1. Fundamental of Optical Engineering Lecture 4

2.  Recall for the four Maxwell’s equation:

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

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

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

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

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

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

9.  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.

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

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

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

13.  From

14.  Phase velocity:  dt=  dz

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

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

17.  Let  be Gaussian beam solution and assume propagation in z-direction

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

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

23.  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)

24.  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?

25.  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?

26.  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.

27. (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?

28.  (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?