Lecture 7- ECE 240a EM Review Maxwell’s 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 Axial Phase Complete Solution Amplitude Factor Beam Divergence Axial Phase Radial Phase 1 ECE 240a Lasers - Fall 2019 Lecture 7
Review - Electromagnetics Lecture 7- ECE 240a EM Review Maxwell’s Equations Maxwell’s Equations Material Properties Helmholtz − ∂ B Equation ∇ × E = Dispersion ∂t Beam ∂ D Optics ∇ × H = ∂t Helmholtz Equation ∇ · B = 0 Paraxial Solutions ∇ · D = 0 Fundamental Mode Beam waist and radius E is the electric field vector (V/m) Axial Phase Complete Solution H is the magnetic field vector (A/m) Amplitude Factor Beam Divergence D is the electric flux density (C/ m 2 ) Axial Phase Radial Phase B is the magnetic flux density (Webers/ m 2 ) 2 ECE 240a Lasers - Fall 2019 Lecture 7
Constitutive Relations Lecture 7- ECE 240a EM Review Maxwell’s Equations Material Properties Helmholtz Equation Dielectric Materials w/no free charge Dispersion Beam Optics Helmholtz = ε 0 E + P D Equation Paraxial B = µ 0 H Solutions Fundamental Mode Beam waist P is the polarization (C/ m 2 ) and radius Axial Phase Complete ε 0 , µ 0 are permittivity and permeability respectively Solution Amplitude Factor Beam Divergence Axial Phase Radial Phase 3 ECE 240a Lasers - Fall 2019 Lecture 7
Linear, Causal Media - Linear Systems Approach Lecture 7- ECE 240a EM Review Maxwell’s Equations Material Properties Helmholtz Equation Dispersion Material as system; E ( r , τ ) as input; P ( r , t ) as output Beam � ∞ Optics Helmholtz P ( r , t ) = ε 0 χ ( r , t − τ ) E ( r , τ ) dτ Equation Paraxial 0 Solutions where χ ( r , t ) is the electric susceptibility Fundamental Mode Beam waist The convolution represents memory of material and radius Axial Phase Materials that exhibit memory effects are called dispersive Complete Solution Amplitude Factor Beam Divergence Axial Phase Radial Phase 4 ECE 240a Lasers - Fall 2019 Lecture 7
Time Harmonic Fields Lecture 7- ECE 240a EM Review Maxwell’s Equations Material Properties Let Helmholtz Equation E ( r , t ) = Re { E ( r , t ) } Dispersion Beam where E ( r , t ) is the complex electric field . Optics Helmholtz E ( r , t ) can be written in terms of as a superposition of time-harmonic Equation Paraxial components using an inverse Fourier transform Solutions Fundamental � ∞ Mode E ( r , t ) = 1 E ( r , ω ) e jωt dω Beam waist and radius 2 π −∞ Axial Phase Complete Solution Convolution relating E and P becomes Amplitude Factor P ( r , ω ) = ε 0 χ ( r , ω ) E ( r , ω ) Beam Divergence Axial Phase Radial Phase 5 ECE 240a Lasers - Fall 2019 Lecture 7
Time -Harmonic Form of Wave Equation Lecture 7- ECE 240a EM Review Maxwell’s Equations Material Properties Helmholtz Equation Dispersion The time harmonic form of the wave equation is the Helmholtz Equation Beam Optics ∇ 2 E + n 2 ( r , ω ) k 2 E = 0 Helmholtz Equation Paraxial k = ω / c 0 is the free-space wavenumber . Solutions Fundamental n 2 ( r , ω ) = 1 + χ ( r , ω ) is the index of refraction Mode Beam waist and radius Axial Phase Solutions are determined by specifying a geometry, choosing an Complete appropriate coordinate system, and applying the boundary conditions. Solution Amplitude Factor Beam Divergence Axial Phase Radial Phase 6 ECE 240a Lasers - Fall 2019 Lecture 7
Plane Waves Lecture 7- ECE 240a EM Review Maxwell’s Equations Material Properties Let the spatial dependence of E and H be given by Helmholtz Equation E 0 e − j β · r ˆ = Dispersion E e H 0 e − j β · r ˆ Beam = H h Optics Helmholtz Equation where β = β x ˆ x + β y ˆ y + β z ˆ z , r = x ˆ x + y ˆ y + z ˆ z is the position vector, Paraxial Solutions E 0 = | E 0 | e jφ e is a complex constant. Fundamental Mode Beam waist The real electric field is given by and radius Axial Phase Complete E = | E 0 | cos ( ωt − β · r + φ e ) ˆ e Solution Amplitude Factor Values of r and t that produce a constant in the argument of the cosine Beam Divergence function define the surface of a plane - plane-wave solutions. Axial Phase Radial Phase 7 ECE 240a Lasers - Fall 2019 Lecture 7
