Calculate the Gaussian curvature of the helicoid and catenoid Calculate the Gaussian curvature of the helicoid and...
Please, solve each part of the problem. The subject is Lasers. a) For a 632nm gaussian beam mode, determine the beam waist for a 10 degree divergence angle. b) Calculate the Rayleigh length only equal convex mirrors, design an optimal laser cavity which will c) Using produce a gaussian beam with a waist of 200 um and the smallest beam radius at output mirror. (By design, you must provide both the radius of curvature and distances between the mirrors). the...
Find the first quadratic form of a surface called helicoid, r(u, v)-(u cos v, u sin v, av), where a is a constant parameter. Then find the angle of intersection of lines on the surface of helicoid given by equations u v0, u0. Find the first quadratic form of a surface called helicoid, r(u, v)-(u cos v, u sin v, av), where a is a constant parameter. Then find the angle of intersection of lines on the surface of helicoid...
3. (4 pts) Find the area of the surface of the helicoid (or spiral ramp) with vector equation (u,v)=u 1 +vcosu 1 +vsin uk, o Su S271, Osvsi.
EXERCISE 4.16. Prove that every compact regular surface has a point of positive Gaussian curvature. HINT: LetpES be a point of maximum distance to the origin. By applying Exercise 1.43 on page 32 to a normal section, conclude that the normal cur- vature of S at p in every direction is where r is the distance from p to the origin. EXERCISE 1.43 Let γ: 1 → Rn be a regular curve. Assume that the function t Iy(t) has a...
Evaluate the surface integral. y ds, S is the helicoid with vector equation r(u, v) = (u cos(V), u sin(), v), OSUS 4,0 SV S.
2. A minimal surface is by definition a critical point of the area functional. Show that the graph of the function z(x, y)tany is a minimal surface. (This surface is called the helicoid.) 2. A minimal surface is by definition a critical point of the area functional. Show that the graph of the function z(x, y)tany is a minimal surface. (This surface is called the helicoid.)
2. [40 pts] Gaussian Beam (Based on Textbook problem 1.4 and 1.5) A diode laser emits a Gaussian beam. The minimum waist diameter is 1 mm. The lasing wavelength is 514 nm. Find (a) Divergence angle 6div (b) The Rayleigh range (c) The diameter of the beam after propagating 10 m (d) 10 m away from the laser output aperture, you want to place a spherically curved mirror to reflect 1.0 mmthe beam back perfectly. Find the radius of curvature...
A biconvex lens has two distinct radii of curvature. One face has a radius of curvature of magnitude R, = 11.8 cm, and the opposite face has a radius of curvature of magnitude R, = 18.9 cm. The lens medium has an index of refraction of n = 1.48. (a) Calculate the focal length (in cm) of the lens for light incident on the side with radius of curvature R,. cm (b) Calculate the focal length (in cm) of the...
Calculate the Gauss and mean curvature for the Surface ou, v) (ucosv, usinv, v)
exercise 4.18(2) proves that every longitude and every latitude is a line of curvature of a surface if revolution EXERCISE 4.23. Let S be the torus obtained by revolving about the axis the circle in the xz-plane with radius 1 centered at (2,0,0). This torus is illustrated in Fig. 4.8. Colored red (respectively green) is the region where 2y4 (respectively r2 +y > 4). Let N be the outward-pointing unit 2- normal field on S. (1) Verify that the unit...