Appendix D: Vector Calculus for Electromagnetic Fields
“The theory of the electromagnetic field is the most successful physical theory we have. It describes precisely how electromagnetic fields propagate and interact with matter.”
Richard Feynman
Vector calculus provides the mathematical language of electromagnetism. While Maxwell’s equations can be written in component form, their true elegance and physical meaning emerge when expressed using vector notation. This mathematical framework doesn’t just simplify calculations—it reveals the fundamental geometric and topological properties of electromagnetic fields that govern all optical phenomena.
Vector calculus forms the mathematical foundation for understanding electromagnetic fields and their propagation through space. Far from being abstract mathematics, these tools provide direct insight into the physical behavior of light, revealing why electromagnetic waves can propagate through vacuum, how energy flows in optical systems, and why certain field configurations are stable while others are not. This appendix will guide you through the essential vector operations, building from basic concepts to Maxwell’s equations and their solutions for optical wave propagation.
1 D.1 Why Vector Calculus Is Essential for Optics ¶ 1.1 D.1.1 The Vector Nature of Electromagnetic Fields ¶ Tip: Unlike scalar quantities like temperature or pressure, electromagnetic fields have both magnitude and direction at each point in space. This directional information is crucial for understanding polarization, wave propagation, and field interactions.
Electromagnetic fields are fundamentally vector fields —quantities that have both magnitude and direction at every point in space. The electric field E ⃗ ( r ⃗ , t ) \vec{E}(\vec{r}, t) E ( r , t ) B ⃗ ( r ⃗ , t ) \vec{B}(\vec{r}, t) B ( r , t )
This vector nature is crucial for optics because:
Polarization : The direction of the electric field vector determines the polarization state
Energy flow : The direction of energy transport is given by the cross product E ⃗ × B ⃗ \vec{E} \times \vec{B} E × B
Wave propagation : The wave vector indicates the direction of phase velocity
Boundary conditions : Field components parallel and perpendicular to interfaces behave differently
1.2 D.1.2 Local vs. Global Properties ¶ Vector calculus allows us to connect local properties (what happens at a point) to global properties (what happens over regions or surfaces):
Table 1: Local-Global Connections
Local Property
Vector Operation
Global Property
Physical Meaning
Field variations
Gradient ∇ f \nabla f ∇ f
Potential differences
Force on charges
Field circulation
Curl ∇ × F ⃗ \nabla \times \vec{F} ∇ × F
Line integrals
Induced electric fields
Field divergence
Divergence ∇ ⋅ F ⃗ \nabla \cdot \vec{F} ∇ ⋅ F
Surface integrals
Source distributions
1.3 D.1.3 Maxwell’s Equations: The Foundation ¶ Maxwell’s equations in their vector form provide a complete description of electromagnetism:
∇ ⋅ E ⃗ = ρ ϵ 0 (Gauss’s law) \nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0} \quad \text{(Gauss's law)} ∇ ⋅ E = ϵ 0 ρ (Gauss’s law) ∇ ⋅ B ⃗ = 0 (No magnetic monopoles) \nabla \cdot \vec{B} = 0 \quad \text{(No magnetic monopoles)} ∇ ⋅ B = 0 (No magnetic monopoles) ∇ × E ⃗ = − ∂ B ⃗ ∂ t (Faraday’s law) \nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t} \quad \text{(Faraday's law)} ∇ × E = − ∂ t ∂ B (Faraday’s law) ∇ × B ⃗ = μ 0 J ⃗ + μ 0 ϵ 0 ∂ E ⃗ ∂ t (Amp e ˋ re-Maxwell law) \nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t} \quad \text{(Ampère-Maxwell law)} ∇ × B = μ 0 J + μ 0 ϵ 0 ∂ t ∂ E (Amp e ˋ re-Maxwell law) 2 D.2 Vector Fields and Coordinate Systems ¶ 2.1 D.2.1 Vector Field Representation ¶ A vector field assigns a vector to each point in space. In optics, we commonly encounter:
Electric field : E ⃗ ( r ⃗ , t ) = E x ( x , y , z , t ) x ^ + E y ( x , y , z , t ) y ^ + E z ( x , y , z , t ) z ^ \vec{E}(\vec{r}, t) = E_x(x,y,z,t)\hat{x} + E_y(x,y,z,t)\hat{y} + E_z(x,y,z,t)\hat{z} E ( r , t ) = E x ( x , y , z , t ) x ^ + E y ( x , y , z , t ) y ^ + E z ( x , y , z , t ) z ^
Magnetic field : B ⃗ ( r ⃗ , t ) = B x ( x , y , z , t ) x ^ + B y ( x , y , z , t ) y ^ + B z ( x , y , z , t ) z ^ \vec{B}(\vec{r}, t) = B_x(x,y,z,t)\hat{x} + B_y(x,y,z,t)\hat{y} + B_z(x,y,z,t)\hat{z} B ( r , t ) = B x ( x , y , z , t ) x ^ + B y ( x , y , z , t ) y ^ + B z ( x , y , z , t ) z ^
Poynting vector : S ⃗ = 1 μ 0 E ⃗ × B ⃗ \vec{S} = \frac{1}{\mu_0}\vec{E} \times \vec{B} S = μ 0 1 E × B
2.2 D.2.2 Coordinate Systems ¶ Cartesian coordinates (x, y, z):
Best for: Rectangular geometries, plane waves, layered media
Unit vectors: x ^ , y ^ , z ^ \hat{x}, \hat{y}, \hat{z} x ^ , y ^ , z ^
Position vector: r ⃗ = x x ^ + y y ^ + z z ^ \vec{r} = x\hat{x} + y\hat{y} + z\hat{z} r = x x ^ + y y ^ + z z ^
Cylindrical coordinates (ρ, φ, z):
Best for: Optical fibers, cylindrical lenses, beam propagation
Unit vectors: ρ ^ , ϕ ^ , z ^ \hat{\rho}, \hat{\phi}, \hat{z} ρ ^ , ϕ ^ , z ^ ρ ^ \hat{\rho} ρ ^ ϕ ^ \hat{\phi} ϕ ^
Position vector: r ⃗ = ρ ρ ^ + z z ^ \vec{r} = \rho\hat{\rho} + z\hat{z} r = ρ ρ ^ + z z ^
Spherical coordinates (r, θ, φ):
Best for: Point sources, spherical mirrors, far-field patterns
Unit vectors: r ^ , θ ^ , ϕ ^ \hat{r}, \hat{\theta}, \hat{\phi} r ^ , θ ^ , ϕ ^
Position vector: r ⃗ = r r ^ \vec{r} = r\hat{r} r = r r ^
A plane wave traveling in the z-direction:
Cartesian : E ⃗ = E 0 cos ( k z − ω t ) x ^ \vec{E} = E_0 \cos(kz - \omega t)\hat{x} E = E 0 cos ( k z − ω t ) x ^ Cylindrical : E ⃗ = E 0 cos ( k z − ω t ) x ^ \vec{E} = E_0 \cos(kz - \omega t)\hat{x} E = E 0 cos ( k z − ω t ) x ^ Spherical : More complex due to coordinate transformation, but reduces to Cartesian form for r ≫ λ r \gg \lambda r ≫ λ
When changing coordinate systems, both vector components and unit vectors transform:
Cartesian to cylindrical :
x ^ = cos ϕ ρ ^ − sin ϕ ϕ ^ \hat{x} = \cos\phi \hat{\rho} - \sin\phi \hat{\phi} x ^ = cos ϕ ρ ^ − sin ϕ ϕ ^
y ^ = sin ϕ ρ ^ + cos ϕ ϕ ^ \hat{y} = \sin\phi \hat{\rho} + \cos\phi \hat{\phi} y ^ = sin ϕ ρ ^ + cos ϕ ϕ ^
z ^ = z ^ \hat{z} = \hat{z} z ^ = z ^ Physical significance : A linearly polarized wave that’s purely x-polarized in Cartesian coordinates has both ρ and φ components in cylindrical coordinates that vary with azimuthal angle φ.
