A computational and statistical framework for multidimensional domain acoustooptic material interrogation

Authors:
H. T. Banks and V. A. Bokil

Journal:
Quart. Appl. Math. **63** (2005), 156-200

MSC (2000):
Primary 78M20, 78A25, 78A02, 78A46, 62F25

DOI:
https://doi.org/10.1090/S0033-569X-05-00949-0

Published electronically:
February 1, 2005

MathSciNet review:
2126573

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Abstract | References | Similar Articles | Additional Information

Abstract: We consider an electromagnetic interrogation technique in two and three dimensions for identifying the dielectric parameters (including the permittivity, the conductivity and the relaxation time) of a Debye medium. In this technique, a travelling acoustic pressure wave in the Debye medium is used as a virtual reflector for an interrogating microwave electromagnetic pulse that is generated in free space. The reflections of the microwave pulse from the air-Debye interface and from the acoustic pressure wave are recorded at a remote antenna. The data is used in an inverse problem to estimate the locally pressure dependent dielectric parameters of the Debye medium. We present a time domain formulation that is solved using finite differences (FDTD) in time and in space. Perfectly matched layer (PML) absorbing boundary conditions are used to absorb outgoing waves at the finite boundaries of the computational domain, preventing spurious reflections from reentering the domain.

Using the method of least squares for the parameter identification problem, we compare two different algorithms (the gradient based Levenberg-Marquardt method and the gradient free, simplex based Nelder-Mead method) in solving an inverse problem to calculate estimates for two or more dielectric parameters. Finally we use statistical error analysis to construct confidence intervals for all the presented estimates, thereby providing a probabilistic statement about the computational procedure with uncertainty aspects of estimates.

