Chapter 2. The Propagation of Rays and Beams

1. Chapter 2. The Propagation of Rays and Beams 2.0 Introduction Propagation of Ray through optical element : Ray (transfer) matrix  Gaussian beam propagation 2.1 Lens Waveguide A ray can be uniquely defined by its distance from the axis (r) and its slope (r’=dr/dz). r’=dr/dz r z

2. Paraxial ray passing through a thin lens of focal length f : Ray matrix for a thin lens Report) Derivation of ray matrices in Table 2-1

3. Table 2-1 Ray Matrices

5. Biperiodic lens sequence (f1, f2,d) In equation form of

6. (2.1-5)  (actually for all elements) trial solution : general solution :

7. Stability condition : The condition that the ray radius oscillates as a function of the cell number s between rmax and –rmax. : q is real  Identical-lens waveguide (f, f,d)  Stability condition :

8. 2.2 Propagation of Rays Between Mirrors stability condition : (n, l : integers) example) n=2, l=1  q=p/2  cosq = b = 1-d/2f = 0 (symmetric confocal)

9. wave front : r s ray path 0 2.3 Rays in Lenslike Media Lenses : optical path across them is a quadratic function of the distance r from the z axis ; phase shift Index of refraction of lenslike medium : <Differential equation for ray propagation> : optical path

10. i) ii) Maxwell equations : if m=1, That is, So, : Differential equation for ray propagation, (2.3-3)

11. For paraxial rays, Focusing distance from the exit plane for the parallel rays : Report) Proof

12. 2.4 Wave Equation in Quadratic Index Media Gaussian beam ? Maxwell’s curl equations (isotrpic, charge free medium) : Scalar wave equation => Put, (monochromatic wave) => Helmholtz equation : where, We limit our derivation to the case in which k2(r) is given by where,

13. Assume, & slow varying approximation => Put, =>

14. 2.5 Gaussian Beams in a Homogeneous Medium In a homogeneous medium, => Otherwise, field cannot be a form of beam. Assume, is must be a complex ! => is pure imaginary. => put, ( : real) At z = z0, : Beam Waist Beam radius at z=0,

15. => => at arbitrary z, : Complex beam radius =>

16. Wave field where, : Beam radius : Radius of curvature of the wave front : Confocal parameter(2z0) or Rayleigh range

17. spread angle : I Gaussian profile Far field (~ spherical wave) Near field (~ plane wave) Gaussian beam

18. 2.6 Fundamental Gaussian beam in a Lenslike Medium - ABCD law For lenslike medium, Introduce s as, Table 2-1 (6) 

19. ao optical elements yo ai yi : ray-transfer matrix Transformation of the Gaussian beam – the ABCD law Matrix method (Ray optics)

20. optical system ABCD law for Gaussian beam : ABCD law for Gaussian beam

21. ? q1 ? example) Gaussian beam focusing

22. - If a strong positive lens is used ; => : f-number => ; The smaller the f# of the lens, the smaller the beam waist at the focused spot. - If => Note) To satisfy this condition, the beam is expanded before being focused.

23. 2.7 A Gaussian Beam in Lens Waveguide Matrix for sequence of thin lenses relating a ray in plane s+1 to the plane s=1 : Stability condition for the Gaussian beam : : Same as condition for stable-ray propagation where,