1Lesson 2 Introduction to Optical Electronics Quantum (Photon) Optics (Ch 12) Resonators (Ch 10) Electromagnetic Optics (Ch 5) Wave Optics (Ch 2 & 3) Ray Optics (Ch 1) Photons & Atoms (Ch 13) Laser Amplifiers (Ch 14) Lasers (Ch 15) Photons in Semiconductors (Ch 16) Semiconductor Photon Detectors (Ch 18) Semiconductor Photon Sources (Ch 17) OpticsPhysicsOptoelectronics

2Lesson 2 Wavefunction (monochromatic)Wave Equation Complex WavefunctionWave Equation Complex AmplitudeHelmholtz Equation Paraxial Wave*Paraxial Helmholtz Equation Wave Optics * A(r) varies slowly with respect to

3Lesson 2 Derivation of the Gaussian Beam

4Lesson 2 Gaussian Amplitude Plot e -1

5Lesson 2 Gaussian Intensity Plot e -2

6Lesson 2 Gaussian Beam Intensity Plots

7Lesson 2 Gaussian Beam Width W(z) 2020 z

8Lesson 2 Wavefronts of a Gaussian Beam

9Lesson 2 Gaussian Beam Radius Of Curvature, R(z)

10Lesson 2 Exercise 3.1-3: Determination of a Beam with Given Width and Curvature 2W02W0 z R 2W2W Given W and R, determine z and W 0.

11Lesson 2 Exercise 3.1-4: Determination of the width and curvature at one point given the width and curvature at another point Given the width (W 1 ) and curvature (R 1 ) at a point, determine the width (W 2 ) and curvature (R 2 ) at a distance d to the right. Given: = 1  m R 1 = 1 m W 1 = 1 mm d = 10 cm to the right Find R 2 and W 2 at d. d R2W2R2W2 R1W1R1W1

12Lesson 2 Exercise 3.1-5: Identification of a Beam with Known Curvatures at Two Points A Gaussian beam has radii of curvature R 1 and R 2 at two points on the beam separated by a distance d. Verify the equations for z 1, z 0 and W 0. d R2R2 R1R1 z1z1 z2z2

13Lesson 2 Transmission through a Thin Lens Gaussian Beam W0W0 W´0W´0 z0z0 z´0z´0 ´0´0 00 z z´z´ WRWR W´ R´ z

14Lesson 2 Exercise A Gaussian beam is transmitted through a thin lens of focal length f. a)Show that the locations of the waists of the incident and transmitted beams, z and z ´, are related by: b)The beam is collimated by making the location of the new waist z ´ as distant as possible from the lens. This is achieved by using the smallest ratio z 0 / f, show that the optimal value of z for collimation is z = f + z 0. W0W0 W´0W´0 z0z0 z´0z´0 z z´z´ WRWR W´R´W´R´ z

15Lesson 2 Transmission Through Optical Components Gaussian Beam W1R1q1W1R1q1 W2R2q2W2R2q2 Applies to thin optical components and to propagation in homogeneous medium of paraxial waves

16Lesson 2 Transmission Through Optical Components Gaussian Beam Find the Radius of Curvature and spot size just to the right of the lens ( R ′ and W ′) if the incident Gaussian Beam is planar on the lens. z W′W′ W1R1q1W1R1q1 W'R'q'W'R'q' W1W1

17Lesson 2 Transmission Through a Thin Lens Gaussian Beam What is the minimum spot size achievable with a thin lens? W 01 W 02 zmzm z W′W′ W1R1q1W1R1q1 W'R'q'W'R'q'

18Lesson 2 Hermite-Gaussian Waves

19Lesson 2 Hermite-Gaussian Waves TEM 0,0 TEM 1,0 TEM 2,0 Amplitude (black) and Intensity (red) distribution along x-axis Higher order modes become wider with ( l,m ): 2 W ( z ) is just a scale factor for higher order modes TEM 0,0 will be most intensely focused beam (Gaussian Beam)

20Lesson 2 Hermite-Gaussian Beams TEM 0,0 TEM 0,1 TEM 0,2 TEM 1,1 TEM 1,2 TEM 2,2