Abstract

An improved two-dimensional phase unwrapping procedure is discussed that uses a weighted least-squares algorithm, a congruence operation, and a filter to unwrap the phase distribution of an electromagnetic beam. These improvements make possible several advances for mirror designs used in gyrotron quasi-optical mode converters. The improved phase unwrapping procedure is demonstrated by applying it to a measured beam and a simulated beam that are used to design mirrors. The unwrapping procedure produces a smooth unwrapped phase that does not change the characteristics of the beam. The smooth unwrapped phase distribution is also used to find an estimate for the wavenumber vector distribution that is needed to design the mirrors.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. P. Perkins and R. J. Vernon, “Mirror design for use in gyrotron quasi-optical mode converters,” IEEE Trans. Plasma Sci. 35, 1747-1757 (2007).
    [CrossRef]
  2. M. P. Perkins, R. Cao, J. M. Neilson, and R. J. Vernon, “A high efficiency launcher and mirror system for use in a 110 GHz TE22,6 mode gyrotron,” Int. J. Infrared Millim. Waves 28, 207-218 (2007).
    [CrossRef]
  3. M. P. Perkins, “The design of beam shaping mirrors for gyrotrons to aid injection of an electromagnetic beam into a corrugated waveguide,” Ph.D. dissertation (University of Wisconsin-Madison, 2004).
  4. B. Z. Katsenelenbaum and V. V. Semenov, “Synthesis of phase correctors shaping a specified field,” J. Radiotechnics Electron. 12, 223-231 (1967).
  5. J. M. Neilson, “Optimal synthesis of quasi-optical launchers for high-power gyrotrons,” IEEE Trans. Plasma Sci. 34, 635-641 (2006).
    [CrossRef]
  6. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping Theory, Algorithms, and Software (Wiley, 1998).
  7. K. Itoh, “Analysis of the phase unwrapping algorithm,” Appl. Opt. 21, 2470 (1982).
    [CrossRef] [PubMed]
  8. D. C. Ghiglia, G. A. Mastin, and L. A. Romero, “Cellular-automata method for phase unwrapping,” J. Opt. Soc. Am. A 4, 267-280 (1987).
    [CrossRef]
  9. H. Takajo and T. Takahashi, “Least-squares phase estimation from the phase difference,” J. Opt. Soc. Am. A 5, 416-425(1988).
    [CrossRef]
  10. R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry--two-dimensional phase unwrapping,” Radio Sci. 23, 713-720 (1988).
    [CrossRef]
  11. D. J. Bone, “Fourier fringe analysis--the two-dimensional phase unwrapping problem,” Appl. Opt. 30, 3627-3632 (1991).
    [CrossRef] [PubMed]
  12. A. Baldi, “Two-dimensional phase unwrapping by quad-tree decomposition,” Appl. Opt. 40, 1187-1194(2001).
    [CrossRef]
  13. M. A. Herraez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, “Fast two-dimensional phase-unwrapping algorithm based on sorting by reliability following a noncontinuous path,” Appl. Opt. 41, 7437-7444 (2002).
    [CrossRef] [PubMed]
  14. J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. 28, 3268-3270 (1989).
    [CrossRef] [PubMed]
  15. J. A. Quiroga, A. Gonzalezcano, and E. Bernabeu, “Phase-unwrapping algorithm-based on an adaptive criterion,” Appl. Opt. 34, 2560-2563 (1995).
    [CrossRef] [PubMed]
  16. R. Cusack, J. M. Huntley, and H. T. Goldrein, “Improved noise-immune phase-unwrapping algorithm,” Appl. Opt. 34, 781-789 (1995).
    [CrossRef] [PubMed]
  17. J. R. Buckland, J. M. Huntley, and S. R. E. Turner, “Unwrapping noisy phase maps by use of a minimum-cost-matching algorithm,” Appl. Opt. 34, 5100-5108 (1995).
    [CrossRef] [PubMed]
  18. C. G. Quan, C. J. Tay, L. J. Chen, and Y. Fu, “Spatial-fringe-modulation-based quality map for phase unwrapping,” Appl. Opt. 42, 7060-7065 (2003).
    [CrossRef] [PubMed]
  19. W. Xu and I. Cumming, “A region-growing algorithm for InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 37, 124-134 (1999).
    [CrossRef]