3 D.3 The Gradient: Measuring Field Variations ¶ 3.1 D.3.1 Definition and Physical Meaning ¶ The gradient of a scalar field f ( r ⃗ ) f(\vec{r}) f ( r )
∇ f = ∂ f ∂ x x ^ + ∂ f ∂ y y ^ + ∂ f ∂ z z ^ \nabla f = \frac{\partial f}{\partial x}\hat{x} + \frac{\partial f}{\partial y}\hat{y} + \frac{\partial f}{\partial z}\hat{z} ∇ f = ∂ x ∂ f x ^ + ∂ y ∂ f y ^ + ∂ z ∂ f z ^ 3.2 D.3.2 Gradient in Different Coordinate Systems ¶ Cartesian :
∇ f = ∂ f ∂ x x ^ + ∂ f ∂ y y ^ + ∂ f ∂ z z ^ \nabla f = \frac{\partial f}{\partial x}\hat{x} + \frac{\partial f}{\partial y}\hat{y} + \frac{\partial f}{\partial z}\hat{z} ∇ f = ∂ x ∂ f x ^ + ∂ y ∂ f y ^ + ∂ z ∂ f z ^ Cylindrical :
∇ f = ∂ f ∂ ρ ρ ^ + 1 ρ ∂ f ∂ ϕ ϕ ^ + ∂ f ∂ z z ^ \nabla f = \frac{\partial f}{\partial \rho}\hat{\rho} + \frac{1}{\rho}\frac{\partial f}{\partial \phi}\hat{\phi} + \frac{\partial f}{\partial z}\hat{z} ∇ f = ∂ ρ ∂ f ρ ^ + ρ 1 ∂ ϕ ∂ f ϕ ^ + ∂ z ∂ f z ^ Spherical :
∇ f = ∂ f ∂ r r ^ + 1 r ∂ f ∂ θ θ ^ + 1 r sin θ ∂ f ∂ ϕ ϕ ^ \nabla f = \frac{\partial f}{\partial r}\hat{r} + \frac{1}{r}\frac{\partial f}{\partial \theta}\hat{\theta} + \frac{1}{r\sin\theta}\frac{\partial f}{\partial \phi}\hat{\phi} ∇ f = ∂ r ∂ f r ^ + r 1 ∂ θ ∂ f θ ^ + r sin θ 1 ∂ ϕ ∂ f ϕ ^ 3.3 D.3.3 Applications in Optics ¶ Electric potential : If the electric field derives from a potential, E ⃗ = − ∇ V \vec{E} = -\nabla V E = − ∇ V
Refractive index gradients : In graded-index media, ∇ n \nabla n ∇ n
d d s ( n d r ⃗ d s ) = ∇ n \frac{d}{ds}\left(n\frac{d\vec{r}}{ds}\right) = \nabla n d s d ( n d s d r ) = ∇ n Phase gradients : The wave vector is the gradient of the phase:
k ⃗ = ∇ ϕ \vec{k} = \nabla \phi k = ∇ ϕ 4 D.4 The Divergence: Measuring Sources and Sinks ¶ 4.1 D.4.1 Definition and Physical Meaning ¶ The divergence of a vector field F ⃗ ( r ⃗ ) \vec{F}(\vec{r}) F ( r )
∇ ⋅ F ⃗ = ∂ F x ∂ x + ∂ F y ∂ y + ∂ F z ∂ z \nabla \cdot \vec{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} ∇ ⋅ F = ∂ x ∂ F x + ∂ y ∂ F y + ∂ z ∂ F z Important: Physical Interpretation : Divergence measures the “outflow” of the vector field from a point. Positive divergence indicates a source, negative divergence indicates a sink, and zero divergence indicates incompressible flow.
4.2 D.4.2 Divergence in Different Coordinate Systems ¶ Cartesian :
∇ ⋅ F ⃗ = ∂ F x ∂ x + ∂ F y ∂ y + ∂ F z ∂ z \nabla \cdot \vec{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} ∇ ⋅ F = ∂ x ∂ F x + ∂ y ∂ F y + ∂ z ∂ F z Cylindrical :
∇ ⋅ F ⃗ = 1 ρ ∂ ( ρ F ρ ) ∂ ρ + 1 ρ ∂ F ϕ ∂ ϕ + ∂ F z ∂ z \nabla \cdot \vec{F} = \frac{1}{\rho}\frac{\partial(\rho F_\rho)}{\partial \rho} + \frac{1}{\rho}\frac{\partial F_\phi}{\partial \phi} + \frac{\partial F_z}{\partial z} ∇ ⋅ F = ρ 1 ∂ ρ ∂ ( ρ F ρ ) + ρ 1 ∂ ϕ ∂ F ϕ + ∂ z ∂ F z Spherical :
∇ ⋅ F ⃗ = 1 r 2 ∂ ( r 2 F r ) ∂ r + 1 r sin θ ∂ ( sin θ F θ ) ∂ θ + 1 r sin θ ∂ F ϕ ∂ ϕ \nabla \cdot \vec{F} = \frac{1}{r^2}\frac{\partial(r^2 F_r)}{\partial r} + \frac{1}{r\sin\theta}\frac{\partial(\sin\theta F_\theta)}{\partial \theta} + \frac{1}{r\sin\theta}\frac{\partial F_\phi}{\partial \phi} ∇ ⋅ F = r 2 1 ∂ r ∂ ( r 2 F r ) + r sin θ 1 ∂ θ ∂ ( sin θ F θ ) + r sin θ 1 ∂ ϕ ∂ F ϕ 4.3 D.4.3 Gauss’s Divergence Theorem ¶ The divergence theorem connects local divergence to global flux:
∫ V ( ∇ ⋅ F ⃗ ) d V = ∮ S F ⃗ ⋅ d A ⃗ \int_V (\nabla \cdot \vec{F}) dV = \oint_S \vec{F} \cdot d\vec{A} ∫ V ( ∇ ⋅ F ) d V = ∮ S F ⋅ d A 4.4 D.4.4 Applications in Optics ¶ Gauss’s law for electricity :
∇ ⋅ E ⃗ = ρ ϵ 0 \nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0} ∇ ⋅ E = ϵ 0 ρ This tells us that electric field lines originate from positive charges and terminate on negative charges.
No magnetic monopoles :
∇ ⋅ B ⃗ = 0 \nabla \cdot \vec{B} = 0 ∇ ⋅ B = 0 Magnetic field lines always form closed loops—they have no beginning or end.
