Lecture 7- ECE 240a Beam Optics Ver Chap. 1-3 Helmholtz Equation - - PowerPoint PPT Presentation

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Lecture 7- ECE 240a Beam Optics Ver Chap. 1-3 Helmholtz Equation - - PowerPoint PPT Presentation

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Lecture 7- ECE 240a EM Review Maxwells Equations Material Properties Helmholtz Equation Dispersion Lecture 7- ECE 240a Beam Optics Ver Chap. 1-3 Helmholtz Equation Paraxial Solutions Fundamental Mode Beam waist and radius

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SLIDE 1

Lecture 7- ECE 240a EM Review

Maxwell’s Equations Material Properties Helmholtz Equation Dispersion

Beam Optics

Helmholtz Equation Paraxial Solutions Fundamental Mode Beam waist and radius Axial Phase Complete Solution Amplitude Factor Beam Divergence Axial Phase Radial Phase

Lecture 7- ECE 240a

Ver Chap. 1-3

ECE 240a Lasers - Fall 2019 Lecture 7 1

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SLIDE 2

Lecture 7- ECE 240a EM Review

Maxwell’s Equations Material Properties Helmholtz Equation Dispersion

Beam Optics

Helmholtz Equation Paraxial Solutions Fundamental Mode Beam waist and radius Axial Phase Complete Solution Amplitude Factor Beam Divergence Axial Phase Radial Phase

Review - Electromagnetics

Maxwell’s Equations ∇ × E = −∂B ∂t ∇ × H = ∂D ∂t ∇ · B = ∇ · D = E is the electric field vector (V/m) H is the magnetic field vector (A/m) D is the electric flux density (C/m2) B is the magnetic flux density (Webers/m2)

ECE 240a Lasers - Fall 2019 Lecture 7 2

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SLIDE 3

Lecture 7- ECE 240a EM Review

Maxwell’s Equations Material Properties Helmholtz Equation Dispersion

Beam Optics

Helmholtz Equation Paraxial Solutions Fundamental Mode Beam waist and radius Axial Phase Complete Solution Amplitude Factor Beam Divergence Axial Phase Radial Phase

Constitutive Relations

Dielectric Materials w/no free charge D = ε0E + P B = µ0H P is the polarization (C/m2) ε0, µ0 are permittivity and permeability respectively

ECE 240a Lasers - Fall 2019 Lecture 7 3

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SLIDE 4

Lecture 7- ECE 240a EM Review

Maxwell’s Equations Material Properties Helmholtz Equation Dispersion

Beam Optics

Helmholtz Equation Paraxial Solutions Fundamental Mode Beam waist and radius Axial Phase Complete Solution Amplitude Factor Beam Divergence Axial Phase Radial Phase

Linear, Causal Media - Linear Systems Approach

Material as system; E(r, τ) as input; P(r, t) as output P(r, t) = ε0

∞

χ(r, t − τ)E(r, τ)dτ where χ(r, t) is the electric susceptibility The convolution represents memory of material Materials that exhibit memory effects are called dispersive

ECE 240a Lasers - Fall 2019 Lecture 7 4

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SLIDE 5

Lecture 7- ECE 240a EM Review

Maxwell’s Equations Material Properties Helmholtz Equation Dispersion

Beam Optics

Helmholtz Equation Paraxial Solutions Fundamental Mode Beam waist and radius Axial Phase Complete Solution Amplitude Factor Beam Divergence Axial Phase Radial Phase

Time Harmonic Fields

Let E(r, t) = Re{E(r, t)} where E(r, t) is the complex electric field. E(r, t) can be written in terms of as a superposition of time-harmonic components using an inverse Fourier transform E(r, t) = 1 2π

∞

−∞

E(r, ω)ejωtdω Convolution relating E and P becomes P(r, ω) = ε0χ(r, ω)E(r, ω)

ECE 240a Lasers - Fall 2019 Lecture 7 5

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SLIDE 6

Lecture 7- ECE 240a EM Review

Maxwell’s Equations Material Properties Helmholtz Equation Dispersion

Beam Optics

Helmholtz Equation Paraxial Solutions Fundamental Mode Beam waist and radius Axial Phase Complete Solution Amplitude Factor Beam Divergence Axial Phase Radial Phase

Time -Harmonic Form of Wave Equation

The time harmonic form of the wave equation is the Helmholtz Equation ∇2E + n2(r, ω)k2E = 0 k = ω/c0 is the free-space wavenumber. n2(r, ω) = 1 + χ(r, ω) is the index of refraction Solutions are determined by specifying a geometry, choosing an appropriate coordinate system, and applying the boundary conditions.

