1 ESO's Compact Laser Guide Star Unit Ottobeuren, Germany Beam optics!

Transcription

1 1 ESO's Compact Laser Guide Star Unit Ottobeuren, Germany Introduction Characteristics Beam optics! ABCD matrices

2 2 Background! A paraxial wave has wavefronts whose normals are paraxial rays.!! Complex amplitude U(r) satisfies Helmholtz equation: complex envelope A(r) must satisfy the paraxial Helmholtz equation.!! A simple solution of the paraxial Helmholtz equation is the paraboloidal wave:!! Today we are going to study another very important solution: the Gaussian beam.! U(r) = A(r)exp( ikz) T 2 A i 2k A z = 0 T 2 = 2 / x / y 2 A(r) A 1 z ρ 2 = x 2 + y 2 2 ρ exp ik 2z

3 3 Paraboloidal vs. Gaussian! Both are solutions of the paraxial Helmholtz equation A(r) A 1 z ρ 2 = x 2 + y 2 2 ρ exp ik 2z Paraboloidal A(r) A 1 q(z) exp ik ρ 2 2q(z) q(z) = z + iz 0 Gaussian

4 4 Many types of lasers emit beams with a Gaussian shape! For all z, the amplitude (intensity) distribution along x and y is a Gaussian curve.! exp x 2 + y 2 W 2 (z) software plot! real HeNe laser!

5 5 The Gaussian beam!! A(r) A 1 q(z) exp ik ρ 2 2q(z) Gaussian beam: ρ 2 = x 2 + y 2 q(z) = z + iz 0 complex envelope! q(z) z 0!q-parameter of the beam!!rayleigh range! For convenience (we will see why soon) we can write q(z) as:! 1 q(z) = 1 = 1 z + iz 0 R(z) i λ πw 2 (z)

6 6 Now let s put everything together! The complex amplitude of the Gaussian beam: U(r) = A(r) e -ikz! W U(r) = A 0 0 W (z) exp $ ρ2 ' & % W 2 ) (z)( exp[ i( kz ζ(z) )] $ exp& ik % ρ 2 ' 2R(z) ) ( amplitude! (real)! phase! (imaginary)! A 0 = A 1 / iz 0 " W (z) =W 0 1+ z % $ ' # z 0 & ( " R(z) = z 1+ z 0 % * $ ' # z & )* ζ(z) = tan 1 z z 0 W 0 = λz 0 π 2 + -,- 2 The beam is fully defined by A 0 and z 0 (and λ)!

7 7 Gaussian beam: amplitude and phase! W U(r) = A 0 0 W (z) exp $ ρ2 ' & % W 2 ) (z)( exp[ i( kz ζ(z) )] $ exp& ik % ρ 2 ' 2R(z) ) ( Amplitude factor! describes beam spread! Longitudinal phase factor! describes phase delay relative to a plane wave or spherical wave! Radial phase factor! phase shift due to measuring a spherical surface along a plane!

8 8 Gaussian beam: amplitude and phase! W U(r) = A 0 0 W (z) exp $ ρ2 ' & % W 2 ) (z)( exp[ i( kz ζ(z) )] $ exp& ik % ρ 2 ' 2R(z) ) ( Amplitude factor! describes beam spread! Longitudinal phase factor! describes phase delay relative to a plane wave or spherical wave! Radial phase factor! phase shift due to measuring a spherical surface along a plane!

9 9 Properties of the Gaussian beam! The intensity is given by:! 2 I(ρ,z) = U(r) 2 # W = I 0 & # 0 % $ W (z) ( exp 2ρ2 & 2 % ' $ W 2 ( I 0 = A o (z)' For any z the intensity is a Gaussian function of ρ! ( ) The Gaussian function has its peak on the z axis (ρ=0), and decreases monotonically as ρ increases.! W(z) is the beam width of the Gaussian distribution; it increases with the axial distance z.! On the beam axis the intensity is! " W I(0,z) = I 0 % 0 $ # W (z) ' & 2 = I 0 ( ) 2 1+ z / z 0

10 10 Intensity vs. distance z! z = 0 y z = z 0 I = I 0 I = I 0 /2 x z = 2 z 0 I = I 0 /5 I(0,z) I 0 ( z 0 / z ) 2, z >> z 0 NB this one is not a gaussian!

11 11 Viewing a Gaussian beam propagation! (YouTube, Propagation of a Gaussian beam, computed with a FDTD code)

12 12 Gaussian beam: power! Using the previous definition for optical power,! P(z) = I(ρ,z)dA = I(ρ,z)2πρdρ A 0 = 1I ( 2 0 πw 2) 0 Total power = (half the peak intensity) (the beam area)! Independent of z (i.e. energy is conserved)!

13 13 A circle of radius W(z) contains ~86% of the total power! power in circle of radius ρ 0 total power! ρ 0 = 1 I(ρ,z)2πρd ρ P 0 = 1 exp 2ρ 2 0 W 2 ( z ) A circle of radius ρ = 1.5 W(z) contains 99% of the total power!

