appl. Opt. 41(7), 1297–1307 (2002).
    [Crossref] [PubMed]
  4. F. Shi, D. Redding, A. Lowman, C. Bowers, L. Burns, P. Petrone, C. Ohara, and S. Basinger, “Segmented Mirror Coarse Phasing with a Dispersed Fringe Sensor: Experiment on NGST's Wavefront Control Testbed,” SPIE 4850, 318–328 (2003).
    [Crossref]
  5. F. Shi, S. Basinger, and D. Redding, “Performance of a Dispersed Fringe Sensor in the presence of segmented mirror aberrations: modeling and simulation,” SPIE 6265, 62650Y (2006).
    [Crossref]
  6. K. L. Baker, E. A. Stappaerts, D. C. Homoelle, M. A. Henesian, E. S. Bliss, C. W. Siders, and C. P. J. Barty, “Interferometric adaptive optics for high power laser pointing, wave-front control and phasing,” accepted into Journal of Micro/Nanolithography MEMS, and MOEMS (2009).
  7. W. Joseph, Goodman, Introduction to Fourier Optics, Boston: McGraw-Hill (1996).
  8. M. Abromowitz and I. A. Stegun, Handbook of Mathematical Functions and Formulas, Graphs, and Mathematical Tables, New York: Dover Publications 231 (1972).
  9. J. Moré, “The Levenberg-Marquardt Algorithm: Implementation and Theory,” in Numerical Analysis, vol. 630, ed. G. A. Watson (Springer-Verlag: Berlin), 105 (1977)

2006 (1)

F. Shi, S. Basinger, and D. Redding, “Performance of a Dispersed Fringe Sensor in the presence of segmented mirror aberrations: modeling and simulation,” SPIE 6265, 62650Y (2006).
[Crossref]

2003 (1)

F. Shi, D. Redding, A. Lowman, C. Bowers, L. Burns, P. Petrone, C. Ohara, and S. Basinger, “Segmented Mirror Coarse Phasing with a Dispersed Fringe Sensor: Experiment on NGST's Wavefront Control Testbed,” SPIE 4850, 318–328 (2003).
[Crossref]

2002 (1)

2000 (1)

1998 (1)

Basinger, S.

F. Shi, S. Basinger, and D. Redding, “Performance of a Dispersed Fringe Sensor in the presence of segmented mirror aberrations: modeling and simulation,” SPIE 6265, 62650Y (2006).
[Crossref]

F. Shi, D. Redding, A. Lowman, C. Bowers, L. Burns, P. Petrone, C. Ohara, and S. Basinger, “Segmented Mirror Coarse Phasing with a Dispersed Fringe Sensor: Experiment on NGST's Wavefront Control Testbed,” SPIE 4850, 318–328 (2003).
[Crossref]

Bowers, C.

F. Shi, D. Redding, A. Lowman, C. Bowers, L. Burns, P. Petrone, C. Ohara, and S. Basinger, “Segmented Mirror Coarse Phasing with a Dispersed Fringe Sensor: Experiment on NGST's Wavefront Control Testbed,” SPIE 4850, 318–328 (2003).
[Crossref]

Burns, L.

F. Shi, D. Redding, A. Lowman, C. Bowers, L. Burns, P. Petrone, C. Ohara, and S. Basinger, “Segmented Mirror Coarse Phasing with a Dispersed Fringe Sensor: Experiment on NGST's Wavefront Control Testbed,” SPIE 4850, 318–328 (2003).
[Crossref]

Chanan, G.

Dekens, F.

Devaney, N.

Kirkman, D.

Lowman, A.

F. Shi, D. Redding, A. Lowman, C. Bowers, L. Burns, P. Petrone, C. Ohara, and S. Basinger, “Segmented Mirror Coarse Phasing with a Dispersed Fringe Sensor: Experiment on NGST's Wavefront Control Testbed,” SPIE 4850, 318–328 (2003).
[Crossref]

Mast, T.

Michaels, S.

Montoya, L.

Nelson, J.

Ohara, C.

F. Shi, D. Redding, A. Lowman, C. Bowers, L. Burns, P. Petrone, C. Ohara, and S. Basinger, “Segmented Mirror Coarse Phasing with a Dispersed Fringe Sensor: Experiment on NGST's Wavefront Control Testbed,” SPIE 4850, 318–328 (2003).
[Crossref]

G. Chanan, C. Ohara, and M. Troy, “Phasing the mirror segments of the Keck telescopes II: the narrow-band phasing algorithm,” Appl. Opt. 39(25), 4706–4714 (2000).
[Crossref]

Petrone, P.