ECE 240a Lasers - Fall 2019 Lecture 7 6

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SLIDE 7

Lecture 7- ECE 240a EM Review

Maxwell’s Equations Material Properties Helmholtz Equation Dispersion

Beam Optics

Helmholtz Equation Paraxial Solutions Fundamental Mode Beam waist and radius Axial Phase Complete Solution Amplitude Factor Beam Divergence Axial Phase Radial Phase

Plane Waves

Let the spatial dependence of E and H be given by E = E0e−jβ·r ˆ e H = H0e−jβ·r ˆ h where β = βx ˆ x + βy ˆ y + βz ˆ z , r = x ˆ x + y ˆ y + z ˆ z is the position vector, E0 = |E0|ejφe is a complex constant. The real electric field is given by E = |E0| cos (ωt − β · r + φe) ˆ e Values of r and t that produce a constant in the argument of the cosine function define the surface of a plane - plane-wave solutions.

ECE 240a Lasers - Fall 2019 Lecture 7 7

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SLIDE 8

Lecture 7- ECE 240a EM Review

Maxwell’s Equations Material Properties Helmholtz Equation Dispersion

Beam Optics

Helmholtz Equation Paraxial Solutions Fundamental Mode Beam waist and radius Axial Phase Complete Solution Amplitude Factor Beam Divergence Axial Phase Radial Phase

Plane Wave Dispersion Relation

Substitute form of into Helmholtz Eq. Note that the spatial operator ∇2 → −β2 Helmholtz equation has a solution when β(ω) = n(ω)k = n(ω)ω c0 = 2πn(ω) λ where λ = nλr is the free-space wavelength. The function β(ω) is called the dispersion relation Values of β is called the propagation constant.

ECE 240a Lasers - Fall 2019 Lecture 7 8

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SLIDE 9

Lecture 7- ECE 240a EM Review

Maxwell’s Equations Material Properties Helmholtz Equation Dispersion

Beam Optics

Helmholtz Equation Paraxial Solutions Fundamental Mode Beam waist and radius Axial Phase Complete Solution Amplitude Factor Beam Divergence Axial Phase Radial Phase

Plane Wave Dispersion Relation -cont.

As the field propagates a distance L in the ˆ β direction, the amplitude of the field is multiplied by ejβ(ω)L corresponding to a phase shift of φL(ω) = β(ω)L if β(ω) is real. This phase shift does not change the functional form of the solution. → Plane wave solutions are the modes or eigenfunctions of an unbounded, linear medium ejβ(ω)L is the eigenvalue. Arbitrary solutions can be constructed as a superposition of these solutions

ECE 240a Lasers - Fall 2019 Lecture 7 9

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SLIDE 10

Lecture 7- ECE 240a EM Review

Maxwell’s Equations Material Properties Helmholtz Equation Dispersion

Beam Optics

Helmholtz Equation Paraxial Solutions Fundamental Mode Beam waist and radius Axial Phase Complete Solution Amplitude Factor Beam Divergence Axial Phase Radial Phase

Intensity and Power

Poynting vector S S S . = E E E × H H H Time Harmonic Field Save(r, t) = 1

2E(r, t) × H∗(r, t)

Intensity (plane-wave) I(r, t) . = Save(r, t) = |E0|2 2η Power P(t) =

  • A

I(r, t)dA

ECE 240a Lasers - Fall 2019 Lecture 7 10

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SLIDE 11

Lecture 7- ECE 240a EM Review

Maxwell’s Equations Material Properties Helmholtz Equation Dispersion

Beam Optics

Helmholtz Equation Paraxial Solutions Fundamental Mode Beam waist and radius Axial Phase Complete Solution Amplitude Factor Beam Divergence Axial Phase Radial Phase

Slowly Varying Solutions to Helmholtz Equation

Start with Helmholtz Equation ∇2E + n2k2E = ∇2E + n2 ω c

2

E = Assume that field is “close -to” a plane wave so that E(r) = E0ψ(r)e−jkz Verdeyen (3.2.5) where k = ωn/c and n is a constant Type of solution is similar to time-dependent Schrödinger equation Now separate spatial dependence ∇2

t E + ∂2E

∂z2 + n2 ω c

2

E = 0

ECE 240a Lasers - Fall 2019 Lecture 7 11

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SLIDE 12

Lecture 7- ECE 240a EM Review

Maxwell’s Equations Material Properties Helmholtz Equation Dispersion

Beam Optics

Helmholtz Equation Paraxial Solutions Fundamental Mode Beam waist and radius Axial Phase Complete Solution Amplitude Factor Beam Divergence Axial Phase Radial Phase