For a spherical wave E ⃗ = A r e i ( k r − ω t ) r ^ \vec{E} = \frac{A}{r}e^{i(kr - \omega t)}\hat{r} E = r A e i ( k r − ω t ) r ^
∇ ⋅ E ⃗ = 1 r 2 ∂ ∂ r ( r 2 A r e i ( k r − ω t ) ) = A r 2 ∂ ∂ r ( r e i ( k r − ω t ) ) \nabla \cdot \vec{E} = \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2 \frac{A}{r}e^{i(kr - \omega t)}\right) = \frac{A}{r^2}\frac{\partial}{\partial r}(re^{i(kr - \omega t)}) ∇ ⋅ E = r 2 1 ∂ r ∂ ( r 2 r A e i ( k r − ω t ) ) = r 2 A ∂ r ∂ ( r e i ( k r − ω t ) ) = A r 2 ( 1 + i k r ) e i ( k r − ω t ) = \frac{A}{r^2}(1 + ikr)e^{i(kr - \omega t)} = r 2 A ( 1 + ik r ) e i ( k r − ω t ) This is non-zero due to the point source at the origin. For a plane wave (no sources), the divergence would be exactly zero.
5 D.5 The Curl: Measuring Rotation and Circulation ¶ 5.1 D.5.1 Definition and Physical Meaning ¶ The curl of a vector field F ⃗ ( r ⃗ ) \vec{F}(\vec{r}) F ( r )
∇ × F ⃗ = ∣ x ^ y ^ z ^ ∂ ∂ x ∂ ∂ y ∂ ∂ z F x F y F z ∣ \nabla \times \vec{F} = \begin{vmatrix} \hat{x} & \hat{y} & \hat{z} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F_x & F_y & F_z \end{vmatrix} ∇ × F = ∣ ∣ x ^ ∂ x ∂ F x y ^ ∂ y ∂ F y z ^ ∂ z ∂ F z ∣ ∣ = ( ∂ F z ∂ y − ∂ F y ∂ z ) x ^ + ( ∂ F x ∂ z − ∂ F z ∂ x ) y ^ + ( ∂ F y ∂ x − ∂ F x ∂ y ) z ^ = \left(\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}\right)\hat{x} + \left(\frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}\right)\hat{y} + \left(\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}\right)\hat{z} = ( ∂ y ∂ F z − ∂ z ∂ F y ) x ^ + ( ∂ z ∂ F x − ∂ x ∂ F z ) y ^ + ( ∂ x ∂ F y − ∂ y ∂ F x ) z ^ 5.2 D.5.2 Curl in Different Coordinate Systems ¶ Cylindrical :
∇ × F ⃗ = ( 1 ρ ∂ F z ∂ ϕ − ∂ F ϕ ∂ z ) ρ ^ + ( ∂ F ρ ∂ z − ∂ F z ∂ ρ ) ϕ ^ + 1 ρ ( ∂ ( ρ F ϕ ) ∂ ρ − ∂ F ρ ∂ ϕ ) z ^ \nabla \times \vec{F} = \left(\frac{1}{\rho}\frac{\partial F_z}{\partial \phi} - \frac{\partial F_\phi}{\partial z}\right)\hat{\rho} + \left(\frac{\partial F_\rho}{\partial z} - \frac{\partial F_z}{\partial \rho}\right)\hat{\phi} + \frac{1}{\rho}\left(\frac{\partial(\rho F_\phi)}{\partial \rho} - \frac{\partial F_\rho}{\partial \phi}\right)\hat{z} ∇ × F = ( ρ 1 ∂ ϕ ∂ F z − ∂ z ∂ F ϕ ) ρ ^ + ( ∂ z ∂ F ρ − ∂ ρ ∂ F z ) ϕ ^ + ρ 1 ( ∂ ρ ∂ ( ρ F ϕ ) − ∂ ϕ ∂ F ρ ) z ^ Spherical :
∇ × F ⃗ = 1 r sin θ ( ∂ ( sin θ F ϕ ) ∂ θ − ∂ F θ ∂ ϕ ) r ^ + 1 r ( 1 sin θ ∂ F r ∂ ϕ − ∂ ( r F ϕ ) ∂ r ) θ ^ + 1 r ( ∂ ( r F θ ) ∂ r − ∂ F r ∂ θ ) ϕ ^ \nabla \times \vec{F} = \frac{1}{r\sin\theta}\left(\frac{\partial(\sin\theta F_\phi)}{\partial \theta} - \frac{\partial F_\theta}{\partial \phi}\right)\hat{r} + \frac{1}{r}\left(\frac{1}{\sin\theta}\frac{\partial F_r}{\partial \phi} - \frac{\partial(rF_\phi)}{\partial r}\right)\hat{\theta} + \frac{1}{r}\left(\frac{\partial(rF_\theta)}{\partial r} - \frac{\partial F_r}{\partial \theta}\right)\hat{\phi} ∇ × F = r sin θ 1 ( ∂ θ ∂ ( sin θ F ϕ ) − ∂ ϕ ∂ F θ ) r ^ + r 1 ( sin θ 1 ∂ ϕ ∂ F r − ∂ r ∂ ( r F ϕ ) ) θ ^ + r 1 ( ∂ r ∂ ( r F θ ) − ∂ θ ∂ F r ) ϕ ^ 5.3 D.5.3 Stokes’ Theorem ¶ Stokes’ theorem connects local curl to circulation around a closed path:
∫ S ( ∇ × F ⃗ ) ⋅ d A ⃗ = ∮ C F ⃗ ⋅ d l ⃗ \int_S (\nabla \times \vec{F}) \cdot d\vec{A} = \oint_C \vec{F} \cdot d\vec{l} ∫ S ( ∇ × F ) ⋅ d A = ∮ C F ⋅ d l 5.4 D.5.4 Applications in Optics ¶ Faraday’s law :
∇ × E ⃗ = − ∂ B ⃗ ∂ t \nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t} ∇ × E = − ∂ t ∂ B A time-varying magnetic field creates circulating electric fields (the principle behind transformers and betatrons).
Ampère-Maxwell law :
∇ × B ⃗ = μ 0 J ⃗ + μ 0 ϵ 0 ∂ E ⃗ ∂ t \nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t} ∇ × B = μ 0 J + μ 0 ϵ 0 ∂ t ∂ E Current and time-varying electric fields create circulating magnetic fields.
For a circularly polarized wave E ⃗ = E 0 ( x ^ + i y ^ ) e i ( k z − ω t ) \vec{E} = E_0(\hat{x} + i\hat{y})e^{i(kz - \omega t)} E = E 0 ( x ^ + i y ^ ) e i ( k z − ω t )
∇ × E ⃗ = ∂ E y ∂ z x ^ − ∂ E x ∂ z y ^ = i k E 0 ( − i x ^ + y ^ ) e i ( k z − ω t ) = i k E 0 ( − y ^ − i x ^ ) e i ( k z − ω t ) \nabla \times \vec{E} = \frac{\partial E_y}{\partial z}\hat{x} - \frac{\partial E_x}{\partial z}\hat{y} = ik E_0(-i\hat{x} + \hat{y})e^{i(kz - \omega t)} = ikE_0(-\hat{y} - i\hat{x})e^{i(kz - \omega t)} ∇ × E = ∂ z ∂ E y x ^ − ∂ z ∂ E x y ^ = ik E 0 ( − i x ^ + y ^ ) e i ( k z − ω t ) = ik E 0 ( − y ^ − i x ^ ) e i ( k z − ω t ) From Faraday’s law:
B ⃗ = − 1 i ω ∇ × E ⃗ = k ω E 0 ( − y ^ − i x ^ ) e i ( k z − ω t ) = E 0 c ( − y ^ − i x ^ ) e i ( k z − ω t ) \vec{B} = -\frac{1}{i\omega}\nabla \times \vec{E} = \frac{k}{\omega}E_0(-\hat{y} - i\hat{x})e^{i(kz - \omega t)} = \frac{E_0}{c}(-\hat{y} - i\hat{x})e^{i(kz - \omega t)} B = − iω 1 ∇ × E = ω k E 0 ( − y ^ − i x ^ ) e i ( k z − ω t ) = c E 0 ( − y ^ − i x ^ ) e i ( k z − ω t ) The magnetic field is perpendicular to the electric field and rotates with it, maintaining circular polarization.
