eso's compact laser guide star unit ottobeuren, germany ... · introduction characteristics...

Introduction Characteristics ABCD matrices

1

Beam optics!

ESO's Compact Laser Guide Star Unit Ottobeuren, Germany www.eso.org


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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 )

€

∇T2A − i2k ∂A

∂z= 0

∇T2 = ∂2 / ∂x 2 + ∂2 / ∂y 2

A(r) ≈ A1zexp −ik ρ2

2z⎡

⎣⎢

⎤

⎦⎥

ρ2 = x 2 + y 2


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Paraboloidal vs. Gaussian!

A(r) ≈ A1zexp −ik ρ2

2z⎡

⎣⎢

⎤

⎦⎥

ρ2 = x 2 + y 2

A(r) ≈ A1q(z )

exp −ik ρ2

2q(z )⎡

⎣⎢

⎤

⎦⎥

q(z ) = z + iz0Paraboloidal Gaussian

Both are solutions of the paraxial Helmholtz equation


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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.!

software plot! real HeNe laser!

∝ exp − x2 + y 2

W 2(z )⎛⎝⎜

⎞⎠⎟


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The Gaussian beam!

A(r) ≈ A1q(z )

exp −ik ρ2

2q(z )⎡

⎣⎢

⎤

⎦⎥

ρ2 = x 2 + y 2 q(z ) = z + iz0Gaussian beam: complex envelope!

q(z) !q-parameter of the beam!z0 !Rayleigh range!

!For convenience (we will see why soon) we can write q(z) as: !

1q(z )

= 1z + iz0

= 1R(z )

− i λπW 2(z )


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Now let’s put everything together!The complex amplitude of the Gaussian beam: U(r) = A(r) e-ikz!

€

U(r) = A0W0

W (z )exp −

ρ2

W 2(z )$

% &

'

( )

×exp −i kz − ζ(z )( )[ ]

×exp −ik ρ2

2R(z )$

% &

'

( )

€

A0 = A1 / iz0

W (z ) =W0 1+zz0

"

# $

%

& '

2

R(z ) = z 1+z0z

" # $

% & ' 2(

) * *

+

, - -

ζ(z ) = tan−1 zz0

W0 =λz0π

The beam is fully defined by A0 and z0 (and λ)!

amplitude!(real)!

phase!(imaginary)!


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Gaussian beam: amplitude and phase!

€

U(r) = A0W0

W (z )exp −

ρ2

W 2(z )$

% &

'

( )

×exp −i kz − ζ(z )( )[ ]

×exp −ik ρ2

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!


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€

U(r) = A0W0

W (z )exp −

ρ2

W 2(z )$

% &

'

( )

×exp −i kz − ζ(z )( )[ ]

×exp −ik ρ2

2R(z )$

% &

'

( )





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Properties of the Gaussian beam!The intensity is given by:!

€

I(ρ,z ) = U(r) 2 = I0W0

W (z )#

$ %

&

' (

2

exp −2ρ2

W 2(z )#

$ %

&

' ( I0 = Ao

2( )•  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!

€

I(0,z ) = I0W0

W (z )"

# $

%

& '

2

=I0

1+ z / z0( )2


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Intensity vs. distance z!

€

I(0,z ) ≈ I0 z0 / z( )2, z >> z0

NB this one is not a gaussian!

z = 0 I = I0

z = z0 I = I0/2

z = 2 z0 I = I0/5

x

y


11

Viewing a Gaussian beam propagation!

(YouTube, Propagation of a Gaussian beam, computed with a FDTD code)


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Gaussian beam: power!Using the previous definition for optical power,!

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P(z ) = I(ρ,z )dAA∫ = I(ρ,z )2πρdρ

0

∞

∫= 1

2 I0 πW02( )

•  Total power = (half the peak intensity) × (the beam area) !•  Independent of z (i.e. energy is conserved)!


13

A circle of radius W(z) contains ~86% of the total power!

power in circle of radius ρ0 total power!

= 1P

I(ρ,z )2πρd ρ0

ρ0∫

= 1− exp − 2ρ02

W 2 z( )⎡

⎣⎢

⎤

⎦⎥

A circle of radius ρ = 1.5 W(z) contains 99% of the total power!