  20. Z. J. Wang and S. S. Li, “Phase unwrapping through a branch-cut-based cut-bridging and window-patching method,” Appl. Opt. 38, 805-814 (1999).
    [CrossRef]
  21. J. A. Quiroga and E. Bernabeu, “Phase-unwrapping algorithm for noisy phase-map processing,” Appl. Opt. 33, 6725-6731(1994).
    [CrossRef] [PubMed]
  22. N. Hubig, S. Suchandt, and N. Adam, “Equivalence of cost generators for minimum cost flow phase unwrapping,” J. Opt. Soc. Am. A 19, 64-70 (2002).
    [CrossRef]
  23. T. J. Flynn, “Two-dimensional phase unwrapping with minimum weighted discontinuity,” J. Opt. Soc. Am. A 14, 2692-2701 (1997).
    [CrossRef]
  24. M. A. Herraez, D. R. Burton, M. J. Lalor, and D. B. Clegg, “Robust, simple and fast algorithm for phase unwrapping,” Appl. Opt. 35, 5847-5852 (1996).
    [CrossRef] [PubMed]
  25. B. Friedlander and J. M. Francos, “Model based phase unwrapping of 2-D signals,” IEEE Trans. Signal Process. 44, 2999-3007 (1996).
    [CrossRef]
  26. J. Strand and T. Taxt, “Two-dimensional phase unwrapping using robust derivative estimation and adaptive integration,” IEEE Trans. Image Process. 11, 1192-1200 (2002).
    [CrossRef]
  27. P. G. Charette and I. W. Hunter, “Robust phase-unwrapping method for phase images with high noise content,” Appl. Opt. 35, 3506-3513 (1996).
    [CrossRef] [PubMed]
  28. K. M. Hung and T. Yamada, “Phase unwrapping by regions using least-squares approach,” Opt. Eng. 37, 2965-2970(1998).
    [CrossRef]
  29. Q. Lin, J. F. Vesecky, and H. A. Zebker, “New approaches in interferometric SAR data processing,” IEEE Trans. Geosci. Remote Sens. 30, 560-567 (1992).
    [CrossRef]
  30. J. Strand, T. Taxt, and A. K. Jain, “Two-dimensional phase unwrapping using a block least-squares method,” IEEE Trans. Image Process. 8, 375-386 (1999).
    [CrossRef]
  31. D. C. Ghiglia and L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” J. Opt. Soc. Am. A 11, 107-117 (1994).
    [CrossRef]
  32. D. C. Ghiglia and L. A. Romero, “Minimum L(p)-norm two-dimensional phase unwrapping,” J. Opt. Soc. Am. A 13, 1999-2013 (1996).
    [CrossRef]
  33. M. D. Pritt, “Phase unwrapping by means of multigrid techniques for interferometric SAR,” IEEE Trans. Geosci. Remote Sens. 34, 728-738 (1996).
    [CrossRef]
  34. S. M. H. Song, S. Napel, N. J. Pelc, and G. H. Glover, “Phase unwrapping of MR phase images using Poisson equation,” IEEE Trans. Image Process. 4, 667-676 (1995).
    [CrossRef]
  35. M. D. Pritt and J. S. Shipman, “Least-squares two-dimensional phase unwrapping using FFTs,” IEEE Trans. Geosci. Remote Sens. 32, 706-708 (1994).
    [CrossRef]
  36. H. A. Zebker and Y. P. Lu, “Phase unwrapping algorithms for radar interferometry: residue-cut, least-squares, and synthesis algorithms,” J. Opt. Soc. Am. A 15, 586-598 (1998).
    [CrossRef]
  37. S. Stramaglia, L. Guerriero, G. Pasquariello, and N. Veneziani, “Mean-field annealing for phase unwrapping,” Appl. Opt. 38, 1377-1383 (1999).
    [CrossRef]
  38. O. Marklund, “Noise-insensitive two-dimensional phase unwrapping method,” J. Opt. Soc. Am. A 15, 42-60 (1998).
    [CrossRef]
  39. J. L. Marroquin and M. Rivera, “Quadratic regularization functionals for phase unwrapping,” J. Opt. Soc. Am. A 12, 2393-2400 (1995).
    [CrossRef]
  40. D. L. Fried, “Least-square fitting a wavefront distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. 67, 370-375 (1977).
    [CrossRef]
  41. R. H. Hudgin, “Wavefront reconstruction for compensated imaging,” J. Opt. Soc. Am. 67, N375-378 (1977).
    [CrossRef]
  42. R. Seara, A. A. Goncalves, and P. B. Uliana, “Filtering algorithm for noise reduction in phase-map images with 2π phase jumps,” Appl. Opt. 37, 2046-2050 (1998).
    [CrossRef]
  43. H. Kogelnik, “Coupling and conversion coefficients for optical modes,” in Symposium on Quasi-Optics (Polytechnic Institute of Brooklyn, 1964), pp. 333-347.

