Nonlinear Optics Lab. Hanyang Univ. 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 r’=dr/dz 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.

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 r max and –r max.  is real  Identical-lens waveguide (f, f, d)  Stability condition :

Nonlinear Optics Lab. Hanyang Univ. 2.2 Propagation of Rays Between Mirrors (, l : integers) stability condition : example) , l=1   =    cos  = b = 1-d/2f = 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 : 0 r s ray path wave front : : optical path

Nonlinear Optics Lab. Hanyang Univ. i) ii) Maxwell equations : if  =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) : Scalar wave equation Put, (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, => Put, => & slow varying approximation

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

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(2z 0 ) or Rayleigh range

Nonlinear Optics Lab. Hanyang Univ. Gaussian beam I Gaussian profile spread angle : 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. Matrix method (Ray optics) yiyi yoyo ii oo optical elements : ray-transfer matrix Transformation of the Gaussian beam – the ABCD law

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 11 ? ?

Nonlinear Optics Lab. Hanyang Univ. - If a strong positive lens is used ;=> - If=> : f-number ; The smaller the f# of the lens, the smaller the beam waist at the focused spot. 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 : where, Stability condition for the Gaussian beam : : Same as condition for stable-ray propagation