6 D.6 The Laplacian: Wave Equations and Diffusion ¶ 6.1 D.6.1 Definition ¶ The Laplacian is the divergence of the gradient:
∇ 2 f = ∇ ⋅ ( ∇ f ) = ∂ 2 f ∂ x 2 + ∂ 2 f ∂ y 2 + ∂ 2 f ∂ z 2 \nabla^2 f = \nabla \cdot (\nabla f) = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2} ∇ 2 f = ∇ ⋅ ( ∇ f ) = ∂ x 2 ∂ 2 f + ∂ y 2 ∂ 2 f + ∂ z 2 ∂ 2 f For vector fields, the Laplacian operates on each component:
∇ 2 F ⃗ = ( ∇ 2 F x ) x ^ + ( ∇ 2 F y ) y ^ + ( ∇ 2 F z ) z ^ \nabla^2 \vec{F} = (\nabla^2 F_x)\hat{x} + (\nabla^2 F_y)\hat{y} + (\nabla^2 F_z)\hat{z} ∇ 2 F = ( ∇ 2 F x ) x ^ + ( ∇ 2 F y ) y ^ + ( ∇ 2 F z ) z ^ 6.2 D.6.2 Laplacian in Different Coordinate Systems ¶ Cartesian :
∇ 2 f = ∂ 2 f ∂ x 2 + ∂ 2 f ∂ y 2 + ∂ 2 f ∂ z 2 \nabla^2 f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2} ∇ 2 f = ∂ x 2 ∂ 2 f + ∂ y 2 ∂ 2 f + ∂ z 2 ∂ 2 f Cylindrical :
∇ 2 f = 1 ρ ∂ ∂ ρ ( ρ ∂ f ∂ ρ ) + 1 ρ 2 ∂ 2 f ∂ ϕ 2 + ∂ 2 f ∂ z 2 \nabla^2 f = \frac{1}{\rho}\frac{\partial}{\partial \rho}\left(\rho\frac{\partial f}{\partial \rho}\right) + \frac{1}{\rho^2}\frac{\partial^2 f}{\partial \phi^2} + \frac{\partial^2 f}{\partial z^2} ∇ 2 f = ρ 1 ∂ ρ ∂ ( ρ ∂ ρ ∂ f ) + ρ 2 1 ∂ ϕ 2 ∂ 2 f + ∂ z 2 ∂ 2 f Spherical :
∇ 2 f = 1 r 2 ∂ ∂ r ( r 2 ∂ f ∂ r ) + 1 r 2 sin θ ∂ ∂ θ ( sin θ ∂ f ∂ θ ) + 1 r 2 sin 2 θ ∂ 2 f ∂ ϕ 2 \nabla^2 f = \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial f}{\partial r}\right) + \frac{1}{r^2\sin\theta}\frac{\partial}{\partial \theta}\left(\sin\theta\frac{\partial f}{\partial \theta}\right) + \frac{1}{r^2\sin^2\theta}\frac{\partial^2 f}{\partial \phi^2} ∇ 2 f = r 2 1 ∂ r ∂ ( r 2 ∂ r ∂ f ) + r 2 sin θ 1 ∂ θ ∂ ( sin θ ∂ θ ∂ f ) + r 2 sin 2 θ 1 ∂ ϕ 2 ∂ 2 f 6.3 D.6.3 Applications in Optics ¶ Wave equation : In free space, electromagnetic fields satisfy:
∇ 2 E ⃗ − μ 0 ϵ 0 ∂ 2 E ⃗ ∂ t 2 = 0 \nabla^2 \vec{E} - \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2} = 0 ∇ 2 E − μ 0 ϵ 0 ∂ t 2 ∂ 2 E = 0
∇ 2 B ⃗ − μ 0 ϵ 0 ∂ 2 B ⃗ ∂ t 2 = 0 \nabla^2 \vec{B} - \mu_0 \epsilon_0 \frac{\partial^2 \vec{B}}{\partial t^2} = 0 ∇ 2 B − μ 0 ϵ 0 ∂ t 2 ∂ 2 B = 0 Helmholtz equation : For monochromatic waves, E ⃗ ( r ⃗ , t ) = E ⃗ ( r ⃗ ) e − i ω t \vec{E}(\vec{r},t) = \vec{E}(\vec{r})e^{-i\omega t} E ( r , t ) = E ( r ) e − iω t
∇ 2 E ⃗ + k 2 E ⃗ = 0 \nabla^2 \vec{E} + k^2 \vec{E} = 0 ∇ 2 E + k 2 E = 0 where k = ω μ 0 ϵ 0 = ω / c k = \omega\sqrt{\mu_0\epsilon_0} = \omega/c k = ω μ 0 ϵ 0 = ω / c
The Gaussian beam is a solution to the paraxial wave equation (approximation to Helmholtz equation):
E ⃗ ( x , y , z ) = E 0 w 0 w ( z ) exp ( − x 2 + y 2 w 2 ( z ) ) exp ( i k z − i k x 2 + y 2 2 R ( z ) + i ζ ( z ) ) \vec{E}(x,y,z) = E_0 \frac{w_0}{w(z)} \exp\left(-\frac{x^2 + y^2}{w^2(z)}\right) \exp\left(ikz - ik\frac{x^2 + y^2}{2R(z)} + i\zeta(z)\right) E ( x , y , z ) = E 0 w ( z ) w 0 exp ( − w 2 ( z ) x 2 + y 2 ) exp ( ik z − ik 2 R ( z ) x 2 + y 2 + i ζ ( z ) ) where:
w ( z ) = w 0 1 + ( z / z R ) 2 w(z) = w_0\sqrt{1 + (z/z_R)^2} w ( z ) = w 0 1 + ( z / z R ) 2
R ( z ) = z [ 1 + ( z R / z ) 2 ] R(z) = z[1 + (z_R/z)^2] R ( z ) = z [ 1 + ( z R / z ) 2 ]
z R = π w 0 2 / λ z_R = \pi w_0^2/\lambda z R = π w 0 2 / λ
ζ ( z ) = arctan ( z / z R ) \zeta(z) = \arctan(z/z_R) ζ ( z ) = arctan ( z / z R )
This solution satisfies the paraxial Helmholtz equation and represents the fundamental mode of laser cavities.