2007

M. P. Perkins and R. J. Vernon, “Mirror design for use in gyrotron quasi-optical mode converters,” IEEE Trans. Plasma Sci. 35, 1747-1757 (2007).
[CrossRef]

M. P. Perkins, R. Cao, J. M. Neilson, and R. J. Vernon, “A high efficiency launcher and mirror system for use in a 110 GHz TE22,6 mode gyrotron,” Int. J. Infrared Millim. Waves 28, 207-218 (2007).
[CrossRef]

2006

J. M. Neilson, “Optimal synthesis of quasi-optical launchers for high-power gyrotrons,” IEEE Trans. Plasma Sci. 34, 635-641 (2006).
[CrossRef]

2003

2002

2001

1999

S. Stramaglia, L. Guerriero, G. Pasquariello, and N. Veneziani, “Mean-field annealing for phase unwrapping,” Appl. Opt. 38, 1377-1383 (1999).
[CrossRef]

W. Xu and I. Cumming, “A region-growing algorithm for InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 37, 124-134 (1999).
[CrossRef]

J. Strand, T. Taxt, and A. K. Jain, “Two-dimensional phase unwrapping using a block least-squares method,” IEEE Trans. Image Process. 8, 375-386 (1999).
[CrossRef]

Z. J. Wang and S. S. Li, “Phase unwrapping through a branch-cut-based cut-bridging and window-patching method,” Appl. Opt. 38, 805-814 (1999).
[CrossRef]

1998

1997

1996

1995

1994

1992

Q. Lin, J. F. Vesecky, and H. A. Zebker, “New approaches in interferometric SAR data processing,” IEEE Trans. Geosci. Remote Sens. 30, 560-567 (1992).
[CrossRef]

1991

1989

1988

H. Takajo and T. Takahashi, “Least-squares phase estimation from the phase difference,” J. Opt. Soc. Am. A 5, 416-425(1988).
[CrossRef]

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry--two-dimensional phase unwrapping,” Radio Sci. 23, 713-720 (1988).
[CrossRef]

1987

1982

1977

1967

B. Z. Katsenelenbaum and V. V. Semenov, “Synthesis of phase correctors shaping a specified field,” J. Radiotechnics Electron. 12, 223-231 (1967).

Adam, N.

Baldi, A.

Bernabeu, E.

Bone, D. J.

Buckland, J. R.

Burton, D. R.

Cao, R.

M. P. Perkins, R. Cao, J. M. Neilson, and R. J. Vernon, “A high efficiency launcher and mirror system for use in a 110 GHz TE22,6 mode gyrotron,” Int. J. Infrared Millim. Waves 28, 207-218 (2007).
[CrossRef]

Charette, P. G.

Chen, L. J.

Clegg, D. B.

Cumming, I.

W. Xu and I. Cumming, “A region-growing algorithm for InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 37, 124-134 (1999).
[CrossRef]

Cusack, R.

Flynn, T. J.

Francos, J. M.

B. Friedlander and J. M. Francos, “Model based phase unwrapping of 2-D signals,” IEEE Trans. Signal Process. 44, 2999-3007 (1996).
[CrossRef]

Fried, D. L.

Friedlander, B.

B. Friedlander and J. M. Francos, “Model based phase unwrapping of 2-D signals,” IEEE Trans. Signal Process. 44, 2999-3007 (1996).
[CrossRef]

Fu, Y.

Gdeisat, M. A.

Ghiglia, D. C.

Glover, G. H.

S. M. H. Song, S. Napel, N. J. Pelc, and G. H. Glover, “Phase unwrapping of MR phase images using Poisson equation,” IEEE Trans. Image Process. 4, 667-676 (1995).
[CrossRef]

Goldrein, H. T.

Goldstein, R. M.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry--two-dimensional phase unwrapping,” Radio Sci. 23, 713-720 (1988).
[CrossRef]

Goncalves, A. A.

