Maxwells Equation In Time

Maxwell’s Equation in the Time-Frequency Distribution H.Kapasi Abstract: We know the Maxwell’s equation in time domain and the frequency domain. There are cases when analysis in time Frequency distribution is needed. In such case Maxwell’s equation doesn't provide enough variables to extract those information. Usually, Numerical methods in the Signal Processing are employed to achieve such goal. Here, Maxwells equation is modified using the operator algebra and Eigen analysis to include the time frequency distributions and frequency transients of any order.

Keywords: Maxwell’s Equation , Time Frequency Distributions(TFD), Frequency Derivatives Distribution (FDD), Frequency Transients. Fourier Transforms. Introduction: Maxwell’s Equation is modified to include the frequency transients of any order. Maxwell’s Equation in the present form in the Time and Frequency Domain doesn't provide enough information as each type excludes the other . Hence, the Time Frequency Distributions cannot be calculated from the Maxwell’s Equation. Various Numerical techniques are used to do TFD analysis. Wigner-Ville Distribution, Gabor Transform, Spectrogram/ Short Time Fourier Transforms (STFT) [1] are among the well known numerical tools. If the Instantaneous frequency (IF) of the signal is predicted [2] then the Maxwell’s equation can be represented easily in the time-frequency domain and the analysis can be done analytically without the need to above mentioned Numerical tools.

Theory and Derivation In the core of Maxwell’s Equation is the Fourier Transform that converts the equation from the time domain into the frequency domain. Fourier Transform operator is inherently in the sinusoidal steady state, which therefore doesn’t provide information of any frequency transients in time. Since Fourier Transform is in sinusoidal steady state, the Maxwell’s equation in the Frequency Domain exhibits the same characteristic. We will change the Fourier transform by the Eigen value problem.

Consider a following Eigenvalue Problem; Wu(t,a)=au(t,a)

(1)

where W is an operator. In theory there could be infinite number of Eigenfunctions corresponding to infinite numbers of the Eigen values. Consider that operator is 1d at = W. The solution of this Eigen value problem will be therefore u(t, a ) = ae j dt

where eat is the Eigen function and a is the Eigen value. Now, a can assume any values but constrains will be that the value should be related to the operator W. If we take the Eigen value to be ω + ω(τ ) where ω(τ ) is the function of time instantaneous radial frequency then the corresponding eigen function will be u(t, a ) = e j (ω +ω(τ ))t . We can prove that operator W is Hermitian and therefore, it guarantees that the Eigen functions are complete and orthogonal. The proof is as follows:

=

1d

∫ g * (t ) j dt f (t )dt

d 1 1 fg * (−∞, −∞ ) − ∫ f (t ) g * (t )dt j j dt ⎛d ⎞ = ∫ f (t ) ⎜⎜ g(t )⎟⎟⎟ * dt ⎜⎝ jdt ⎠ =

(2) The Eigen Value problem is therefore jht

W u(h, t ) = he , h = ω + ω(τ )

With the solution

u(t, a ) =

(3)

u(t,a ) = ce j (ω +ω(τ ))t , Normalization value can be given by

1 j (ω +ω(τ ))t . Any signal can hence be expressed in terms of the e 2π

complete set of Eigenfunctions.

s(t ) = =

∫ u(h, t )F (h )dh

1 2π

∫e

jht

F (h )dh (4)

Where F (h ) is the transform of signal s(t ) . The Inverse transform can be expressed by

F (h ) =

∫e =∫e =

∫ u * (h, t )s(t )dt

− jht

s(t )dt

− j ( ω ± ω ( τ ))t

s(t )dt (5)

Thus, the transform of the function with an Eigenfunction is established. Fourier Transform can be considered as a special case of the above transform. We know that Fourier Transform of the Maxwell’s equation is given by

∂E ∂t ⇒ ∇× H = j ωεE ∇× H = ε

Using, the transform variable h , the equation will be

∂E ∂t ⇒ ∇× H ' = jh εE ' ∇× H = ε

= j (ω ± ω(τ ))εE ' (6) Similarly, Second Maxwell’s equation can be given by

∂H ∂t ⇒ ∇× E ' = −jh µH ' ∇× E = −µ

= −j (ω ± ω(τ ))µH ' (7) This result can be analyzed by the Helmholtz’s wave equation which is derived from the Maxwell’s equation. One-dimensional Helmholtz equation for propagation in the x-direction, which permits waves to propagate both in positive and negative direction, is given by;

∇2E ' = −h 2µεE ' ⇒ ∇2Ex ' = −h 2µεEx ',(x − component ) ∂2Ex ' ∂2Ex ' ∂2Ex ' ⇒ + + = −h 2µεEx ' 2 2 2 ∂x ∂y ∂z (8) Consider that Ex ' is independent of x and y,

∂ 2Ex ' = −h 2µεEx ' 2 ∂z ⎛∂ ⎞⎛ ∂ ⎞ ⇒ ⎜⎜ + jh µε⎟⎟⎟ ⎜⎜ − jh µε⎟⎟⎟ Ex ' = 0 ⎝ ∂z ⎠⎝ ∂z ⎠ ⇒

(9) The solutions of this equation will be of the form e

± jh εµz

= e ± j (ω ± ω (τ ))

εµz

. Thus, the

result of the h-transform compared to the Fourier transform in the solution of wave

equation, is the addition /subtraction of the phase factor ω(τ )z to ωz .Here, h can also be expressed as the frequency derivative function of any order and the results can be analyzed against the time. There are many methods for estimating the Instantaneous frequency of the signal [2]. Narrow angle analytic approximation of the Helmholtz wave is shown in [5]. The figure shows narrow angle gaussian beam propagation at different instances of time as a function of frequency. As shown in the figure, the beam spread is not constant and changes with time depending upon the instantaneous frequency variation.

Conclusion: It has been shown that the Fourier Operator can be changed by the operator algebra method and the validity of the Eigen value is analysed. The Maxwell’s Equation which is in the frequency domain is modified to include the time-frequency distribution of any order..

Authors Affiliation: H. Kapasi is with the School of Engineering, University of Warwick, UK. Corresponding author’s e-mail address: [email protected]

References: [1]

Boalem Boashash, “Time Frequency Signal Analysis and Processing: A

Comprehensive Reference,” Elsiver, 2003. [2] Boalem Boashash, “Estimating and interpreting the instantenous frequency of the signal-part 1: Fundamentals,” Proc. IEEE, vol 80, pp. 520-538, April 1992. [3] L. Cohen, “Time Frequency Distributions,” ,” Proc. IEEE, vol 77, pp. 941-981, July 1989. [4] D.Gabor, “Theory of Communication,” J. IEE, vol. 93(3),pp. 429-457, November 1946. [5] A.Yariv, “Introduction to Optical Electronics”,Second Edition, Holt, Rinehart and Winston, 1971.

Captions Figure 1 Caption: Narrow Angle Gaussian Beam Propagation with IF, ω(τ ) = τ at the time instant τ = 0 Figure 2 Caption: Narrow Angle Gaussian Beam Propagation with IF, ω(τ ) = τ at the time instant τ = 5 Figure 3 Caption: Narrow Angle Gaussian Beam Propagation with IF, ω(τ ) = τ at the time instant τ = 10

Figure 4 Caption: Narrow Angle Gaussian Beam Propagation with IF, ω(τ ) = τ at the time instant τ = 15