7 D.7 Vector Identities and Relationships ¶ 7.1 D.7.1 Fundamental Vector Identities ¶ These identities are essential for manipulating Maxwell’s equations:
Table 2: Essential Vector Identities
Identity
Mathematical Form
Physical Significance
Divergence of curl
∇ ⋅ ( ∇ × F ⃗ ) = 0 \nabla \cdot (\nabla \times \vec{F}) = 0 ∇ ⋅ ( ∇ × F ) = 0
Magnetic field has no divergence
Curl of gradient
∇ × ( ∇ f ) = 0 \nabla \times (\nabla f) = 0 ∇ × ( ∇ f ) = 0
Conservative fields
Vector Laplacian
∇ 2 F ⃗ = ∇ ( ∇ ⋅ F ⃗ ) − ∇ × ( ∇ × F ⃗ ) \nabla^2 \vec{F} = \nabla(\nabla \cdot \vec{F}) - \nabla \times (\nabla \times \vec{F}) ∇ 2 F = ∇ ( ∇ ⋅ F ) − ∇ × ( ∇ × F )
Separates longitudinal and transverse parts
Product rules
∇ ⋅ ( f F ⃗ ) = f ( ∇ ⋅ F ⃗ ) + F ⃗ ⋅ ( ∇ f ) \nabla \cdot (f\vec{F}) = f(\nabla \cdot \vec{F}) + \vec{F} \cdot (\nabla f) ∇ ⋅ ( f F ) = f ( ∇ ⋅ F ) + F ⋅ ( ∇ f )
Field interactions with media
Taking the curl of Faraday’s law:
∇ × ( ∇ × E ⃗ ) = − ∇ × ∂ B ⃗ ∂ t = − ∂ ∂ t ( ∇ × B ⃗ ) \nabla \times (\nabla \times \vec{E}) = -\nabla \times \frac{\partial \vec{B}}{\partial t} = -\frac{\partial}{\partial t}(\nabla \times \vec{B}) ∇ × ( ∇ × E ) = − ∇ × ∂ t ∂ B = − ∂ t ∂ ( ∇ × B ) Substituting Ampère’s law (in vacuum, J ⃗ = 0 \vec{J} = 0 J = 0
∇ × ( ∇ × E ⃗ ) = − μ 0 ϵ 0 ∂ 2 E ⃗ ∂ t 2 \nabla \times (\nabla \times \vec{E}) = -\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2} ∇ × ( ∇ × E ) = − μ 0 ϵ 0 ∂ t 2 ∂ 2 E Using the vector identity:
∇ ( ∇ ⋅ E ⃗ ) − ∇ 2 E ⃗ = − μ 0 ϵ 0 ∂ 2 E ⃗ ∂ t 2 \nabla(\nabla \cdot \vec{E}) - \nabla^2 \vec{E} = -\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2} ∇ ( ∇ ⋅ E ) − ∇ 2 E = − μ 0 ϵ 0 ∂ t 2 ∂ 2 E In charge-free regions, ∇ ⋅ E ⃗ = 0 \nabla \cdot \vec{E} = 0 ∇ ⋅ E = 0 wave equation :
∇ 2 E ⃗ = μ 0 ϵ 0 ∂ 2 E ⃗ ∂ t 2 \nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2} ∇ 2 E = μ 0 ϵ 0 ∂ t 2 ∂ 2 E 7.3 D.7.3 Boundary Conditions ¶ At interfaces between different media, Maxwell’s equations impose boundary conditions:
Tangential electric field : n ^ × ( E ⃗ 2 − E ⃗ 1 ) = 0 \hat{n} \times (\vec{E}_2 - \vec{E}_1) = 0 n ^ × ( E 2 − E 1 ) = 0 Normal electric displacement : n ^ ⋅ ( D ⃗ 2 − D ⃗ 1 ) = σ f \hat{n} \cdot (\vec{D}_2 - \vec{D}_1) = \sigma_f n ^ ⋅ ( D 2 − D 1 ) = σ f Tangential magnetic field : n ^ × ( B ⃗ 2 − B ⃗ 1 ) = μ 0 K ⃗ f \hat{n} \times (\vec{B}_2 - \vec{B}_1) = \mu_0 \vec{K}_f n ^ × ( B 2 − B 1 ) = μ 0 K f Normal magnetic field : n ^ ⋅ ( B ⃗ 2 − B ⃗ 1 ) = 0 \hat{n} \cdot (\vec{B}_2 - \vec{B}_1) = 0 n ^ ⋅ ( B 2 − B 1 ) = 0
where σ f \sigma_f σ f K ⃗ f \vec{K}_f K f
8 D.8 Applications to Wave Propagation ¶ 8.1 D.8.1 Plane Wave Solutions ¶ For a plane wave E ⃗ = E ⃗ 0 e i ( k ⃗ ⋅ r ⃗ − ω t ) \vec{E} = \vec{E}_0 e^{i(\vec{k} \cdot \vec{r} - \omega t)} E = E 0 e i ( k ⋅ r − ω t )
Divergence : ∇ ⋅ E ⃗ = i k ⃗ ⋅ E ⃗ = 0 \nabla \cdot \vec{E} = i\vec{k} \cdot \vec{E} = 0 ∇ ⋅ E = i k ⋅ E = 0 transverse : k ⃗ ⊥ E ⃗ \vec{k} \perp \vec{E} k ⊥ E
Curl : ∇ × E ⃗ = i k ⃗ × E ⃗ \nabla \times \vec{E} = i\vec{k} \times \vec{E} ∇ × E = i k × E i k ⃗ × E ⃗ = i ω B ⃗ i\vec{k} \times \vec{E} = i\omega \vec{B} i k × E = iω B B ⃗ = k ⃗ × E ⃗ ω = k ⃗ × E ⃗ c ∣ k ⃗ ∣ \vec{B} = \frac{\vec{k} \times \vec{E}}{\omega} = \frac{\vec{k} \times \vec{E}}{c|\vec{k}|} B = ω k × E = c ∣ k ∣ k × E
Key results :
E ⃗ ⊥ B ⃗ \vec{E} \perp \vec{B} E ⊥ B
E ⃗ ⊥ k ⃗ \vec{E} \perp \vec{k} E ⊥ k B ⃗ ⊥ k ⃗ \vec{B} \perp \vec{k} B ⊥ k
∣ B ⃗ ∣ = ∣ E ⃗ ∣ / c |\vec{B}| = |\vec{E}|/c ∣ B ∣ = ∣ E ∣/ c
k ⃗ , E ⃗ , B ⃗ \vec{k}, \vec{E}, \vec{B} k , E , B
8.2 D.8.2 Energy and Momentum ¶ Poynting vector (energy flow):
S ⃗ = 1 μ 0 E ⃗ × B ⃗ \vec{S} = \frac{1}{\mu_0}\vec{E} \times \vec{B} S = μ 0 1 E × B For a plane wave: ∣ S ⃗ ∣ = ∣ E ⃗ ∣ 2 μ 0 c = ∣ E ⃗ ∣ 2 Z 0 |\vec{S}| = \frac{|\vec{E}|^2}{\mu_0 c} = \frac{|\vec{E}|^2}{Z_0} ∣ S ∣ = μ 0 c ∣ E ∣ 2 = Z 0 ∣ E ∣ 2
where Z 0 = μ 0 / ϵ 0 = 377 Ω Z_0 = \sqrt{\mu_0/\epsilon_0} = 377\,\Omega Z 0 = μ 0 / ϵ 0 = 377 Ω
Electromagnetic momentum density :
g ⃗ = ϵ 0 E ⃗ × B ⃗ = S ⃗ c 2 \vec{g} = \epsilon_0 \vec{E} \times \vec{B} = \frac{\vec{S}}{c^2} g = ϵ 0 E × B = c 2 S 8.3 D.8.3 Spherical Waves ¶ For spherical waves emanating from a point source:
E ⃗ ( r , θ , ϕ ) = f ( θ , ϕ ) r e i ( k r − ω t ) θ ^ + g ( θ , ϕ ) r e i ( k r − ω t ) ϕ ^ \vec{E}(r,\theta,\phi) = \frac{f(\theta,\phi)}{r} e^{i(kr - \omega t)} \hat{\theta} + \frac{g(\theta,\phi)}{r} e^{i(kr - \omega t)} \hat{\phi} E ( r , θ , ϕ ) = r f ( θ , ϕ ) e i ( k r − ω t ) θ ^ + r g ( θ , ϕ ) e i ( k r − ω t ) ϕ ^ The radial component vanishes in the far field due to the transversality condition ∇ ⋅ E ⃗ = 0 \nabla \cdot \vec{E} = 0 ∇ ⋅ E = 0