Gonzalezcano, A.

Guerriero, L.

Herraez, M. A.

Hubig, N.

Hudgin, R. H.

Hung, K. M.

K. M. Hung and T. Yamada, “Phase unwrapping by regions using least-squares approach,” Opt. Eng. 37, 2965-2970(1998).
[CrossRef]

Hunter, I. W.

Huntley, J. M.

Itoh, K.

Jain, A. K.

J. Strand, T. Taxt, and A. K. Jain, “Two-dimensional phase unwrapping using a block least-squares method,” IEEE Trans. Image Process. 8, 375-386 (1999).
[CrossRef]

Katsenelenbaum, B. Z.

B. Z. Katsenelenbaum and V. V. Semenov, “Synthesis of phase correctors shaping a specified field,” J. Radiotechnics Electron. 12, 223-231 (1967).

Kogelnik, H.

H. Kogelnik, “Coupling and conversion coefficients for optical modes,” in Symposium on Quasi-Optics (Polytechnic Institute of Brooklyn, 1964), pp. 333-347.

Lalor, M. J.

Li, S. S.

Lin, Q.

Q. Lin, J. F. Vesecky, and H. A. Zebker, “New approaches in interferometric SAR data processing,” IEEE Trans. Geosci. Remote Sens. 30, 560-567 (1992).
[CrossRef]

Lu, Y. P.

Marklund, O.

Marroquin, J. L.

Mastin, G. A.

Napel, S.

S. M. H. Song, S. Napel, N. J. Pelc, and G. H. Glover, “Phase unwrapping of MR phase images using Poisson equation,” IEEE Trans. Image Process. 4, 667-676 (1995).
[CrossRef]

Neilson, J. M.

M. P. Perkins, R. Cao, J. M. Neilson, and R. J. Vernon, “A high efficiency launcher and mirror system for use in a 110 GHz TE22,6 mode gyrotron,” Int. J. Infrared Millim. Waves 28, 207-218 (2007).
[CrossRef]

J. M. Neilson, “Optimal synthesis of quasi-optical launchers for high-power gyrotrons,” IEEE Trans. Plasma Sci. 34, 635-641 (2006).
[CrossRef]

Pasquariello, G.

Pelc, N. J.

S. M. H. Song, S. Napel, N. J. Pelc, and G. H. Glover, “Phase unwrapping of MR phase images using Poisson equation,” IEEE Trans. Image Process. 4, 667-676 (1995).
[CrossRef]

Perkins, M. P.

M. P. Perkins, R. Cao, J. M. Neilson, and R. J. Vernon, “A high efficiency launcher and mirror system for use in a 110 GHz TE22,6 mode gyrotron,” Int. J. Infrared Millim. Waves 28, 207-218 (2007).
[CrossRef]

M. P. Perkins and R. J. Vernon, “Mirror design for use in gyrotron quasi-optical mode converters,” IEEE Trans. Plasma Sci. 35, 1747-1757 (2007).
[CrossRef]

M. P. Perkins, “The design of beam shaping mirrors for gyrotrons to aid injection of an electromagnetic beam into a corrugated waveguide,” Ph.D. dissertation (University of Wisconsin-Madison, 2004).

Pritt, M. D.

M. D. Pritt, “Phase unwrapping by means of multigrid techniques for interferometric SAR,” IEEE Trans. Geosci. Remote Sens. 34, 728-738 (1996).
[CrossRef]

M. D. Pritt and J. S. Shipman, “Least-squares two-dimensional phase unwrapping using FFTs,” IEEE Trans. Geosci. Remote Sens. 32, 706-708 (1994).
[CrossRef]

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping Theory, Algorithms, and Software (Wiley, 1998).

Quan, C. G.

Quiroga, J. A.

Rivera, M.

Romero, L. A.

Seara, R.

Semenov, V. V.

B. Z. Katsenelenbaum and V. V. Semenov, “Synthesis of phase correctors shaping a specified field,” J. Radiotechnics Electron. 12, 223-231 (1967).

Shipman, J. S.

M. D. Pritt and J. S. Shipman, “Least-squares two-dimensional phase unwrapping using FFTs,” IEEE Trans. Geosci. Remote Sens. 32, 706-708 (1994).
[CrossRef]

Song, S. M. H.

