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Fundamental of Optical Engineering Lecture 4

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Recall for the four Maxwell’s equation:

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The wave equation is derived from the assumptions of ◦ Non-magnetic material, ◦ Uniform dielectric medium

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◦ No current or Therefore, the simplified form of Maxwell’s equation can be written as

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From (2): From (1): Since, we end up with Similarly, we have

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Equation (3) and (4) are wave equations of the form where is a function of x,y,z, and t.

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Generally, we only are interested in electric field. The wave equation may be written as Assume that the wave propagates in only z- direction

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Then assume that E(z,t) = E(z)E(t) and put it into (5) or

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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.

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The general solutions of these equations are Constants C 1, C 2, D 1, and D 2 could be found by the boundary conditions.

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We now can express the general solution E(z,t) as

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A wave travelling from left to right has a function of the form

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From

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Phase velocity: dt= dz

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Write the expression of a plane wave traveling in z-direction that has maximum amplitude of unity and a wavelength of 514.4 nm.

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The time average power density: For plane wave propagation in z-direction, using Maxwell’s equations and definition of s, we find that

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Let be Gaussian beam solution and assume propagation in z-direction

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Geometrical optics may be employed to determine the beam waist location in Gaussian problems.

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Consider a HeNe laser with λ = 0.63 μm. Calculate the radius of curvature for mirrors in the figure below.

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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)

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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 z 0 = 0. What is the beam half width to e -2 relative power density at z = 1m, 100 m, 10 km, and 1,000 km?

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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?

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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.

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(a) For what value of L will the smallest spot be obtained for some value of z 0 > 0? What is the value of z 0 corresponding to the location of that spot? What is the diameter of that spot?

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(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?

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