An oscillating electric dipole p ⃗ ( t ) = p 0 cos ( ω t ) z ^ \vec{p}(t) = p_0 \cos(\omega t)\hat{z} p ( t ) = p 0 cos ( ω t ) z ^
E ⃗ ( r , θ , t ) = μ 0 ω 2 p 0 sin θ 4 π r cos ( ω t − k r + ϕ ) θ ^ \vec{E}(r,\theta,t) = \frac{\mu_0 \omega^2 p_0 \sin\theta}{4\pi r} \cos(\omega t - kr + \phi) \hat{\theta} E ( r , θ , t ) = 4 π r μ 0 ω 2 p 0 sin θ cos ( ω t − k r + ϕ ) θ ^ B ⃗ ( r , θ , t ) = μ 0 ω 2 p 0 sin θ 4 π c r cos ( ω t − k r + ϕ ) ϕ ^ \vec{B}(r,\theta,t) = \frac{\mu_0 \omega^2 p_0 \sin\theta}{4\pi c r} \cos(\omega t - kr + \phi) \hat{\phi} B ( r , θ , t ) = 4 π cr μ 0 ω 2 p 0 sin θ cos ( ω t − k r + ϕ ) ϕ ^ The radiated power pattern is: P ( θ ) ∝ sin 2 θ P(\theta) \propto \sin^2\theta P ( θ ) ∝ sin 2 θ
This shows why dipole antennas have directional radiation patterns and why Rayleigh scattering depends on sin 2 θ \sin^2\theta sin 2 θ
9 D.9 Applications to Guided Waves ¶ 9.1 D.9.1 Cylindrical Waveguides ¶ For waves propagating in the z-direction with cylindrical symmetry, we separate variables:
E ⃗ ( ρ , ϕ , z , t ) = E ⃗ ( ρ , ϕ ) e i ( β z − ω t ) \vec{E}(\rho,\phi,z,t) = \vec{E}(\rho,\phi) e^{i(\beta z - \omega t)} E ( ρ , ϕ , z , t ) = E ( ρ , ϕ ) e i ( β z − ω t ) The wave equation in cylindrical coordinates becomes:
1 ρ ∂ ∂ ρ ( ρ ∂ E z ∂ ρ ) + 1 ρ 2 ∂ 2 E z ∂ ϕ 2 + γ 2 E z = 0 \frac{1}{\rho}\frac{\partial}{\partial \rho}\left(\rho\frac{\partial E_z}{\partial \rho}\right) + \frac{1}{\rho^2}\frac{\partial^2 E_z}{\partial \phi^2} + \gamma^2 E_z = 0 ρ 1 ∂ ρ ∂ ( ρ ∂ ρ ∂ E z ) + ρ 2 1 ∂ ϕ 2 ∂ 2 E z + γ 2 E z = 0 where γ 2 = k 2 − β 2 \gamma^2 = k^2 - \beta^2 γ 2 = k 2 − β 2 β \beta β
Solutions : Bessel functions for the radial dependence, sinusoidal functions for the azimuthal dependence.
9.2 D.9.2 Optical Fibers ¶ In step-index optical fibers, we have:
Weakly guiding approximation : n 1 − n 2 ≪ n 1 n_1 - n_2 \ll n_1 n 1 − n 2 ≪ n 1
The fundamental mode (LP₀₁) has approximately Gaussian transverse profile:
E ( ρ ) ∝ e − ρ 2 / w 2 E(\rho) \propto e^{-\rho^2/w^2} E ( ρ ) ∝ e − ρ 2 / w 2 where w w w
V = 2 π a λ n 1 2 − n 2 2 V = \frac{2\pi a}{\lambda}\sqrt{n_1^2 - n_2^2} V = λ 2 πa n 1 2 − n 2 2 Single-mode condition : V < 2.405 V < 2.405 V < 2.405
9.3 D.9.3 Slab Waveguides ¶ For a symmetric slab waveguide (thickness 2 a 2a 2 a n 1 n_1 n 1 n 2 n_2 n 2
TE modes (E z = 0 E_z = 0 E z = 0
tan ( k a ) = γ k (symmetric modes) \tan(ka) = \frac{\gamma}{k} \quad \text{(symmetric modes)} tan ( ka ) = k γ (symmetric modes)
cot ( k a ) = − γ k (antisymmetric modes) \cot(ka) = -\frac{\gamma}{k} \quad \text{(antisymmetric modes)} cot ( ka ) = − k γ (antisymmetric modes) where k 2 = ω 2 n 1 2 / c 2 − β 2 k^2 = \omega^2 n_1^2/c^2 - \beta^2 k 2 = ω 2 n 1 2 / c 2 − β 2 γ 2 = β 2 − ω 2 n 2 2 / c 2 \gamma^2 = \beta^2 - \omega^2 n_2^2/c^2 γ 2 = β 2 − ω 2 n 2 2 / c 2
10 D.10 Advanced Applications ¶ For subwavelength structures, we can define effective permittivity and permeability tensors:
D ⃗ = ϵ ↔ ⋅ E ⃗ , B ⃗ = μ ↔ ⋅ H ⃗ \vec{D} = \overleftrightarrow{\epsilon} \cdot \vec{E}, \quad \vec{B} = \overleftrightarrow{\mu} \cdot \vec{H} D = ϵ ⋅ E , B = μ ⋅ H Wire arrays : Effective plasma frequency
ω p 2 = 2 π c 2 a 2 ln ( a / r ) \omega_p^2 = \frac{2\pi c^2}{a^2 \ln(a/r)} ω p 2 = a 2 ln ( a / r ) 2 π c 2 Split-ring resonators : Magnetic resonance at
ω m = 1 L C \omega_m = \frac{1}{\sqrt{LC}} ω m = L C 1 10.2 D.10.2 Nonlinear Optics ¶ In nonlinear media, the polarization depends nonlinearly on the field:
P ⃗ = ϵ 0 χ ( 1 ) E ⃗ + ϵ 0 χ ( 2 ) E ⃗ E ⃗ + ϵ 0 χ ( 3 ) E ⃗ E ⃗ E ⃗ + ⋯ \vec{P} = \epsilon_0 \chi^{(1)} \vec{E} + \epsilon_0 \chi^{(2)} \vec{E}\vec{E} + \epsilon_0 \chi^{(3)} \vec{E}\vec{E}\vec{E} + \cdots P = ϵ 0 χ ( 1 ) E + ϵ 0 χ ( 2 ) E E + ϵ 0 χ ( 3 ) E E E + ⋯ This leads to coupled wave equations:
∇ 2 E ⃗ − μ 0 ϵ 0 ∂ 2 E ⃗ ∂ t 2 = μ 0 ∂ 2 P ⃗ N L ∂ t 2 \nabla^2 \vec{E} - \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2} = \mu_0 \frac{\partial^2 \vec{P}_{NL}}{\partial t^2} ∇ 2 E − μ 0 ϵ 0 ∂ t 2 ∂ 2 E = μ 0 ∂ t 2 ∂ 2 P N L 10.3 D.10.3 Quantum Optics ¶ The electromagnetic field operators satisfy commutation relations:
[ E ^ i ( r ⃗ ) , E ^ j ( r ⃗ ′ ) ] = 0 [\hat{E}_i(\vec{r}), \hat{E}_j(\vec{r}')] = 0 [ E ^ i ( r ) , E ^ j ( r ′ )] = 0
[ E ^ i ( r ⃗ ) , B ^ j ( r ⃗ ′ ) ] = i ℏ c ϵ i j k ∇ k δ 3 ( r ⃗ − r ⃗ ′ ) [\hat{E}_i(\vec{r}), \hat{B}_j(\vec{r}')] = i\hbar c \epsilon_{ijk} \nabla_k \delta^3(\vec{r} - \vec{r}') [ E ^ i ( r ) , B ^ j ( r ′ )] = i ℏ c ϵ ijk ∇ k δ 3 ( r − r ′ ) These lead to uncertainty principles and quantum effects like photon antibunching and squeezed light.