S. M. H. Song, S. Napel, N. J. Pelc, and G. H. Glover, “Phase unwrapping of MR phase images using Poisson equation,” IEEE Trans. Image Process. 4, 667-676 (1995).
[CrossRef]

Stramaglia, S.

Strand, J.

J. Strand and T. Taxt, “Two-dimensional phase unwrapping using robust derivative estimation and adaptive integration,” IEEE Trans. Image Process. 11, 1192-1200 (2002).
[CrossRef]

J. Strand, T. Taxt, and A. K. Jain, “Two-dimensional phase unwrapping using a block least-squares method,” IEEE Trans. Image Process. 8, 375-386 (1999).
[CrossRef]

Suchandt, S.

Takahashi, T.

Takajo, H.

Taxt, T.

J. Strand and T. Taxt, “Two-dimensional phase unwrapping using robust derivative estimation and adaptive integration,” IEEE Trans. Image Process. 11, 1192-1200 (2002).
[CrossRef]

J. Strand, T. Taxt, and A. K. Jain, “Two-dimensional phase unwrapping using a block least-squares method,” IEEE Trans. Image Process. 8, 375-386 (1999).
[CrossRef]

Tay, C. J.

Turner, S. R. E.

Uliana, P. B.

Veneziani, N.

Vernon, R. J.

M. P. Perkins and R. J. Vernon, “Mirror design for use in gyrotron quasi-optical mode converters,” IEEE Trans. Plasma Sci. 35, 1747-1757 (2007).
[CrossRef]

M. P. Perkins, R. Cao, J. M. Neilson, and R. J. Vernon, “A high efficiency launcher and mirror system for use in a 110 GHz TE22,6 mode gyrotron,” Int. J. Infrared Millim. Waves 28, 207-218 (2007).
[CrossRef]

Vesecky, J. F.

Q. Lin, J. F. Vesecky, and H. A. Zebker, “New approaches in interferometric SAR data processing,” IEEE Trans. Geosci. Remote Sens. 30, 560-567 (1992).
[CrossRef]

Wang, Z. J.

Werner, C. L.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry--two-dimensional phase unwrapping,” Radio Sci. 23, 713-720 (1988).
[CrossRef]

Xu, W.

W. Xu and I. Cumming, “A region-growing algorithm for InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 37, 124-134 (1999).
[CrossRef]

Yamada, T.

K. M. Hung and T. Yamada, “Phase unwrapping by regions using least-squares approach,” Opt. Eng. 37, 2965-2970(1998).
[CrossRef]

Zebker, H. A.

H. A. Zebker and Y. P. Lu, “Phase unwrapping algorithms for radar interferometry: residue-cut, least-squares, and synthesis algorithms,” J. Opt. Soc. Am. A 15, 586-598 (1998).
[CrossRef]

Q. Lin, J. F. Vesecky, and H. A. Zebker, “New approaches in interferometric SAR data processing,” IEEE Trans. Geosci. Remote Sens. 30, 560-567 (1992).
[CrossRef]

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry--two-dimensional phase unwrapping,” Radio Sci. 23, 713-720 (1988).
[CrossRef]

Appl. Opt.

D. J. Bone, “Fourier fringe analysis--the two-dimensional phase unwrapping problem,” Appl. Opt. 30, 3627-3632 (1991).
[CrossRef] [PubMed]

J. A. Quiroga and E. Bernabeu, “Phase-unwrapping algorithm for noisy phase-map processing,” Appl. Opt. 33, 6725-6731(1994).
[CrossRef] [PubMed]

R. Seara, A. A. Goncalves, and P. B. Uliana, “Filtering algorithm for noise reduction in phase-map images with 2π phase jumps,” Appl. Opt. 37, 2046-2050 (1998).
[CrossRef]

Z. J. Wang and S. S. Li, “Phase unwrapping through a branch-cut-based cut-bridging and window-patching method,” Appl. Opt. 38, 805-814 (1999).
[CrossRef]

R. Cusack, J. M. Huntley, and H. T. Goldrein, “Improved noise-immune phase-unwrapping algorithm,” Appl. Opt. 34, 781-789 (1995).
[CrossRef] [PubMed]

J. A. Quiroga, A. Gonzalezcano, and E. Bernabeu, “Phase-unwrapping algorithm-based on an adaptive criterion,” Appl. Opt. 34, 2560-2563 (1995).
[CrossRef] [PubMed]

