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ENE 492. Fundamental of Optical Engineering Lecture 4. Wave Equation. Recall for the four Maxwell’s equation:. Wave Equation. The wave equation is derived from the assumptions of Non-magnetic material, Uniform dielectric medium. Wave Equation. No current or
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ENE 492 Fundamental of Optical Engineering Lecture 4
Wave Equation • Recall for the four Maxwell’s equation:
Wave Equation • The wave equation is derived from the assumptions of • Non-magnetic material, • Uniform dielectric medium
Wave Equation • No current or Therefore, the simplified form of Maxwell’s equation can be written as
Wave Equation • From (2): • From (1): • Since , we end up with • Similarly, we have
Wave Equation • Equation (3) and (4) are wave equations of the form where is a function of x,y,z, and t.
Wave Equation • Generally, we only are interested in electric field. The wave equation may be written as • Assume that the wave propagates in only z-direction
Wave Equation • Then assume that E(z,t) = E(z)E(t) and put it into (5) or
Wave Equation • We clearly see that the left side of (6) is dependent on ‘z’ only, and the right side of (6) is on ‘t’ only. • Both sides must be equal to the same constant, which we arbitrarily denote as -2.
Wave Equation • The general solutions of these equations are • Constants C1, C2, D1, and D2 could be found by the boundary conditions.
Wave Equation • We now can express the general solution E(z,t) as
Wave Equation • A wave travelling from left to right has a function of the form
Wave Equation • From
Wave Equation • Phase velocity: dt=dz
Example • Write the expression of a plane wave traveling in z-direction that has maximum amplitude of unity and a wavelength of 514.4 nm.
Power flow • The time average power density: • For plane wave propagation in z-direction, using Maxwell’s equations and definition of s, we find that
Gaussian Beam • Let be Gaussian beam solution and assume propagation in z-direction
Gaussian Beam • Geometrical optics may be employed to determine the beam waist location in Gaussian problems.
Example • Consider a HeNe laser with λ = 0.63 μm. Calculate the radius of curvature for mirrors in the figure below.
Example • Calculate beam width at mirrors from the previous example and at a distance of 1m, 1 km, and 1,000 km from center of laser (assuming that mirrors do not deform beam)
Example • Consider a colliminated Nd:YAG laser beam (λ=1.06 μm) with a diameter to e-2 relative power density of 10 cm at the beam waist with z0 = 0. What is the beam half width to e-2 relative power density at z = 1m, 100 m, 10 km, and 1,000 km?
Example • From the previous example, what is power density on beam axis at each distance, assuming the total power is 5 W? What is the divergence angle of beam to e-2 and e-4 relative power density?
Example • Two identical thin lenses with f = 15 cm and D = 5 cm are located in plane z = 0 and z = L. A Gaussian beam of diameter 0.5 cm to e-2 relavtive power density for λ = 0.63 μm is incident on the first lens. The value of L is constained such that the e-2 relative power density locus is contained within the aperture of the second lens.
Example (a) For what value of L will the smallest spot be obtained for some value of z0 > 0? What is the value of z0 corresponding to the location of that spot? What is the diameter of that spot?
Example • (b) For what value of L will the smallest spot size be obtained on the surface of the moon at a distance of 300,000 km? What is the beam diameter on the moon surface?