11 D.11 Computational Methods ¶ 11.1 D.11.1 Finite Difference Methods ¶ Discretize Maxwell’s equations on a grid:
∂ E x ∂ t = 1 ϵ ( H z ( y + Δ y ) − H z ( y ) Δ y − H y ( z + Δ z ) − H y ( z ) Δ z ) \frac{\partial E_x}{\partial t} = \frac{1}{\epsilon}\left(\frac{H_z(y+\Delta y) - H_z(y)}{\Delta y} - \frac{H_y(z+\Delta z) - H_y(z)}{\Delta z}\right) ∂ t ∂ E x = ϵ 1 ( Δ y H z ( y + Δ y ) − H z ( y ) − Δ z H y ( z + Δ z ) − H y ( z ) ) Yee algorithm : Stagger electric and magnetic field components in space and time for stability.
11.2 D.11.2 Finite Element Methods ¶ Expand fields in basis functions:
E ⃗ ( r ⃗ ) = ∑ i E i N ⃗ i ( r ⃗ ) \vec{E}(\vec{r}) = \sum_i E_i \vec{N}_i(\vec{r}) E ( r ) = i ∑ E i N i ( r ) Substitute into wave equation and apply Galerkin method to obtain matrix equation:
[ K ] { E } = ω 2 [ M ] { E } [K]\{E\} = \omega^2[M]\{E\} [ K ] { E } = ω 2 [ M ] { E } 11.3 D.11.3 Method of Moments ¶ For scattering problems, solve the integral equation:
E ⃗ i n c ( r ⃗ ) = − ∫ G ↔ ( r ⃗ , r ⃗ ′ ) ⋅ J ⃗ ( r ⃗ ′ ) d 3 r ′ \vec{E}^{inc}(\vec{r}) = -\int \overleftrightarrow{G}(\vec{r},\vec{r}') \cdot \vec{J}(\vec{r}') d^3r' E in c ( r ) = − ∫ G ( r , r ′ ) ⋅ J ( r ′ ) d 3 r ′ where G ↔ \overleftrightarrow{G} G
12 D.12 Summary and Physical Insights ¶ 12.1 D.12.1 Fundamental Principles ¶ Vector calculus reveals the deep structure of electromagnetism:
Table 3: Core Physical Insights
Vector Operation
Maxwell Equation
Physical Principle
Divergence
∇ ⋅ E ⃗ = ρ / ϵ 0 \nabla \cdot \vec{E} = \rho/\epsilon_0 ∇ ⋅ E = ρ / ϵ 0
Electric charges are sources of field
Curl
∇ × E ⃗ = − ∂ B ⃗ / ∂ t \nabla \times \vec{E} = -\partial\vec{B}/\partial t ∇ × E = − ∂ B / ∂ t
Changing magnetic field induces electric field
Wave equation
∇ 2 E ⃗ = μ 0 ϵ 0 ∂ 2 E ⃗ / ∂ t 2 \nabla^2 \vec{E} = \mu_0\epsilon_0 \partial^2\vec{E}/\partial t^2 ∇ 2 E = μ 0 ϵ 0 ∂ 2 E / ∂ t 2
Fields propagate as waves
Transversality
k ⃗ ⋅ E ⃗ = 0 \vec{k} \cdot \vec{E} = 0 k ⋅ E = 0
EM waves are transverse
12.2 D.12.2 Symmetries and Conservation Laws ¶ Gauge invariance : Physics is unchanged by gauge transformations
A ⃗ → A ⃗ + ∇ Λ , ϕ → ϕ − ∂ Λ ∂ t \vec{A} \rightarrow \vec{A} + \nabla \Lambda, \quad \phi \rightarrow \phi - \frac{\partial \Lambda}{\partial t} A → A + ∇Λ , ϕ → ϕ − ∂ t ∂ Λ Energy conservation : From Poynting’s theorem
∂ u ∂ t + ∇ ⋅ S ⃗ = − J ⃗ ⋅ E ⃗ \frac{\partial u}{\partial t} + \nabla \cdot \vec{S} = -\vec{J} \cdot \vec{E} ∂ t ∂ u + ∇ ⋅ S = − J ⋅ E Momentum conservation : Electromagnetic momentum density g ⃗ = ϵ 0 E ⃗ × B ⃗ \vec{g} = \epsilon_0 \vec{E} \times \vec{B} g = ϵ 0 E × B
12.3 D.12.3 Connection to Other Physics ¶ Relativity : Maxwell’s equations are Lorentz covariant—they have the same form in all inertial frames.
Quantum mechanics : The vector potential appears in the covariant derivative and leads to the Aharonov-Bohm effect.
Condensed matter : Effective medium theories use vector calculus to relate microscopic and macroscopic properties.