J. R. Buckland, J. M. Huntley, and S. R. E. Turner, “Unwrapping noisy phase maps by use of a minimum-cost-matching algorithm,” Appl. Opt. 34, 5100-5108 (1995).
[CrossRef] [PubMed]

M. A. Herraez, D. R. Burton, M. J. Lalor, and D. B. Clegg, “Robust, simple and fast algorithm for phase unwrapping,” Appl. Opt. 35, 5847-5852 (1996).
[CrossRef] [PubMed]

P. G. Charette and I. W. Hunter, “Robust phase-unwrapping method for phase images with high noise content,” Appl. Opt. 35, 3506-3513 (1996).
[CrossRef] [PubMed]

S. Stramaglia, L. Guerriero, G. Pasquariello, and N. Veneziani, “Mean-field annealing for phase unwrapping,” Appl. Opt. 38, 1377-1383 (1999).
[CrossRef]

A. Baldi, “Two-dimensional phase unwrapping by quad-tree decomposition,” Appl. Opt. 40, 1187-1194(2001).
[CrossRef]

K. Itoh, “Analysis of the phase unwrapping algorithm,” Appl. Opt. 21, 2470 (1982).
[CrossRef] [PubMed]

J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. 28, 3268-3270 (1989).
[CrossRef] [PubMed]

M. A. Herraez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, “Fast two-dimensional phase-unwrapping algorithm based on sorting by reliability following a noncontinuous path,” Appl. Opt. 41, 7437-7444 (2002).
[CrossRef] [PubMed]

C. G. Quan, C. J. Tay, L. J. Chen, and Y. Fu, “Spatial-fringe-modulation-based quality map for phase unwrapping,” Appl. Opt. 42, 7060-7065 (2003).
[CrossRef] [PubMed]

IEEE Trans. Geosci. Remote Sens.

W. Xu and I. Cumming, “A region-growing algorithm for InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 37, 124-134 (1999).
[CrossRef]

Q. Lin, J. F. Vesecky, and H. A. Zebker, “New approaches in interferometric SAR data processing,” IEEE Trans. Geosci. Remote Sens. 30, 560-567 (1992).
[CrossRef]

M. D. Pritt, “Phase unwrapping by means of multigrid techniques for interferometric SAR,” IEEE Trans. Geosci. Remote Sens. 34, 728-738 (1996).
[CrossRef]

M. D. Pritt and J. S. Shipman, “Least-squares two-dimensional phase unwrapping using FFTs,” IEEE Trans. Geosci. Remote Sens. 32, 706-708 (1994).
[CrossRef]

IEEE Trans. Image Process.

J. Strand and T. Taxt, “Two-dimensional phase unwrapping using robust derivative estimation and adaptive integration,” IEEE Trans. Image Process. 11, 1192-1200 (2002).
[CrossRef]

S. M. H. Song, S. Napel, N. J. Pelc, and G. H. Glover, “Phase unwrapping of MR phase images using Poisson equation,” IEEE Trans. Image Process. 4, 667-676 (1995).
[CrossRef]

J. Strand, T. Taxt, and A. K. Jain, “Two-dimensional phase unwrapping using a block least-squares method,” IEEE Trans. Image Process. 8, 375-386 (1999).
[CrossRef]

IEEE Trans. Plasma Sci.

M. P. Perkins and R. J. Vernon, “Mirror design for use in gyrotron quasi-optical mode converters,” IEEE Trans. Plasma Sci. 35, 1747-1757 (2007).
[CrossRef]

J. M. Neilson, “Optimal synthesis of quasi-optical launchers for high-power gyrotrons,” IEEE Trans. Plasma Sci. 34, 635-641 (2006).
[CrossRef]

IEEE Trans. Signal Process.

B. Friedlander and J. M. Francos, “Model based phase unwrapping of 2-D signals,” IEEE Trans. Signal Process. 44, 2999-3007 (1996).
[CrossRef]

Int. J. Infrared Millim. Waves

M. P. Perkins, R. Cao, J. M. Neilson, and R. J. Vernon, “A high efficiency launcher and mirror system for use in a 110 GHz TE22,6 mode gyrotron,” Int. J. Infrared Millim. Waves 28, 207-218 (2007).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Radiotechnics Electron.

