EE 485, Winter 2004, Lih Y. Lin
Chapter 3 Beam Optics
- An important paraxial wave solution that satisfies Helmholtz equation is Gaussian beam. Example: Laser.
3.1 The Gaussian Beam A. Complex Amplitude * W ρ 2 U (r) = A 0 exp− Amplitude factor 0 W (z) W 2 (z) −1 z × exp− jkz − tan Longitudinal phase (3.1-7) z0 kρ 2 × exp− j Radial phase 2R(z) 2 = + z W (z) W0 1 (3.1-8) z0 z 2 R(z) = z1+ 0 (3.1-9) z πW 2 z ≡ 0 (3.1-11) 0 λ λ → Knowing W0 and (or z0 ), a Gaussian beam is determined! Ref: Verdeyen, “Laser Electronics,” Chapter 3, Prentice-Hall
B. Properties of Gaussian Beam
Intensity and power
1 EE 485, Winter 2004, Lih Y. Lin
2 W 2ρ 2 I(ρ, z) = I 0 exp− (3.1-12) 0 W (z) W 2 (z) 1 P = I ()πW 2 (3.1-14) 2 0 0
Beam radius and divergence 1 I(ρ, z) = I(0, z) when ρ = W (z) e2 2 = + z W (z) W0 1 : Beam radius (3.1-17) z0
W0 : Waist radius
2W0 : Spot size = W (z0 ) 2W0 λ θ θ = : Divergence angle (= in Fig. 3.2 shown above) 0 π W0 2 (3.1-19) λ → Highly directional beam requires short and large W0 .
Depth of focus (Confocal parameter)
2 EE 485, Winter 2004, Lih Y. Lin
2πW 2 2z = 0 (3.1-21) 0 λ
Longitudinal phase
− z ϕ(z) = kz − tan 1 z0 → Second term: phase retardation ϕ −1 = = c > Phase velocity v p c ! ωz λ − z 1− tan 1 π 2 z z0
Wavefront bending Surface of constant phase velocity: 2 ρ λ − z z + = qλ + ξ , ξ ≡ tan 1 π 2R 2 z0 → Parabolic surface with radius of curvature R
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3.2 Transmission through Optical Components Gaussian beam remains a Gaussian beam after transmitting through a set of circularly symmetrical optical components aligned with the beam axis. Only the beam waist and curvature are altered.
A. Transmission through a Thin Lens
1 = 1 − 1 R' R R' (3.2-2) W '= W W W '= (3.2-3) 0 2 πW 2 1+ λR'
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R' z'= (3.2-4) λR' 2 1+ πW 2 = Waist radius W0 ' MW0 (3.2-5) Waist locatioin (z'− f ) = M 2 (z − f ) (3.2-6) = 2 Depth of focus 2z0 ' M (2z0 ) (3.2-7) 2θ Divergence 2θ '= 0 (3.2-8) 0 M M Magnification M = r (3.2-9) 1+ r 2 f z M ≡ , r ≡ 0 (3.2-9a) r z − f z − f Example: A planar wave transmitting through a thin lens is focused at distance z'= f .
B. Beam Shaping
Waist of incident Gaussian beam is at lens location. R'= − f W W '= 0 (3.2-13) 0 2 z 1+ 0 f → To focus into a small spot, we need large incident beam width, short focal length, short wavelength.
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= f z' 2 (3.2-14) f 1+ z0 F number of a lens f F ≡ # D = D 2W0 : Diameter of the lens 4 Focal spot size 2W '= λF (3.2-17) 0 π #
C. Reflection from a Spherical Mirror Same as transmission through a thin lens, f = − R 2 = W2 W1 (3.2-19) 1 1 2 = + (3.2-20) R2 R1 R
D. Transmission through an Arbitrary Optical System The ABCD Law
Aq + B q = 1 (3.2-9) 2 + Cq1 D
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Like in the case of ray-transfer matrix, the ABCD matrix of a cascade of optical components (or systems) is a product of the ABCD matrices of the individual components (or systems).
3.3 Hermite-Gaussian Beams Modulated version of Gaussian beam → Intensity distribution not Gaussian, but same wavefronts and angular divergence as the Gaussian beam.
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