14 14 W(z) is the beam radius, with a minimum W 0 at z = 0! " ρ = W(z) = beam radius or beam width! W (z) =W 0 1+ z % $ ' # z 0 & 2 Minimum value (beam waist) happens for z = 0: W(0) = W 0! Waist diameter 2W 0 is called spot size! Beam width has the value 2W 0 at z = z 0! Width increases linearly for z >> z 0!

15 15 Beam divergence! For large z the beam width increases linearly,! " W (z) =W 0 1+ z % $ ' # z 0 & 2 W 0 z 0 z = θ 0 z θ 0 = W 0 z 0 = λ πw 0 We define the divergence angle! 2θ 0 = 4 λ π 2W 0 It varies linearly with the wavelength λ! It is inversely proportional to the spot size 2W 0!

16 16 The Lunar Laser ranging Experiment! Apollo 15 retro-reflectors cδt D= 2 What kind of laser beam would you choose to send to the Moon: Wide or narrow? Red or green? Introduction Characteristics ABCD matrices 2θ 0 = 4 λ π 2W0

17 17 Gaussian beam: amplitude and phase! W U(r) = A 0 0 W (z) exp $ ρ2 ' & % W 2 ) (z)( exp[ i( kz ζ(z) )] $ exp& ik % ρ 2 ' 2R(z) ) ( Amplitude factor! describes beam spread! Longitudinal phase factor! describes phase delay relative to a plane wave or spherical wave! Radial phase factor! phase shift due to measuring a spherical surface along a plane!

18 18 The phase of a Gaussian beam has three components! ϕ(ρ,z) = [ kz ζ(z) ] + k ρ 2 2R(z) kz!phase of a plane wave propagating along z!!!!!phase delay specific of the Gaussian beam, that makes!it different from either a plane or a spherical wave.!this is called the Gouy effect.! term responsible for wavefront bending i.e. shift from a plane to a spherical wavefront at off-axis points.

19 19 The phase components ζ(z) and kρ 2 /2R(z) vary slowly with z! ϕ(ρ,z) = [ kz ζ(z) ] + k ρ 2 2R(z) ζ(z) 1/R(z)

20 20 Gaussian beam: wavefronts! $ ϕ(ρ,z) = k & z + % ρ2 ' ) ζ(z) = 2πq 2R(z) ( Since ζ(z) and R(z) vary slowly, they may be considered constant for low values of ρ: ζ(z) ζ and R(z) R, leading to! z + ρ2 2R qλ + ζλ / 2π R(z) = z 1+ z 2 0 z This represents a paraboloidal surface of radius of curvature R = R(z)!

21 21 R(z) is the radius of curvature. It has a minimum ±2z 0 at z = ±z 0.! Note that the sign convention for wavefronts and for optical surfaces is the opposite!

22 A Gaussian beam may be described by its complex q-parameter! 22 If we know q(z):!! then! On the other hand:!!! so we have! q(z) = z + iz 0 Re[ q(z) ] = z = distance to beam waist Im[ q(z) ] = z 0 = Rayleigh length 1 q(z) = 1 R(z) i λ πw 2 (z) R(z) = Re[ 1 q(z) ] 1 = radius of curvature W (z) = λ π Im 1 q(z) 1 = beam radius

23 23 Exercise! Use the definition of q(z) and the last two expressions to arrive at the definitions for the beam size and the radius of curvature.! q(z) = z + iz 0 R(z) = Re[ 1 q(z) ] 1 = radius of curvature W (z) = λ π Im[ 1 q(z) ] 1 = beam radius " W (z) =W 0 1+ z % $ ' # z 0 & ( " R(z) = z 1+ z 0 % * $ ' # z & )* 2 + -,- 2 W 0 = λz 0 π

24 24 Gaussian beam transmitted through a thin lens! We just need to multiply the complex amplitude of the Gaussian beam by the complex transmittance of the lens. The resulting phase is:! & ϕ(ρ,z) = kz ζ + k ρ2 ) ( ' 2R + * k ρ2 2f = kz + k ρ2 2 R, ζ R 1, = 1 R 1 f

25 25 Useful demonstrations! Gaussian beam propagation through two lenses! (

26 26 Transmission through an arbitrary optical system! Remember the ABCD matrices from ray optics? Consider an arbitrary paraxial optical system characterized by an [ABCD] matrix:! q 2 = Aq 1 + B Cq 1 + D The ABCD law! The same rules for cascading optical components apply.!

27 27 Example: Gaussian beam focusing by a lens! Use the ABCD law to show that the beam diameter at the focus of a lens is given by! W 2 = λf πw 1 > What are the best wavelengths for micro-lithography (semiconductor manufacturing) or micromachining?!

28 28 UV lasers are used for microlithography! In photolithography, light from an ultraviolet laser is used to transfer an image from a mask onto a silicon substrate. In direct laser writing, pulsed laser light is used for printing 2D or 3D patterns directly.