1. Chapter II The Propagation of Rays and Beams
Lecture 2
Chapter II The Propagation of Rays and Beams
Highlights
Optical ray propagation through
a variety of optical media
The propagation of Gaussian beam
Homogeneous and isotropic
Thin lenses
Dielectric interface
Curved mirrors
Characteristics
ABCD law
2 x 2 matrices
Stability condition of periodic optical system

2.   §2.1 Lens Waveguide Ray Matrices +: Ray output direction above
A paraxial ray passing through a thin lens of focal length f
tanri’ ~ ri’ f
Focal Plane z
(ri, ri’)
(ro, ro’)
ri’f
tan-1ri’
+: Ray output
direction above
z axis
-: Ray output
direction below  
Output side
Input side

3. §2.1 Lens Waveguide Exercises
1. Spherical Mirror: Radius of curvature R
ri’
ro’ a b z R
2. Straight sections ( d ) and a thin lens ( f ) f z
(ri, ri’)
(ro, ro’) d
(r2, r2’) z1 z2 z3

4. Most close to output side Most close to input side
§2.1 Lens Waveguide
Exercises
Ray matrices for multi-element cascade system
Most close to output side
Most close to input side
3. Biperiodic lens sequence f2 z
(rs+1, rs+1’) d
(rs, rs’) f1 s
s +1

5. §2.1 Lens Waveguide
Stability condition

6. §2.2 Propagation of Rays Between Mirrorsd
(r1, r1’)
(r2, r2’)
(r3 r3’)
(r4, r4’)
(r5, r5’) z

7. §2.2 Propagation of Rays Between Mirrors
Stability condition

8. §2.2 Propagation of Rays Between Mirrors Examples
1. Plane parallel cavity
Rays parallel to z axis are trapped, otherwise escape. Trapped rays are closed after one times reflection.
2. Concentric cavity
Rays pass through concentric point are trapped, otherwise escape. Trapped rays are closed after one times reflection.
3. Confocal cavity
All rays as long as reflected back are trapped. Trapped rays are closed after two times reflection.

9. §2.2 Propagation of Rays Between Mirrors
Stability condition for coaxial spherical cavity
Unstability condition
Metastability condition

10. §2.3 Rays in Lenslike Mediaz
(r0, r0’)
Ray Equation:
For g2 > 0

11. §2.3 Rays in Lenslike Media
For g2 < 0
Physical situations
Spread
1. Absorption medium:
Focus
Spread
2. Optical Pumped SL rods:
Focus
3. Dielectric waveguide:
4. Optical fiber:
5. Optical waveguide:

12. §2.4 Wave Equation in Quadratic Index Media
The mostly encountered optical beam is Gaussian beam.
Isotropic, charge-free medium
& Neglect the right side

13. §2.4 Wave Equation in Quadratic Index Media
We allow for a possible dependence of e on position r and the
possibility of losses (s > 0) or gain (s < 0) in the medium
k2 is some constant characteristic of the medium

14. §2.4 Wave Equation in Quadratic Index Media
To obtain above equation, we neglect all second order derivate terms.
If above equation holds for all r, we then have:
This is a basic wave propagation equation for Gaussian beams.

15. §2.5 Gaussian Beams in a Homogenous Medium
q0 is an arbitrary integration constant
The choice of imaginary q0 will be found to
lead to physically meaningful waves whose
energy density is confined near the z axis.

16. §2.5 Gaussian Beams in a Homogenous Medium
The first term of right side:
Where we used:
The second term of right side:

17. §2.5 Gaussian Beams in a Homogenous Medium
We define the following parameters:

18. §2.5 Gaussian Beams in a Homogenous Medium
纵向相位
振幅因子
径向相位
上面的结果就是基摸 (fundamental mode)高斯光束的解。

19. §2.5 Gaussian Beams in a Homogenous Medium
Z=0 w0 z z0 q R
Spot size: distance which the amplitude is down by factor of 1/e
Radius of curvature:
Reyleigh length
Beam spread angle

20. Homework
章节后习题2.1、2.3、2.6。