Evaluate Derivatives

The derivatives are ∇2

t E = E0

  • ∇2

t ψ

e−jkz ∂E ∂z = E0

  • −jkψ + ∂ψ

∂z

  • e−jkz

∂2E ∂z2 = E0

  • −k2ψ − j2k ∂ψ

∂z + ∂2ψ ∂z2

  • e−jkz

Plug back into Helmholtz equation ∇2

t E + ∂2E

∂z2 + n2 ω c

2

E = 0 ∇2

t ψ − j2k ∂ψ

∂z + ∂2ψ ∂z2 = 0 Equation is exact.

ECE 240a Lasers - Fall 2019 Lecture 7 12

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SLIDE 13

Lecture 7- ECE 240a EM Review

Maxwell’s Equations Material Properties Helmholtz Equation Dispersion

Beam Optics

Helmholtz Equation Paraxial Solutions Fundamental Mode Beam waist and radius Axial Phase Complete Solution Amplitude Factor Beam Divergence Axial Phase Radial Phase

Slowly Varying (Paraxial) Approximation

Paraxial (small-angle) field where ψ(r) varies slowly. If ψ(r) constant - then plane wave (angle =0) For ψ(r) slowing varying, near plane wave solutions with small angles Therefore z-dependence is ∂2ψ ∂z2 ≪ 2k ∂ψ ∂z Helmholtz equation is then ∇2

t ψ(r) + −j2k ∂ψ(r)

∂z = 0 Paraxial Helmholtz equation

ECE 240a Lasers - Fall 2019 Lecture 7 13

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SLIDE 14

Lecture 7- ECE 240a EM Review

Maxwell’s Equations Material Properties Helmholtz Equation Dispersion

Beam Optics

Helmholtz Equation Paraxial Solutions Fundamental Mode Beam waist and radius Axial Phase Complete Solution Amplitude Factor Beam Divergence Axial Phase Radial Phase

Axial Symmetric (TEM00) Solutions

Assume that ψ(r) = ψ(r, φ, z) has no φ dependence (axially symmetric) Then ∇2

t ψ = 1

r ∂ ∂r

  • r ∂ψ

∂r

  • − jk ∂ψ

∂z = 0 Guess solution ψ0 = exp

  • −j
  • P(z) + kr2

2q(z)

  • ECE 240a Lasers - Fall 2019 Lecture 7

14

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SLIDE 15

Lecture 7- ECE 240a EM Review

Maxwell’s Equations Material Properties Helmholtz Equation Dispersion

Beam Optics

Helmholtz Equation Paraxial Solutions Fundamental Mode Beam waist and radius Axial Phase Complete Solution Amplitude Factor Beam Divergence Axial Phase Radial Phase

Form Solution

−jk ∂ψ ∂z =

  • −2kP ′(z) + kr2q′(z)

q2(z)

  • ψ0

∂ψ0 ∂r = −j kr q(z)ψ0 → r ∂ψ ∂r = −j kr2 q(z)ψ0

∂ ∂r

  • r ∂ψ

∂r

  • = ∂

∂r

  • −j kr2

q(z) ψ0

  • =

∂ψ0 ∂r

  • −j kr2

q(z)

  • + ∂

∂r

  • −j kr2

q(z)

  • ψ0

=

  • −j kr2

q(z)

  • −j kr

q(z)

  • ψ0 − 2j kr

q(z) ψ0

1 r ∂ ∂r

  • r ∂ψ

∂r

  • =
  • − k2r2

q2(z) − 2j kr q(z)

  • ψ0

ECE 240a Lasers - Fall 2019 Lecture 7 15

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SLIDE 16

Lecture 7- ECE 240a EM Review

Maxwell’s Equations Material Properties Helmholtz Equation Dispersion

Beam Optics

Helmholtz Equation Paraxial Solutions Fundamental Mode Beam waist and radius Axial Phase Complete Solution Amplitude Factor Beam Divergence Axial Phase Radial Phase

Solution - cont.