**[ABK04]**R. Albanese, H. T. Banks, and G. M. Kepler,*Experimental results involving acousto-optical interactions*, Tech. Report in preparation, CRSC, N. C. State University, 2004.**[ABR02]**R. A. Albanese, H. T. Banks, and J. K. Raye,*Nondestructive evaluation of materials using pulsed microwave interrogating signals and acoustic wave induced reflections*, Inverse Problems**18**(2002), 1935-1958. MR**1955927 (2003k:78021)****[AMP94]**R. A. Albanese, R. L. Medina, and J. W. Penn,*Mathematics, medicine and microwaves*, Inverse Problems**10**(1994), 995-1007. MR**1296358****[And67]**J. C. Anderson,*Dielectrics*, Chapman and Hall, London, 1967.**[APM89]**R. A. Albanese, J. W. Penn, and R. L. Medina,*Short-rise-time microwave pulse propagation through dispersive biological media*, J. Optical Society of America A**6**(1989), 1441-1446.**[BB]**H. T. Banks and K. L. Bihari,*Modeling and estimating uncertainty in parameter estimation*, Tech. Report CRSC-TR99-40, N. C. State University, Dec. 1999;*Inverse Problems*17: 95-111, 2001. MR**1818494 (2002d:35215)****[BB04a]**H. T. Banks and J. M. Bardsley,*Parameter identification for a dispersive dielectric in 2d electromagnetics: Forward and inverse methodology with statistical considerations*, Tech. Report CRSC-TR03-48, N. C. State University, Dec. 2003;*Intl. J. Comp. and Num. Anal. and Applic.,*to appear, 2004.**[BB04b]**H. T. Banks and V. A. Bokil,*Parameter identification for dispersive dielectrics using pulsed microwave interrogating signals and acoustic wave induced reflections in two and three dimensions*, Tech. Report CRSC-TR04-27, N. C. State University, July 2004.**[BBL00]**H. T. Banks, M. W. Buksas, and T. Lin,*Electromagnetic material interrogation using conductive interfaces and acoustic wavefronts*, Frontiers in Applied Mathematics, vol. FR21, SIAM, Philadelphia, PA, 2000. MR**1787981 (2001k:78036)****[Ber94]**J. P. Berenger,*A perfectly matched layer for the absorption of electromagnetic waves*, J. Comput. Phys.**114**(1994), no. 2, 185-200. MR**1294924 (95e:78002)****[BF95]**J. G. Blaschak and J. Franzen,*Precursor propagation in dispersive media from short-rise-time pulses at oblique incidence*, J. Optical Society of America A**12**(1995), 1501-1512.**[BG04]**H. T. Banks and N. L. Gibson,*Well-posedness in maxwell systems with distributions of polarization relaxation parameters*, Tech. Report CRSC-TR04-01, N. C. State University, Jan. 2004;*Applied Math. Letters,*to appear, 2004.**[BGW03]**H. T. Banks, N. L. Gibson, and W. P. Winfree,*Electromagnetic crack detection inverse problems using terahertz interrogating propagating signals*, Tech. Report CRSC-TR03-40, N. C. State University, October 2003.**[BK89]**H. T. Banks and K. Kunisch,*Estimation techniques for distributed parameter systems*, Birkhauser, Boston, 1989. MR**1045629 (91b:93085)****[Bri60]**L. Brillouin,*Wave propagation and group velocity*, Academic Press, New York and London, 1960. MR**0108217 (21:6933)****[DG95]**M. Davidian and D. M. Giltinan,*Nonlinear models for repeated measurement data*, Monographs on Statistics and Applied Probability, vol. 62, Chapman and Hall, New York, 1995.**[FMS03]**E. C. Fear, P. M. Meaney, and M. A. Stuchly,*Microwaves for breast cancer detection*, IEEE Potentials (2003), 12-18.**[Ged96]**S. D. Gedney,*An anisotropic perfectly matched layer absorbing media for the truncation of FDTD lattices*, IEEE Trans. Antennas Propagat.**44**(1996), no. 12, 1630-1639.**[Jac99]**J. D. Jackson,*Classical electromagnetics*, John Wiley and Sons, New York, 1999.**[JN84]**N. S. Jayant and P. Noll,*Digital coding of waveforms*, Prentice Hall, Englewood Cliffs, 1984.**[Kel99]**C. T. Kelley,*Iterative methods for optimization*, Frontiers in Applied Mathematics, vol. FR18, SIAM, Philadelphia, PA, 1999. MR**1678201 (2000h:90003)****[Kor97]**A. Korpel,*Acousto-optics*, 2nd ed., Marcel-Dekker, New York, 1997.**[MI68]**P. Morse and K. Ingard,*Theoretical acoustics*, McGraw-Hill, New York, 1968.**[MP95]**R. Mittra and U. Pekel,*A new look at the perfectly matched layer (PML) concept for the reflectionless absorption of electromagnetic waves*, IEEE Microwave Guided Wave Lett.**5**(1995), no. 3, 84-86.**[Pet94]**P. Petropoulos,*Stability and phase error analysis of FDTD in dispersive dielectrics*, IEEE Trans. Antennas Propagat.**42**(1994), no. 1, 62-69.**[Rap95]**C. M. Rappaport,*Perfectly matched absorbing boundary conditions based on anisotropic lossy mapping of space*, IEEE Microwave Guided Wave Lett.**5**(1995), no. 3, 90-92.**[SKLL95]**Z. S. Sacks, D. M. Kinsland, R. Lee, and J. F. Lee,*A perfectly matched anisotropic absorber for use as an absorbing boundary condition*, IEEE Trans. Antennas Propagat.**43**(1995), 1460-1463.**[Taf95]**A. Taflove,*Computational electrodynamics: The finite-difference time-domain method*, Artech House, Norwood, MA, 1995. MR**1338377 (96f:78001)****[Taf98]**-,*Advances in computational electrodynamics: The finite-difference time-domain method*, Artech House, Norwood, MA, 1998. MR**1639352 (99c:78001)**

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Additional Information

**H. T. Banks**

Affiliation:
Center for Research in Scientific Computation, North Carolina State University, Raleigh, N.C. 27695-8205

Email:
htbanks@ncsu.edu

**V. A. Bokil**

Affiliation:
Center for Research in Scientific Computation, North Carolina State University, Raleigh, N.C. 27695-8205

Email:
vabokil@ncsu.edu

DOI:
https://doi.org/10.1090/S0033-569X-05-00949-0

Keywords:
Electromagnetic-acoustic interaction,
Debye dielectric materials,
pulsed antenna source microwaves,
inverse problems,
FDTD,
statistical inference.

Received by editor(s):
August 14, 2004

Published electronically:
February 1, 2005

Article copyright:
© Copyright 2005
by the authors