Chapter 2 The Propagation of Rays and Beams

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 Nonlinear Optics Lab. Hanyang Univ.

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 Nonlinear Optics Lab. Hanyang Univ.

Table 2 -1 Ray Matrices Nonlinear Optics Lab. Hanyang Univ.

Nonlinear Optics Lab. Hanyang Univ.

Biperiodic lens sequence (f 1, f 2, d) In equation form of Nonlinear Optics Lab. Hanyang Univ.

(2. 1 -5) (actually for all elements) trial solution : general solution : Nonlinear Optics Lab. Hanyang Univ.

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 : Nonlinear Optics Lab. Hanyang Univ.

2. 2 Propagation of Rays Between Mirrors stability condition : (n, l : integers) example) n=2, l=1 q=p/2 cosq = b = 1 -d/2 f = 0 (symmetric confocal) Nonlinear Optics Lab. Hanyang Univ.

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> wave front : r 0 s ray path Nonlinear Optics Lab. : optical path Hanyang Univ.

i) ii) Maxwell equations : if m=1, That is, So, : Differential equation for ray propagation, (2. 3 -3) Nonlinear Optics Lab. Hanyang Univ.

For paraxial rays, Focusing distance from the exit plane for the parallel rays : Report) Proof Nonlinear Optics Lab. Hanyang Univ.

2. 4 Wave Equation in Quadratic Index Media Gaussian beam ? Maxwell’s curl equations (isotrpic, charge free medium) => Put, : Scalar wave equation (monochromatic wave) => Helmholtz equation : where, We limit our derivation to the case in which k 2(r) is given by where, Nonlinear Optics Lab. Hanyang Univ.

Assume, & slow varying approximation => Put, => Nonlinear Optics Lab. Hanyang Univ.

2. 5 Gaussian Beams in a Homogeneous Medium In a homogeneous medium, => v Otherwise, field cannot be a form of beam. is must be a complex ! => Assume, => put, ( is pure imaginary. : real) At z = z 0, v Beam radius at z=0, : Beam Waist Nonlinear Optics Lab. Hanyang Univ.

at arbitrary z, : Complex beam radius => => => Nonlinear Optics Lab. Hanyang Univ.

Wave field where, : Beam radius : Radius of curvature of the wave front : Confocal parameter(2 z 0) or Rayleigh range Nonlinear Optics Lab. Hanyang Univ.

Gaussian beam spread angle : I Gaussian profile Near field (~ plane wave) Far field (~ spherical wave) Nonlinear Optics Lab. Hanyang Univ.

2. 6 Fundamental Gaussian beam in a Lenslike Medium - ABCD law For lenslike medium, Introduce s as, Table 2 -1 (6) Nonlinear Optics Lab. Hanyang Univ.

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

ABCD law for Gaussian beam optical system ABCD law for Gaussian beam : Nonlinear Optics Lab. Hanyang Univ.

example) Gaussian beam focusing ? q 1 ? Nonlinear Optics Lab. Hanyang Univ.

- 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. Nonlinear Optics Lab. Hanyang Univ.

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, Nonlinear Optics Lab. Hanyang Univ.