Group into powers 1 r ∂ ∂r

  • r ∂ψ

∂r

  • − jk ∂ψ

∂z

  • k2

q2(z)

  • q′(z) − 1

r2

  • − 2k
  • P ′(z) + j

1 q(z)

  • r
  • ψ0 = 0

Each power must be zero → two differential equations q′(z) = 1 P ′(z) = −j 1 q(z) Solve for first equation q(z) = z + jz0 where z0is a constant. Why complex? ψ0 =∝ exp

  • −j kr2

2q(z)

  • If q(z)real amplitude does not change but phase does.

ECE 240a Lasers - Fall 2019 Lecture 7 16

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SLIDE 17

Lecture 7- ECE 240a EM Review

Maxwell’s Equations Material Properties Helmholtz Equation Dispersion

Beam Optics

Helmholtz Equation Paraxial Solutions Fundamental Mode Beam waist and radius Axial Phase Complete Solution Amplitude Factor Beam Divergence Axial Phase Radial Phase

Look at ψ0(z = 0)

ψ0(z = 0) = exp

  • −kr2

2z0

  • Decay as function of r

exp [−jP(z = 0)]

  • Phase term

Define distance when argument to exp =1 w2 . = 2z0 k = λ0z0 nπ

  • r

z0 . = πnw2 λ0 where n is the index of refraction. Parameter w0 is called beam waist or spot size Parameter z0 called Rayleigh range

ECE 240a Lasers - Fall 2019 Lecture 7 17

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SLIDE 18

Lecture 7- ECE 240a EM Review

Maxwell’s Equations Material Properties Helmholtz Equation Dispersion

Beam Optics

Helmholtz Equation Paraxial Solutions Fundamental Mode Beam waist and radius Axial Phase Complete Solution Amplitude Factor Beam Divergence Axial Phase Radial Phase

Beam Width and Radius

Separate q(z) into real and imaginary parts 1 q(z) = 1 z + jz0 = z z2 + z2 − j z0 z2 + z2 = 1 R(z) − j λ0 πnw2(z) Look at ψ(z) where

ψ0(z) = exp

  • −j
  • P (z) +

kr2 2q(z)

  • = exp
  • −

kz0r2 2 z2 + z2

  • real decaying part

exp

  • −j

kzr2 2 z2 + z2

  • radial phase

exp [−jP (z)]

  • axial phase

Waist or spot size now a function of z w2(z) = 2 z2 + z2

  • kz0

= 2z0 k

  • 1 +

z

z0

2

= w2

  • 1 +
  • λ0z

πnw2

  • Radius of curvature of beam is then

R(z) = 1 z

  • z2 + z2
  • = z
  • 1 +

z0

z

2

= z

  • 1 +
  • πnw2

λ0z

2

ECE 240a Lasers - Fall 2019 Lecture 7 18

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SLIDE 19

Lecture 7- ECE 240a EM Review

Maxwell’s Equations Material Properties Helmholtz Equation Dispersion

Beam Optics

Helmholtz Equation Paraxial Solutions Fundamental Mode Beam waist and radius Axial Phase Complete Solution Amplitude Factor Beam Divergence Axial Phase Radial Phase

Axial (Longitudinal) Phase

Solve differential equation for axial phase term P ′(z) = −j 1 q(z) = −j 1 z + jz0 Integrating we have jP(z) =

z

dz′ z′ + jz0 = ln(z′ + jz0)

  • z

0 = ln

  • 1 − j

z

z0

  • Write in terms of magnitude and phase

1 − j

z

z0

  • =
  • 1 +

z

z0

21/2

exp

  • −j tan−1 z

z0

  • So that

exp [−jP(z)] = exp

  • ln
  • 1 +

z

z0

21/2

exp

  • −j tan−1 z

z0

  • =
  • 1 +

z

z0

2−1/2

exp

  • j tan−1 z

z0

  • ECE 240a Lasers - Fall 2019 Lecture 7

19

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SLIDE 20

Lecture 7- ECE 240a EM Review

Maxwell’s Equations Material Properties Helmholtz Equation Dispersion

Beam Optics

Helmholtz Equation Paraxial Solutions Fundamental Mode Beam waist and radius Axial Phase Complete Solution Amplitude Factor Beam Divergence Axial Phase Radial Phase

Final Solution

E(x, y, z) E0 = w0 w(z) exp

  • −

r2 w2(z)

  • Amplitude Factor

× exp

  • −j kr2

2R(z)