F. Shi, D. Redding, A. Lowman, C. Bowers, L. Burns, P. Petrone, C. Ohara, and S. Basinger, “Segmented Mirror Coarse Phasing with a Dispersed Fringe Sensor: Experiment on NGST's Wavefront Control Testbed,” SPIE 4850, 318–328 (2003).
[Crossref]

Redding, D.

F. Shi, S. Basinger, and D. Redding, “Performance of a Dispersed Fringe Sensor in the presence of segmented mirror aberrations: modeling and simulation,” SPIE 6265, 62650Y (2006).
[Crossref]

F. Shi, D. Redding, A. Lowman, C. Bowers, L. Burns, P. Petrone, C. Ohara, and S. Basinger, “Segmented Mirror Coarse Phasing with a Dispersed Fringe Sensor: Experiment on NGST's Wavefront Control Testbed,” SPIE 4850, 318–328 (2003).
[Crossref]

Schumacher, A.

Shi, F.

F. Shi, S. Basinger, and D. Redding, “Performance of a Dispersed Fringe Sensor in the presence of segmented mirror aberrations: modeling and simulation,” SPIE 6265, 62650Y (2006).
[Crossref]

F. Shi, D. Redding, A. Lowman, C. Bowers, L. Burns, P. Petrone, C. Ohara, and S. Basinger, “Segmented Mirror Coarse Phasing with a Dispersed Fringe Sensor: Experiment on NGST's Wavefront Control Testbed,” SPIE 4850, 318–328 (2003).
[Crossref]

Troy, M.

Appl. Opt. (3)

SPIE (2)

F. Shi, D. Redding, A. Lowman, C. Bowers, L. Burns, P. Petrone, C. Ohara, and S. Basinger, “Segmented Mirror Coarse Phasing with a Dispersed Fringe Sensor: Experiment on NGST's Wavefront Control Testbed,” SPIE 4850, 318–328 (2003).
[Crossref]

F. Shi, S. Basinger, and D. Redding, “Performance of a Dispersed Fringe Sensor in the presence of segmented mirror aberrations: modeling and simulation,” SPIE 6265, 62650Y (2006).
[Crossref]

Other (4)

K. L. Baker, E. A. Stappaerts, D. C. Homoelle, M. A. Henesian, E. S. Bliss, C. W. Siders, and C. P. J. Barty, “Interferometric adaptive optics for high power laser pointing, wave-front control and phasing,” accepted into Journal of Micro/Nanolithography MEMS, and MOEMS (2009).

W. Joseph, Goodman, Introduction to Fourier Optics, Boston: McGraw-Hill (1996).

M. Abromowitz and I. A. Stegun, Handbook of Mathematical Functions and Formulas, Graphs, and Mathematical Tables, New York: Dover Publications 231 (1972).

J. Moré, “The Levenberg-Marquardt Algorithm: Implementation and Theory,” in Numerical Analysis, vol. 630, ed. G. A. Watson (Springer-Verlag: Berlin), 105 (1977)

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Figures (10)

Fig. 1
Fig. 1 Pupil layout and ideal far-field energy distribution for the advanced radiography capability on the National Ignition Facility. The pupil layout is displayed in Fig. 1(a) and it represents four beam pairs with each of the beam pairs containing 1.99 kJ centered around 1.053 μm in a 5 ps pulse. Figure 1(b) shows the far-field pattern generated from this pupil assuming that all of the beams are pistoned correctly and have a perfect Strehl ratio.

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Fig. 2
Fig. 2 Integrated intensity along one axis of the far field pattern of two rectangular apertures with a differential phase shift. These figures provide a comparison between simulations, solid black line, and analytic calculations, dashed blue line, corresponding to Eq. (4) and they were generated assuming δ = 0.25a. They represent a differential phase shift between the apertures of 0, 1.65, 3.31 and 4.96 rad for Fig. 2(a), 2(b), 2(c) and 2(d), respectively.

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Fig. 3
Fig. 3 Experimental setup used to test the far-field distribution, ensquared energy and center-of-mass for two rectangular apertures with a differential phase shift. The test bed contains a 1053 nm laser, lenses, L, beam splitters, BS, a beam block, BB, a pixilated deformable mirror, MEMS, and a far-field camera.

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Fig. 4
Fig. 4 Measured far-field distributions obtained by placing a differential phase shift between two sides of a pixilated MEMS device They represent a differential phase shift between the apertures of 0, 1.65, 3.31 and 4.96 rad for Fig. 4(a), 4(b), 4(c) and 4(d), respectively.

