(10)

(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 ωm 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, J0(d) = 1 and J1(d) = sin(d/2) which leads to:

(29)

If we use our suggested approximation instead of the Bessel function identities in Eout, 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 Eout we apply it for small δ where, cosδ = 1 and sin2(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.

REFERENCES

  1. A. V. Alekseev and N. V. Sushilov, “Analytic Solutions of Bloch and Maxwell-Bloch Equations in the Case of Arbitrary Field Amplitude and Phase Modulation” Physical Review A, Vol. 46, No. 1, 1992, pp. 351-355. doi:10.1103/PhysRevA.46.351
  2. N. Nayak and G. S. Agarwal, “Absorption and Fluorescence in Frequency-Modulated Fields under Conditions of Strong Modulation and Saturation” Physical Review A, Vol. 31, No. 5, 1985, pp. 3175-3182. doi:10.1103/PhysRevA.31.3175
  3. A. Hund, “Frequency Modulation,” McGraw-Hill, New York, 1942.
  4. J. D. Jackson, “Classical Electrodynamics,” 3rd edition, Wiley, New York, 1998.
  5. J. G. Proakis and M. Salehi, “Communication Systems Engineering,” Prentice Hall, Upper Saddle River, 2001.
  6. N. M. Blachman, “Noise and Its Effect on Communication,” 2nd Edition, Krieger Publishing Co., Malabar, 1982.
  7. G. N. Watson, “A Treatise on the Theory of Bessel Functions,” 2nd edition, Cambridge University Press, Cambridge, 1995.
  8. S. Saadeh, J. Shultz and G. Salamo, “Experimental Observation of Chirped Continuous Pulse-train Soliton Solutions to the Maxwell-Bloch Equations,” Optics Express, Vol. 8, No. 2, 2001, pp. 153-158. doi:10.1364/OE.8.000153
  9. L. A. Pipes, “Applied Mathematics for Engineers and Physicists,” 2nd Edition, McGraw-Hill, New York, 1958.
  10. M. Abramowitz and I. A. Stegun, “Handbook of Mathematical Functions,” National Bureau of Standards, Washington DC, 1964.
  11. N. M. Blachman and S. H. Mousavinezhad, “Trigonometric Approximation for Bessel Functions,” IEEE Transactions on Aerospace and Electronic Systems, Vol. 22, No. 1, 1986, pp. 2-7. doi:10.1109/TAES.1986.310686
  12. A. Yarvis, “Introduction to Optical Electronics,” 2nd Edition, Holt McDougal, Geneva, 1977.

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