(11)

(12)

The second postulate is:

(13)

If this is a valid assumption, then:

(14)

(15)

(16)

If that is right then:

(17)

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

(18)

(19)

(20)

(21)

(22)

(23)

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

3. Testing the Suggested Expression

The new expression is tested in two approaches. The first approach is to analytically compare the result to the established phase modulation of light [12]. The second approach to compare the results with computer simulations.

3.1. Phase Modulation of Light

Amnon Yariv discuses phase modulation of light in chapter 9 of his book Introduction to Optical Electronics [12]. The electric field for the modulated light is given by:

(24)

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

(25)

The above book uses the Bessel function identities

(26)

(27)

(28)

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:

(29)

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

(30)

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

(31)

(32)

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,

(33)

If we use the identity:

(34)

we get

(a)(b)(c)(d)(e)

Figure 1. Comparison between exact results and proposed expression for.

(a)(b)(c)(d)(e)

Figure 2. Comparison between exact results and proposed expression for.

(35)

Which exactly matches the proposed solution?

3.2. Computer Generated Numerical Comparison

The graphs of Figure 1, compare the computer generated numerical results for the exact equation: at different values of d with that of the new proposed mathematical identity. Figure 2 shows the comparison for.

4. Conclusion

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.

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