13 D.13 Worked Examples ¶ 13.1 D.13.1 Rectangular Waveguide Analysis ¶ Find the field components for the TE₁₀ mode in a rectangular waveguide with dimensions a × b a \times b a × b
Solution:
For TE modes, E z = 0 E_z = 0 E z = 0 H z H_z H z
Boundary conditions : E x = E y = 0 E_x = E_y = 0 E x = E y = 0
Ansatz : H z ( x , y ) = H 0 cos ( π x a ) e i ( β z − ω t ) H_z(x,y) = H_0 \cos\left(\frac{\pi x}{a}\right) e^{i(\beta z - \omega t)} H z ( x , y ) = H 0 cos ( a π x ) e i ( β z − ω t )
From Maxwell’s equations :
E x = i β γ 2 ∂ H z ∂ y = 0 E_x = \frac{i\beta}{\gamma^2} \frac{\partial H_z}{\partial y} = 0 E x = γ 2 i β ∂ y ∂ H z = 0
E y = − i β γ 2 ∂ H z ∂ x = i β π H 0 γ 2 a sin ( π x a ) e i ( β z − ω t ) E_y = -\frac{i\beta}{\gamma^2} \frac{\partial H_z}{\partial x} = \frac{i\beta \pi H_0}{\gamma^2 a} \sin\left(\frac{\pi x}{a}\right) e^{i(\beta z - \omega t)} E y = − γ 2 i β ∂ x ∂ H z = γ 2 a i β π H 0 sin ( a π x ) e i ( β z − ω t ) Dispersion relation : γ 2 = ( π / a ) 2 \gamma^2 = (\pi/a)^2 γ 2 = ( π / a ) 2
β 2 = ω 2 c 2 − π 2 a 2 \beta^2 = \frac{\omega^2}{c^2} - \frac{\pi^2}{a^2} β 2 = c 2 ω 2 − a 2 π 2 Cutoff frequency : ω c = π c / a \omega_c = \pi c/a ω c = π c / a
For ω > ω c \omega > \omega_c ω > ω c ω < ω c \omega < \omega_c ω < ω c
13.2 D.13.2 Electromagnetic Wave Scattering ¶ Analyze the scattering of a plane wave by a dielectric sphere using vector spherical harmonics.
Solution:
Incident wave : E ⃗ i n c = E 0 e i k z x ^ \vec{E}^{inc} = E_0 e^{ikz} \hat{x} E in c = E 0 e ik z x ^
Expand in spherical harmonics : The incident, scattered, and internal fields are expanded as:
E ⃗ = ∑ l , m [ a l m M ⃗ l m ( 1 ) + b l m N ⃗ l m ( 1 ) ] \vec{E} = \sum_{l,m} [a_{lm} \vec{M}_{lm}^{(1)} + b_{lm} \vec{N}_{lm}^{(1)}] E = l , m ∑ [ a l m M l m ( 1 ) + b l m N l m ( 1 ) ] where M ⃗ \vec{M} M N ⃗ \vec{N} N
Boundary conditions : At r = R r = R r = R
Mie coefficients : These conditions determine the expansion coefficients a l a_l a l b l b_l b l
Scattering cross-section :
σ s c a t = 2 π k 2 ∑ l = 1 ∞ ( 2 l + 1 ) ( ∣ a l ∣ 2 + ∣ b l ∣ 2 ) \sigma_{scat} = \frac{2\pi}{k^2} \sum_{l=1}^{\infty} (2l+1)(|a_l|^2 + |b_l|^2) σ sc a t = k 2 2 π l = 1 ∑ ∞ ( 2 l + 1 ) ( ∣ a l ∣ 2 + ∣ b l ∣ 2 ) Rayleigh limit (k R ≪ 1 kR \ll 1 k R ≪ 1 σ s c a t ∝ ( k R ) 4 ∝ λ − 4 \sigma_{scat} \propto (kR)^4 \propto \lambda^{-4} σ sc a t ∝ ( k R ) 4 ∝ λ − 4
13.3 D.13.3 Gaussian Beam Propagation ¶ A Gaussian beam with waist w 0 w_0 w 0 f f f
Solution:
ABCD matrix for lens : ( 1 0 − 1 / f 1 ) \begin{pmatrix} 1 & 0 \\ -1/f & 1 \end{pmatrix} ( 1 − 1/ f 0 1 )
Gaussian beam parameter : q = z + i z R q = z + iz_R q = z + i z R z R = π w 0 2 / λ z_R = \pi w_0^2/\lambda z R = π w 0 2 / λ
Transformation rule : q o u t = A q i n + B C q i n + D q_{out} = \frac{Aq_{in} + B}{Cq_{in} + D} q o u t = C q in + D A q in + B
For thin lens : q o u t = q i n − q i n / f + 1 q_{out} = \frac{q_{in}}{-q_{in}/f + 1} q o u t = − q in / f + 1 q in
If beam waist is at lens (z = 0 z = 0 z = 0 q i n = i z R q_{in} = iz_R q in = i z R
q o u t = i z R − i z R / f + 1 = i z R f f − i z R q_{out} = \frac{iz_R}{-iz_R/f + 1} = \frac{iz_R f}{f - iz_R} q o u t = − i z R / f + 1 i z R = f − i z R i z R f Output waist location : z o u t = Re ( q o u t ) = z R 2 f f 2 + z R 2 z_{out} = \text{Re}(q_{out}) = \frac{z_R^2 f}{f^2 + z_R^2} z o u t = Re ( q o u t ) = f 2 + z R 2 z R 2 f
Output waist size : w o u t = w 0 f f 2 + z R 2 w_{out} = w_0 \frac{f}{\sqrt{f^2 + z_R^2}} w o u t = w 0 f 2 + z R 2 f
Minimum waist occurs when f = z R f = z_R f = z R w m i n = w 0 / 2 w_{min} = w_0/\sqrt{2} w min = w 0 / 2
14 D.14 Practice Problems ¶ 14.1 D.14.1 Vector Operations ¶ 14.2 D.14.2 Electromagnetic Waves ¶ 14.3 D.14.3 Waveguide Modes ¶ For a step-index optical fiber with core radius a = 5 a = 5 a = 5 n 1 = 1.46 n_1 = 1.46 n 1 = 1.46 n 2 = 1.45 n_2 = 1.45 n 2 = 1.45
14.4 D.14.4 Boundary Conditions ¶ 15 D.15 Final Thoughts: The Mathematical Beauty of Electromagnetism ¶ Vector calculus provides more than computational tools—it reveals the geometric and topological structure underlying electromagnetic phenomena. The elegance of Maxwell’s equations in vector form reflects deep symmetries in space and time that govern how information and energy propagate through the universe.
“Maxwell’s equations are the most beautiful equations in physics. They connect electricity, magnetism, and light in a unified theory that has stood unchanged for over 150 years.”
Freeman Dyson
The mathematical framework we’ve explored connects to fundamental concepts throughout physics:
Gauge theory : The vector potential and gauge transformations appear in particle physics
Relativity : Maxwell’s equations are the same in all inertial frames
Quantum field theory : Electromagnetic fields become quantum operators
Topology : Field configurations can have topological properties (monopoles, skyrmions)
Developing Vector Intuition : The key to mastering vector calculus in optics is developing geometric intuition:
Gradient : Points “uphill,” perpendicular to level surfaces
Divergence : Measures “spreading out” from a point
Curl : Measures “twisting” around an axis
Laplacian : Measures how much a point differs from its neighbors
Practice visualizing these operations geometrically, and the mathematical formalism will become a natural language for describing electromagnetic phenomena.
As you continue studying optics, remember that every optical phenomenon—from simple refraction to complex laser dynamics—emerges from the vector calculus operations we’ve explored. Master these mathematical tools, and you’ll have the foundation for understanding how light behaves in any situation, from the cosmic to the quantum scale.
Tip: Connecting Mathematics to Physics : Always ask these questions when working with vector calculus:
What does this gradient represent physically?
Why is the divergence zero (or non-zero) here?
What physical process creates this curl?
How do boundary conditions reflect the underlying physics?
The mathematics is not separate from the physics—it is the physics, expressed in its most precise and universal form.
Vector calculus stands as one of the greatest achievements in mathematical physics, providing a unified language for describing electromagnetic phenomena across all scales and applications. Master these concepts, and you’ll have one of the most powerful tools in all of science at your command.