B. Z. Katsenelenbaum and V. V. Semenov, “Synthesis of phase correctors shaping a specified field,” J. Radiotechnics Electron. 12, 223-231 (1967).

Opt. Eng.

K. M. Hung and T. Yamada, “Phase unwrapping by regions using least-squares approach,” Opt. Eng. 37, 2965-2970(1998).
[CrossRef]

Radio Sci.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry--two-dimensional phase unwrapping,” Radio Sci. 23, 713-720 (1988).
[CrossRef]

Other

M. P. Perkins, “The design of beam shaping mirrors for gyrotrons to aid injection of an electromagnetic beam into a corrugated waveguide,” Ph.D. dissertation (University of Wisconsin-Madison, 2004).

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping Theory, Algorithms, and Software (Wiley, 1998).

H. Kogelnik, “Coupling and conversion coefficients for optical modes,” in Symposium on Quasi-Optics (Polytechnic Institute of Brooklyn, 1964), pp. 333-347.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

Schematic diagram of a four-mirror internal quasi-optical mode converter in a gyrotron and the coordinate system being used.

Fig. 2
Fig. 2

(a) Amplitude contours of E x for measured beam over a plane from 3 dB to 30 dB in 3 dB increments. (b) Corresponding wrapped phase of E x .

Fig. 3
Fig. 3

(a) Amplitude contours of E φ for a simulated beam over a toroid from 3 dB to 30 dB in 3 dB increments. (b) Corresponding wrapped phase of E φ .

Fig. 4
Fig. 4

(a) Wrapped phase of s ( x ) = exp ( j 5 x 2 ) . (b) Integer values of g ( x ) needed to unwrap s ( x ) . (c) Unwrapped phase of s ( x ) .

Fig. 5
Fig. 5

Residues of charge + 1 ( + ) and 1 ( x ) for the wrapped phases of (a) the measured beam and (b) the simulated beam.

Fig. 6
Fig. 6

Discontinuities greater than π in the unwrapped congruent phase after applying Eq. (9) for (a) the measured beam and (b) the simulated beam.

Fig. 7
Fig. 7

Discontinuities in the unwrapped congruent phase greater than π for the simulated beam after manipulation.

Fig. 8
Fig. 8

Final unwrapped phase distributions with the linear tilt in the z direction subtracted off for (a) the measured beam and (b) the simulated beam.

Fig. 9
Fig. 9

Final unwrapped phase distributions rewrapped for (a) the measured beam and (b) the simulated beam.

Equations (13)

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

δ ( x , z ) = 1 2 k k ( x , z ) · n ^ ( x , z ) | k ( x , z ) | θ ( x , z ) n y ( x , z ) .
ψ ( r ) = tan 1 ( Im ( s ( r ) ) Re ( s ( r ) ) ) ,
φ ( r ) = ψ ( r ) + 2 π g ( r ) ,
φ ( r ) = C φ · d r + φ ( r 0 ) ,
Δ 1 = ψ ( m , n + 1 ) ψ ( m , n ) + 2 π × integer, Δ 2 = ψ ( m + 1 , n + 1 ) ψ ( m , n + 1 ) + 2 π × integer, Δ 3 = ψ ( m + 1 , n ) ψ ( m + 1 , n + 1 ) + 2 π × integer, Δ 4 = ψ ( m , n ) ψ ( m + 1 , n ) + 2 π × integer.
q = n = 1 4 Δ n .
Δ = U ( x , z ) | φ ( x , z ) x ψ ( x , z ) x | 2 + V ( x , z ) | φ ( x , z ) z ψ ( x , z ) z | 2 d x d z .
exp ( j φ ( x , z ) ) = exp ( j ψ ( x , z ) ) .
φ c = ψ + 2 π × round ( φ ( ψ t ) 2 π ) , 0 < t < 2 π ,
| c | 2 = | a ( x , z ) 2 exp [ j ( φ filtered ( x , z ) ψ ( x , z ) ) ] d A a ( x , z ) 2 d A | 2 ,
Surf ( x , z ) = [ φ filtered ( x , z ) k y ( x , z ) k ] .
k 2 ( x , z ) = k Re ( E ( x , z ) × H * ( x , z ) ) | Re ( E ( x , z ) × H * ( x , z ) ) | .
c k = k 1 ( x , z ) · k 2 ( x , z ) d A k 1 ( x , z ) 2 d A k 2 ( x , z ) 2 d A .

Metrics