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W(z) is the beam radius,with a minimum W0 at z = 0!

ρ = W(z) = beam radius or beam width!

€

W (z ) =W0 1+zz0

"

# $

%

& '

2

• Minimum value (beam waist) happens for z = 0: W(0) = W0!• Waist diameter 2W0 is called spot size!•  Beam width has the value √2W0 at z = z0!• Width increases linearly for z >> z0!


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Beam divergence!For large z the beam width increases linearly,!

€

W (z ) =W0 1+zz0

"

# $

%

& '

2

≈W0

z0z = θ0z

We define the divergence angle!

θ0 =W0

z0= λπW0

•  It varies linearly with the wavelength λ!•  It is inversely proportional to the spot size 2W0!

2θ0 =4π

λ2W0


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The Lunar Laser ranging Experiment!

D =cΔt2

What kind of laser beam would you choose to send to the Moon: Wide or narrow? Red or green?

Apollo 15 retro-reflectors

2θ0 =4π

λ2W0


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€

U(r) = A0W0

W (z )exp −

ρ2

W 2(z )$

% &

'

( )

×exp −i kz − ζ(z )( )[ ]

×exp −ik ρ2

2R(z )$

% &

'

( )





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The phase of a Gaussian beam has three components!

€

ϕ(ρ,z ) = kz − ζ(z )[ ] + k ρ2

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.

!


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The phase components ζ(z) and kρ2/2R(z) vary slowly with z !

€

ϕ(ρ,z ) = kz − ζ(z )[ ] + k ρ2

2R(z )

1/R(z)

ζ(z)


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Gaussian beam: wavefronts!

€

ϕ(ρ,z ) = k z +ρ2

2R(z )$

% &

'

( ) − ζ(z ) = 2πq

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

This represents a paraboloidal surface of radius of curvature R = R(z)!

R(z ) = z 1+ z0z

⎛⎝⎜

⎞⎠⎟2⎡

⎣⎢⎢

⎤

⎦⎥⎥


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R(z) is the radius of curvature.It has a minimum ±2z0 at z = ±z0.!

Note that the sign convention for wavefronts and for optical surfaces is the opposite!


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On the other hand:!!!so we have!

A Gaussian beam may be described by its complex q-parameter!

If we know q(z):!!then!

€

q(z ) = z + iz0

€

1q(z )

=1

R(z )− i λ

πW 2(z )€

Re q(z )[ ] = z = distance to beam waist

Im q(z )[ ] = z0 = Rayleigh length

R(z ) = Re 1 q(z )[ ]−1 = radius of curvature

W (z ) = λπ

Im − 1q(z )

⎡⎣⎢

⎤⎦⎥

−1

= beam radius


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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.!

€

W (z ) =W0 1+zz0

"

# $

%

& '

2

R(z ) = z 1+z0z

" # $

% & ' 2(

) * *

+

, - -

W0 =λz0π

€

q(z ) = z + iz0

R(z ) = Re 1 q(z )[ ]−1= radius of curvature

W (z ) =λπ

Im −1 q(z )[ ]−1= beam radius


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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 ρ

2

2f

= kz + k ρ2

2 , R − ζ

1, R

=1R−1f


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Useful demonstrations!Gaussian beam propagation through two lenses!(http://demonstrations.wolfram.com)!


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Transmission through an arbitrary optical system!

Remember the ABCD matrices from ray optics? Consider an arbitrary paraxial optical system characterized by an [ABCD] matrix:!

€

q2 =Aq1 +BCq1 +D

The ABCD law!

The same rules for cascading optical components apply.!


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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!

€

W2 =λfπW1

> What are the best wavelengths for micro-lithography (semiconductor manufacturing) or micromachining?!


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UV lasers are used for microlithography !

In direct laser writing, pulsed laser light is used for printing

2D or 3D patterns directly.

In photolithography, light from an ultraviolet laser is used to transfer an image from a mask onto a silicon substrate.

eso's compact laser guide star unit ottobeuren, germany ... · introduction characteristics...

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