## Simulation study on the detection of size, shape and position of three different scatterers using Non-standard time domain time inverse Maxwell’s algorithm

Optics Express, Vol. 18, Issue 5, pp. 4148-4157 (2010)

http://dx.doi.org/10.1364/OE.18.004148

Acrobat PDF (199 KB)

### Abstract

Inverse method has wide application on medical diagnosis where non-destructive evaluation is the key factor .Back scattered waves or echoes generated from the forward moving waves has information about its geometry, size and location. In this paper we have investigated how well different geometries of the object is determined from the back scattered waves by a high accuracy Non-Standard Finite Difference Time Inverse (NSFD-TI) Maxwell’s algorithm and how the refractive index of the object plays a deterministic role on its size.

© 2010 OSA

## 1. Introduction

## 2. Model and Calculation Methods

*H*is the magnetic field,

*B*is the magnetic induction,

*E*the electric field,

*D*the electric displacement,

*J*the electric current density, and

*ρ*is the density of free charge. Materials, such as iron or glass, modify the electric and magnetic fields.

*B*describes how the material affects the magnetic field, and

*D*describes how it affects electric field. The vector position is

*x*=

*V*is

*J*= 0. Also to an excellent approximationwhere,

*μ*, the magnetic permeability, is a constant. In general, the electrical permittivity,

*ε*depends upon the frequency, but when this dependence is weak, it is a good approximation to takeThe permittivity is different for different materials, however so it is a function of position. For example, in glass

*H*. Expanding

*H*and

*E*is evaluated at a different position. For example to compute

7. J. B. Cole, “High-Accuracy Yee algorithm Based on Nonstandard Finite Difference: New Developments and Verifications,” IEEE Trans. Antenn. Propag. **50**(9), 1185–1191 (2002). [CrossRef]

*γ*which is also a function of

*μ*), given by the expression

*μ*is constant. To reverse the time direction we need to reverse the sequence of the above computations; doing so we get,andHere, we observed that we basically get the same equations as we already derived for forward propagating wave. This depicts that even if we iterate the wave in time reverse direction the nature of the wave remain same. For this purpose we have applied the periodic boundary conditions instead of Mur boundary conditions because the later causes field explosion when we iterate in time reverse condition [5

5. R. Sorrentino, L. Roselli, and P. Mezzanotte, “Time reversal in finite difference time domain method,” IEEE Microw. Guid. Wave Lett. **3**(11), 402–404 (1993). [CrossRef]

## Algorithm Stability

*A*be an algorithm to solve for an unknown vector

*ψ*. Let

*D*-dimensional FDTD algorithm is stable if

*λ*we get,Thus the most general solution is:The condition for the stability will be

*D*= Dimension

*γ*. For the stability we optimize the value of

*γ*by

*γ*given above, we get Max (

*k*, stability is guaranteed by inserting the maximum value of

*θ*, so it is obvious that

## 3. Verifications and Practical Tests

*θ*) from the

*h*), then we remove the scatterer and iterate the forward moving wave in the time reverse in order to trace the location and shape of the scatterer and examine how well it will match with the actual location and size. For this experiment, we have confined our observation on TE mode only, i.e. we will compute x-intensity and y-intensity as depicted in Fig. 3 .

## 4. Results and discussions

## Acknowledgements

## References and links

1. | L. W. Schmerr, Jr., in |

2. | R. Mickens, E, in |

3. | K. S. Kunz, and R. J. Luebbers, in The |

4. | J. B. Cole, S. Banerjee, and M. Haftel, in |

5. | R. Sorrentino, L. Roselli, and P. Mezzanotte, “Time reversal in finite difference time domain method,” IEEE Microw. Guid. Wave Lett. |

6. | P. W. Barber, and S. C. Hill, in |

7. | J. B. Cole, “High-Accuracy Yee algorithm Based on Nonstandard Finite Difference: New Developments and Verifications,” IEEE Trans. Antenn. Propag. |

8. | H. Kudo, |

**OCIS Codes**

(200.0200) Optics in computing : Optics in computing

(070.7345) Fourier optics and signal processing : Wave propagation

**ToC Category:**

Scattering

**History**

Original Manuscript: October 7, 2009

Revised Manuscript: December 14, 2009

Manuscript Accepted: December 16, 2009

Published: February 17, 2010

**Virtual Issues**

Vol. 5, Iss. 6 *Virtual Journal for Biomedical Optics*

**Citation**

Kisalaya Chakrabarti and James B. Cole, "Simulation study on the detection of size, shape and position of three different scatterers using Non-standard time domain time inverse Maxwell’s algorithm," Opt. Express **18**, 4148-4157 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-5-4148

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### References

- L. W. Schmerr, Jr., in Fundamentals of Ultrasonic Nondestructive Evaluation: A Modeling Approach, (Plenum Press, New York, 1998).
- R. Mickens, E, in Nonstandard finite difference models of differential equations, (World Scientific Publishing Co., Inc., River Edge, NJ, 1994).
- K. S. Kunz, and R. J. Luebbers, in The Finite Difference Time Domain Method for Electromagnetics, (CRC Press, New York, 1993).
- J. B. Cole, S. Banerjee, and M. Haftel, in Advances in the Applications of Nonstandard Finite Difference Schemes, ed. R. E. Mickens (World Scientific Singapore, 2007), Chap.4.
- R. Sorrentino, L. Roselli, and P. Mezzanotte, “Time reversal in finite difference time domain method,” IEEE Microw. Guid. Wave Lett. 3(11), 402–404 (1993). [CrossRef]
- P. W. Barber, and S. C. Hill, in Light Scattering by Particles: Computational Methods, (World Scientific, Singapore, 1990).
- J. B. Cole, “High-Accuracy Yee algorithm Based on Nonstandard Finite Difference: New Developments and Verifications,” IEEE Trans. Antenn. Propag. 50(9), 1185–1191 (2002). [CrossRef]
- H. Kudo, et al., “Numerical Dispersion and Stability Condition of the Nonstandard FDTD Method” Electronics and Communications in Japan, 85, 22–30(2002), http://www3.interscience.wiley.com/cgi-bin/fulltext/93514073/PDFSTART .

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