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Fig. 5
Fig. 5 Integrated intensity along one axis of the far field pattern of two rectangular apertures with a differential phase shift. These figures provide a comparison between measurements, solid black line, and analytic equations fit to the measured data, open blue squares, corresponding to Eq. (4). They represent a differential phase shift between the apertures of 0, 1.65, 3.31 and 4.96 rad for Fig. 5(a), 5(b), 5(c) and 5(d), respectively.

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Fig. 6
Fig. 6 The Levenberg-Marquardt fit to the measured data displayed in Fig. 5 with the analytic equation given in Eq. (3). The thin blue line represents the applied phase to the MEMS device, the squares represent the fit for each of the 100 far-field distributions(five at each of the 20 applied phases) and the thick red line represents the phase variance between the applied phase and the five fit phases at each of the 20 applied phases.

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Fig. 7
Fig. 7 Fractional energy contained within a square in the far-field distribution. The blue line represents the analytic expression derived in Eq. (5). The measured ensquared energy is displayed as squares for each of the 100 measured far-field distributions. The square was chosen to encompass the first lobe of the far-field distribution which represents a choice of β = π in Eq. (5).

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Fig. 8
Fig. 8 Linearly interpolated fit to the measured data displayed in Fig. 7 with the analytic equation given in Eq. (5). The thin blue line represents the applied phase to the MEMS device, the squares represent the fit for each of the 100 far-field distributions (five at each of the 20 applied phases) and the thick red line represents the phase variance between the applied phase and the five fit phases at each of the 20 applied phases.

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Fig. 9
Fig. 9 Normalized center-of-mass of the far-field distribution as a function of relative phase shift between the two rectangular apertures. The blue line represents the analytic expression calculated in Eq. (6). The measured values for each of the recorded 100 far-field distributions are displayed by squares.

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Fig. 10
Fig. 10 Linearly interpolated fit to the measured data displayed in Fig. 9 with the analytic equation given in Eq. (6). The thin blue line represents the applied phase to the MEMS device, the squares represent the fit for each of the 100 far-field distributions (five at each of the 20 applied phases) and the thick red line represents the phase variance between the applied phase and the five fit phases at each of the 20 applied phases.

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Equations (7)

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E(x,y)={Aexp(iφ0) −a≤x≤δ; −b≤y≤bAexp(iφ1) δ<x≤a; −b≤y≤b ,0             x<−a; x>a; y<-b; y>b
EFF(kxa,kyb)=A∫−bbeikyydy[∫−aδeikxxeiφdx+∫δaeikxxdx] ,
IFF(β,ζ)∬IFF(β,ζ)dβdζ=sinc2(ζ)4π2β2{4−2cos[β(1+δ/a)]−2cos[β(1−δ/a)]−2cos(φ)+2cos[φ−β(1+δ/a)] +2cos[φ−β(1−δ/a)] −2cos(φ−2β)} ,
∫IFF(β,ζ)dζ∬IFF(β,ζ)dβdζ=14π1β2{4−2cos[β(1+δ/a)]−2cos[β(1−δ/a)]−2cos(φ)+2cos[φ−β(1+δ/a)] +2cos[φ−β(1−δ/a)]−2cos(φ−2β)} ,
EnsquaredEnergy = 18π2[−2η+2cos(2η)η+4Si(2η)]{2η[−4+2cos(φ)]+[−2+2cos(φ)](1+δa)[−2cos[η(1+δ/a)][η(1+δ/a)]−2Si[η(1+δ/a)]] +[−2+2cos(φ)] ,(1−δa)[−2cos[η(1−δ/a)][η(1−δ/a)]−2Si[η(1−δ/a)]]−4cos(φ)[−cos(2η)η−2Si(2η)]}
∬βIFF(β,ζ)dβdζ2π∬IFF(β,ζ)dβdζ=2πsin(φ){Si[α(1+δa)]Si[α(1−δa)]−Si(2α)}{2α[−4+2cos(φ)]+[−2+2cos(φ)](1+δa)[−2cos[α(1+δ/a)][α(1+δ/a)]−2Si[α(1+δ/a)]] ,[−2+2cos(φ)](1−δa)[−2cos[α(1−δ/a)][α(1−δ/a)]−2Si[α(1−δ/a)]]−4cos(φ)[−cos(2α)α−2Si(2α)]}−1
∬βIFF(β,ζ)dβdζ2π∬IFF(β,ζ)dβdζ=14πsin(φ) .

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