_{1}

^{*}

A new mathematical identity is suggested to describe narrow band phase modulation and other similar physical problems instead of using the Bessel function. Bessel functions are extensively used in mathematical physics [1,2], electromagnetic wave propagation and scattering [3,4], and communication system theory [3,5,6]. Such phenomena must often be approximated by appropriate formulas since there is no closed form solution or expression, which usually leads to complex mathematical solutions [5,7]. Comparisons are made between the exact solution numerically calculated and graphed with the new mathematical identities’ prediction of phase modulation behavior. The proposed mathematical identity matches the results very well, leading to simpler analysis of such physical behavior.

Frequency or phase modulation is an efficient form of communication, where information is transmitted over a carrier signal by changing the instantaneous frequency [

The expressions and where δ and ω_{m} are the index and frequency of modulation, appear frequently in physics research and literature. They play a key role in the mathematical treatment of narrowband frequency modulation in communication and in antennas radiation analysis, design and studies [3,4].

Usually and are replaced by Bessel function series, which adds complexity to the mathematical treatment. In many cases this can lead to using mathematical approximations that reduce the accuracy of the analysis.

Previous published work done in collaboration with Prof. Salamo from the University of Arkansas dealt with laser beam propagation through two level samples [

Our first postulate is:

To verify the accuracy of our assumption, the end result undergoes vigorous numerical and analytical testing.

If the proposed postulate is true then it follows:

by substituting equation (4) in equation (5),

The second postulate is:

If this is a valid assumption, then:

If that is right then:

by substituting equation (16) in equation (17),

From the above derivation, we get our new identities in equations (4), (12), (16) and (23).

The new expression is tested in two approaches. The first approach is to analytically compare the result to the established phase modulation of light [

Amnon Yariv discuses phase modulation of light in chapter 9 of his book Introduction to Optical Electronics [

where d is the phase modulation index and ω_{m} is the phase modulation frequency.

The above book uses the Bessel function identities

However, it is generally assumed that only the first three terms are significant. This agrees with our experiments in which only the first three terms in the series are detected.

For small modulation, i.e. d < 1, J_{0}(d) = 1 and J_{1}(d) = sin(d/2) which leads to:

If we use our suggested approximation instead of the Bessel function identities in E_{out}, then:

By using the new identities of equations (4) and (12) in equation (30), we get:

To check this formula against the previous formula of E_{out}_{ } we apply it for small δ where, cosδ = 1 and sin^{2}(d/2) = 0. In this case,

If we use the identity:

we get

Which exactly matches the proposed solution?

The graphs of

The derivation of the four new mathematical identities in equations (4), (12), (16) and (23) that describe narrow band phase modulation is presented. The proposed identity can also be used for similar physical phenomena’s that use the Bessel function. The mathematical identity was shown to match analytical and numerical results very well. The new identities greatly reduce computation time and complexity of analytical treatment of such physical behavior.