Plane Wave Dispersion Relation Lecture 7- ECE 240a EM Review Maxwell’s Equations Material Properties Substitute form of into Helmholtz Eq. Helmholtz Equation Note that the spatial operator ∇ 2 → − β 2 Dispersion Beam Optics Helmholtz equation has a solution when Helmholtz Equation Paraxial Solutions β ( ω ) = n ( ω ) k = n ( ω ) ω = 2 πn ( ω ) Fundamental Mode c 0 λ Beam waist and radius where λ = nλ r is the free-space wavelength. Axial Phase Complete Solution The function β ( ω ) is called the dispersion relation Amplitude Factor Beam Divergence Values of β is called the propagation constant . Axial Phase Radial Phase 8 ECE 240a Lasers - Fall 2019 Lecture 7
Plane Wave Dispersion Relation -cont. Lecture 7- ECE 240a EM Review Maxwell’s Equations Material Properties As the field propagates a distance L in the ˆ β direction, the amplitude of Helmholtz the field is multiplied by e jβ ( ω ) L corresponding to a phase shift of Equation Dispersion φ L ( ω ) = β ( ω ) L if β ( ω ) is real. Beam Optics Helmholtz This phase shift does not change the functional form of the solution. → Equation Paraxial Solutions Plane wave solutions are the modes or eigenfunctions of an Fundamental Mode unbounded, linear medium Beam waist and radius Axial Phase e jβ ( ω ) L is the eigenvalue . Complete Solution Amplitude Factor Arbitrary solutions can be constructed as a superposition of these Beam Divergence solutions Axial Phase Radial Phase 9 ECE 240a Lasers - Fall 2019 Lecture 7
Intensity and Power Lecture 7- ECE 240a EM Review Maxwell’s Equations Poynting vector Material Properties Helmholtz S . Equation S S = E E E × H H H Dispersion Beam Time Harmonic Field Optics Helmholtz S ave ( r , t ) = 1 2 E ( r , t ) × H ∗ ( r , t ) Equation Paraxial Solutions Intensity (plane-wave) Fundamental Mode Beam waist and radius = S ave ( r , t ) = | E 0 | 2 I ( r , t ) . Axial Phase 2 η Complete Solution Amplitude Power Factor � Beam Divergence P ( t ) = I ( r , t ) dA Axial Phase A Radial Phase 10 ECE 240a Lasers - Fall 2019 Lecture 7
Slowly Varying Solutions to Helmholtz Equation Lecture 7- ECE 240a EM Review Maxwell’s Start with Helmholtz Equation Equations Material Properties ∇ 2 E + n 2 k 2 E = 0 Helmholtz Equation ∇ 2 E + n 2 � ω � 2 Dispersion = E 0 c Beam Optics Helmholtz Assume that field is “close -to” a plane wave so that Equation Paraxial Solutions E ( r ) = E 0 ψ ( r ) e − jkz Verdeyen (3.2.5) Fundamental Mode Beam waist where k = ωn / c and n is a constant and radius Axial Phase Complete Solution Type of solution is similar to time-dependent Schrödinger equation Amplitude Factor Beam Now separate spatial dependence Divergence Axial Phase t E + ∂ 2 E ∂z 2 + n 2 � ω � 2 Radial Phase ∇ 2 E = 0 c 11 ECE 240a Lasers - Fall 2019 Lecture 7
Evaluate Derivatives Lecture 7- ECE 240a EM Review Maxwell’s The derivatives are Equations � t ψ � ∇ 2 ∇ 2 e − jkz Material t E = E 0 Properties Helmholtz Equation � � ∂ E − jkψ + ∂ψ Dispersion e − jkz ∂z = E 0 Beam ∂z Optics Helmholtz � � ∂ 2 E ∂z + ∂ 2 ψ Equation − k 2 ψ − j 2 k ∂ψ e − jkz ∂z 2 = E 0 Paraxial Solutions ∂z 2 Fundamental Mode Plug back into Helmholtz equation Beam waist and radius Axial Phase t E + ∂ 2 E ∂z 2 + n 2 � ω � 2 ∇ 2 Complete E = 0 Solution c Amplitude Factor Beam ∂z + ∂ 2 ψ Divergence t ψ − j 2 k ∂ψ ∇ 2 ∂z 2 = 0 Axial Phase Radial Phase Equation is exact. 12 ECE 240a Lasers - Fall 2019 Lecture 7
Slowly Varying (Paraxial) Approximation Lecture 7- ECE 240a EM Review Maxwell’s Equations Paraxial (small-angle) field where ψ ( r ) varies slowly. Material Properties Helmholtz If ψ ( r ) constant - then plane wave (angle =0) Equation Dispersion Beam For ψ ( r ) slowing varying, near plane wave solutions with small angles Optics Helmholtz Therefore z-dependence is Equation Paraxial Solutions ∂ 2 ψ ∂z 2 ≪ 2 k ∂ψ Fundamental Mode ∂z Beam waist and radius Axial Phase Helmholtz equation is then Complete Solution Amplitude Factor t ψ ( r ) + − j 2 k ∂ψ ( r ) ∇ 2 = 0 Beam Divergence ∂z Axial Phase Radial Phase Paraxial Helmholtz equation 13 ECE 240a Lasers - Fall 2019 Lecture 7
Axial Symmetric (TEM 00 ) Solutions Lecture 7- ECE 240a EM Review Maxwell’s Equations Material Properties Helmholtz Equation Dispersion Assume that ψ ( r ) = ψ ( r , φ , z ) has no φ dependence (axially symmetric) Beam Optics Then Helmholtz � � t ψ = 1 ∂ r ∂ψ − jk ∂ψ Equation ∇ 2 ∂z = 0 Paraxial r ∂r ∂r Solutions Fundamental Mode Guess solution � � �� Beam waist P ( z ) + kr 2 and radius ψ 0 = exp − j Axial Phase 2 q ( z ) Complete Solution Amplitude Factor Beam Divergence Axial Phase Radial Phase 14 ECE 240a Lasers - Fall 2019 Lecture 7
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