  • Radial phase

× exp

  • −j
  • kz − tan−1 z

z0

  • Longitudinal (axial) phase

Beam spot size w2(z) = 2 z2 + z2

  • kz0

= 2z0 k

  • 1 +

z

z0

2

= w2

  • 1 +
  • λ0z

πnw2

  • Radial phase

R(z) = 1 z

  • z2 + z2
  • = z
  • 1 +

z0

z

2

= z

  • 1 +
  • πnw2

λ0z

2

ECE 240a Lasers - Fall 2019 Lecture 7 20

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SLIDE 21

Lecture 7- ECE 240a EM Review

Maxwell’s Equations Material Properties Helmholtz Equation Dispersion

Beam Optics

Helmholtz Equation Paraxial Solutions Fundamental Mode Beam waist and radius Axial Phase Complete Solution Amplitude Factor Beam Divergence Axial Phase Radial Phase

Amplitude Factor

w0 w(z) exp

  • −

r2 w2(z)

  • w2(z) =

2 z2 + z2

  • kz0

= 2z0 k

  • 1 +

z

z0

2

= w2

  • 1 +
  • λ0z

πnw2

  • On-axis intensity (W/area)

I(r, z) = I0

  • w0

w(z)

2

exp

  • − 2r2

w2(z)

  • where I0 = E2

2η

On axis intensity I(0, z) = I0

  • w0

w(z)

2

= I0 1 + (z/z0)2 Intensity falls off as 1/z2 for z ≫ z0 just like any other field

ECE 240a Lasers - Fall 2019 Lecture 7 21

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SLIDE 22

Lecture 7- ECE 240a EM Review

Maxwell’s Equations Material Properties Helmholtz Equation Dispersion

Beam Optics

Helmholtz Equation Paraxial Solutions Fundamental Mode Beam waist and radius Axial Phase Complete Solution Amplitude Factor Beam Divergence Axial Phase Radial Phase

Total Power

Power P = 1 2η

  • A

|E(r)|2 rdrdφ = E2 2η w2 w2(z)

2π ∞

exp

  • −

r2 w2(z)

  • rdrdφ

= E2 2η

  • πw2

2

  • ECE 240a Lasers - Fall 2019 Lecture 7

22

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SLIDE 23

Lecture 7- ECE 240a EM Review

Maxwell’s Equations Material Properties Helmholtz Equation Dispersion

Beam Optics

Helmholtz Equation Paraxial Solutions Fundamental Mode Beam waist and radius Axial Phase Complete Solution Amplitude Factor Beam Divergence Axial Phase Radial Phase

Beam Divergence

Wo θ W(z) z0 Max R(z) @ z=z0 Waist at z=0 θ ≈ λ d

For z ≫ z0 define the divergence as θ ≈ w0 z0 = λ πnw0 Spreads w/angle proportional to λ; inverse with waist w0

ECE 240a Lasers - Fall 2019 Lecture 7 23

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SLIDE 24

Lecture 7- ECE 240a EM Review

Maxwell’s Equations Material Properties Helmholtz Equation Dispersion

Beam Optics

Helmholtz Equation Paraxial Solutions Fundamental Mode Beam waist and radius Axial Phase Complete Solution Amplitude Factor Beam Divergence Axial Phase Radial Phase

Axial Phase

φ = −j

  • kz − tan−1 z

z0

  • where k = ωn/c.

For z ≫ z0, tan−1 z z0

  • → π

2 phase is linear in z (plane wave) For z = 0, φ = 0

ECE 240a Lasers - Fall 2019 Lecture 7 24

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SLIDE 25

Lecture 7- ECE 240a EM Review

Maxwell’s Equations Material Properties Helmholtz Equation Dispersion

Beam Optics

Helmholtz Equation Paraxial Solutions Fundamental Mode Beam waist and radius Axial Phase Complete Solution Amplitude Factor Beam Divergence Axial Phase Radial Phase

Radial Phase

Radial phase exp

  • −j kr2

2R(z)

  • R(z) = 1

z

  • z2 + z2
  • = z
  • 1 +

z0

z

2

= z

  • 1 +
  • πnw2

λ0z

2

Parameter R(z) is a measure of radius of curvature of wavefront Plane wave at z = 0 and z = ∞; max curvature at z = z0 Plane at z = 0 is where

Spot size is minimum Wavefront is planar

ECE 240a Lasers - Fall 2019 Lecture 7 25