Classical Light – Theodore B Norris (1)

ENt = (r1r2e−iδ)N−1(−t1t2e−iδ/2)Einc. The total transmitted field is the sum Etr = ” ∞ ∑ l=0 (r1r2e−iδ)l # (−t1t2e−iδ/2)Einc. (6.33) The mirror reflectivities may be close to unity, but they can never equal (or exceed) it, so r1,r2 < 1. If we define x = r1r2e−iδ, then |x| < 1, and the infinite series converges: ∞ ∑ l=0 (x)l = 1−x , (6.34) so the total transmitted field is Etr(ω) = " −t1t2e−iδ/2 1−r1r2e−iδ # Einc(ω) = HFP(ω) Einc(ω), (6.35) where the factor in brackets, HFP(ω), is the frequency-domain transfer function of the Fabry- Perot interferometer.

The frequency dependence of HFP is due to the frequency dependence of the phase factor δ as given by Eq. (6.31). The reflected field is found in a similar fashion. There is an initial reflection of the incident field from the first mirror, followed by an infinite series of partial waves making an integer number of round trips, so we find Erefl(ω) = r1Einc + ” ∞ ∑ l=0 (r1r2e−iδ)l # (−t2 1r2e−iδ)Einc = ” r1 − t2 1r2e−iδ 1−r1r2e−iδ # Einc(ω). (6.36) The second method of calculating the transmission and reflection is via transfer matrices, which relate the total fields in the different regions of the system.

The transfer matrices for the mirrors are given by Eq. (3.107): t1,2  −r1,2 r1,2 −1  . (6.37) Chapter 6 Interferometers The transfer matrix for propagation across the length of the cavity is given by Eq. (3.69), so the system matrix is M = i t2  −r2 r2 −1  e−iδ2 eiδ/2  i t1  −r1 r1 −1  .

(6.38) We know from Eq. (3.152) that the transmission of a system is related to its system transfer and scattering matrices by = HFP(ω). (6.39) From Eq. (6.38), matrix multiplication yields t1t2 (−r1r2e−iδ/2 +eiδ/2). (6.40) Computing M−1 22 gives directly the result expressed by Eq. (6.35), which we obtained using the partial wave method. Let us now consider the transmitted intensity, given by the square magnitude of Eq. (6.35): Itr(ω) Iinc(ω) = −t1t2e−iδ/2 1−r1r2e−iδ/2 .

The University of Michigan Copyright © 2025 Theodore B. Norris This book is published by Michigan Publishing under an agreement with the author. It is made available free of charge in electronic form to any student or instructor interested in the subject matter. Published in the United States of America by Michigan Publishing. Manufactured in the United States of America. ISBN 978-1-60785-919-2 (print) ISBN 978-1-60785-920-8 (OA) The Free ECE Textbook Initiative is sponsored by the ECE Department and the College of Engineering of the University of Michigan.

To Penny and Jessica iv Notes on the Cover Images The cover illustrates several of the important optical and wave phenomena analyzed in this textbook. The front cover shows the generation of a white light beam using a femtosecond laser in the author’s laboratory. A near-infrared (and therefore invisible) laser beam consisting of a train of 100-fs pulses from a Ti:sapphire laser is incident onto a sapphire crystal (upper right in the photo).

White light is generated via nonlinear processes in the sapphire. The beam is collimated by a lens, and propagates from the lens towards the lower left of the photograph. Multiple optical processes are involved in the generation of a white light continuum. The nonlinear optical processes of self-phase modulation, self-steepening, and self-focusing are analyzed in Chapter 11. The dispersion of the pulse propagation in the medium is treated in Chapter 8. Methods for the generation of the femtosecond pulses used to generate the white light are presented in Chapter 13.

The diffraction and focusing of optical beams is analyzed in Chapter 7. The description of light propagation in the geometrical optics approximation is developed in Chapter 5. The back cover illustrates a number of optical and wave phenomena. The blue sky at the top is a manifestation of Rayleigh scattering of light in the atmosphere, while the red- orange color of the sunset and the sky near the horizon is a consequence of attenuation due to Rayleigh scattering and optical absorption.

These phenomena may be explained by the classical theory of light-matter interactions, which is extensively developed in Chapter 4. The specular reflection of the sun on the water surface is a consequence of the boundary conditions on light propagation at interfaces, which is studied in Chapter 3.

This is a short excerpt from the opening of “” by Unknown, quoted for review and introduction purposes. All rights belong to the copyright holders.

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