Five boundary conditions at an interface

Boundaries, Near-field Optics Boundaries, Near-field Optics

Five boundary conditions at an interface

Fresnel Equations : Transmission and Reflection Coefficients Transmittance and Reflectance

Brewster’s condition Æ a consequence of Impedance matching

Evanescent Waves

BOUNDARY CONDITIONS BOUNDARY CONDITIONS

1. The frequency of light does not change across the border,

unless one of the media happens to be a nonlinear medium. In

nonlinear media higher harmonics are generated.

2. The wavelength either expands or contracts, according to the ratio of the indices of refraction:

BOUNDARY CONDITIONS BOUNDARY CONDITIONS

3. Speaking in terms of electromagnetic theory, the ratio of light energy,

contained in the form of electric energy, to that in the form of magnetic energy is changed at the boundary depending on the intrinsic impedance of the medium.

Energy density stored in an electric field Energy density stored in a magnetic field

(

)

i i

E E ε ε ε

u = i = 2

1 ₂

, ε

(

)

i i

H Hⁱ

u = μ μ = μ μ 2

1 ₂

,

o i i

i i

i

i H n

E η

ε

η ≡ = μ = ¹ Intrinsic impedance of the medium i :

i i i

H i E

i u

u u

μ

= ε

=

, ,

(vacuum) ≡ =120π(Ω) =377Ω ε

η μ

o o o

Index of refraction : n_i = ε_i /ε_o = ε_ri

i i

vi

μ ε

= 1

i i

i H

B = μ

i i

i v

B = E

i o ri

o ri o i

i i

i i i

i i i i i

i i

i i

i

i v n

B E H

E H

E η

ε ε

μ μ ε

μ μ

ε μ μ

μ μ μ

η = = = = = = = =

The impedance η_i has to be changed as soon as the light crosses the boundary.

o

i μ

μ =

BOUNDARY CONDITIONS BOUNDARY CONDITIONS

5. The normal components of D and B are continuous,

where D and B are the electric and magnetic flux densities.

4. The tangential components of E and H are continuous across the boundary.

TM wave

xm xd

E = E H _ym = H _yd ε_mE_zm = ε_d E_zd

• At the boundary (continuity of the tangential E_x, H_y, and the normal D_z):

Remind that the boundary condition at the metal-dielectric interface!

(Maxwell’s continuity conditions)

BOUNDARY CONDITIONS BOUNDARY CONDITIONS

Suppose that at a particular instance and at a particular location of the boundary,

the oscillation of the incident wave is at its maximum;

then both reflected and transmitted waves have to be at their maxima.

In other words,

the wavelengths along the interface surface

must have the same temporal and spatial variation.

3 2

1 z z

z

λ λ

λ = =

constant 2 =

=

zi

i λ

β π Propagation constant :

β, k, phase, or momentum matching

But all mean the same thing: wavelength matching at the boundary!

Snell’s law :

TRANSMISSION AND REFLECTION COEFFICIENTS

(Normal Incidence)

TRANSMISSION AND REFLECTION COEFFICIENTS

(Normal Incidence)

At normal incidence, both the E and H fields are parallel to the interface.

Electric (magnetic) field transmission coefficient : t_E (t_H) Electric (magnetic) field reflection coefficient : r_E (r_H)

From the boundary Condition 4,

Example 2.1 Example 2.1

Calculate the coefficients of reflection and transmission associated with the air-glass interface for normal incidence. ( n₁ = 1, n₂ = 1.5 )

Can the transmission coefficient be larger than 1?

What does a negative value for the reflection coefficient mean?

We set the positive directions of E₁and E₃are the same

but those of H₁and H₃are opposite

Boundary condition for E is satisfied : E = (4/5)E₁ Boundary condition for H is satisfied : H = (6/5)H₁

Example 2.1 Example 2.1

E₃= (4/5)E₁ : H₃= (6/5)H₁

When light is incident from a medium of a higher intrinsic impedance (lower index of refraction) onto that of a lower intrinsic impedance (higher index of refraction),

(n_incident < ntransmitted)

the transmitted E field decreases,

whereas thetransmitted H field increases (larger than unity!).

Transmittance T : power ratio of transmitted light to incident light.

Reflectance R : power ratio of reflected light to incident light.

The light power is for unit area (A = 1, for simplicity)

In terms of the H field, In terms of the E field,

The same R and T.

4%

96%

TRANSMISSION AND REFLECTION COEFFICIENTS

(Arbitrary incident angle)

TRANSMISSION AND REFLECTION COEFFICIENTS

(Arbitrary incident angle)

In-plane (of incident) polarization Parallel polarization

TM polarization p polarization

Out-of-plane (of incident) polarization Perpendicular polarization

TE polarization s polarization

TRANSMISSION AND REFLECTION COEFFICIENTS

(Arbitrary incident angle for TM polarization)

TRANSMISSION AND REFLECTION COEFFICIENTS

(Arbitrary incident angle for TM polarization)

TM polarization The tangential components of the E and H fields

on both sides of the boundary are equal;

From the continuity of the normal component of D, and from the equality θ₁ = θ₃,

Fresnel’s equations for TM wave

TRANSMISSION AND REFLECTION COEFFICIENTS

(Arbitrary incident angle for TE polarization)

TRANSMISSION AND REFLECTION COEFFICIENTS

(Arbitrary incident angle for TE polarization)

The tangential components of the E and H fields on both sides of the boundary are equal;

TE polarization Fresnel’s equations for TE wave

TRANSMISSION AND REFLECTION COEFFICIENTS

(Fresnel Equations)

TRANSMISSION AND REFLECTION COEFFICIENTS

(Fresnel Equations)

Note that This is a direct consequence of Maxwell’s continuity condition, as can be seen by multiplying both sides by the incident field E₁. Æ Continuity of E fields.

(Ex) Reflection and transmission coefficients as a function of incident angle for air to glass interface (n

= 1.0, n

= 1.5).

(Ex) Reflection and transmission coefficients as a function of incident angle for air to glass interface (n

= 1.0, n

= 1.5).

Transmission coeff.

Reflection coeff.

Brewster angle only for TM pol.

For a glass lens used to collect a diverging light source such as the emission from a light-emitting diode (LED).

Best case for minimal increase in reflection toward the edge of a meniscus lens.

TRANSMISSION AND REFLECTION COEFFICIENTS

(Impedance approach for arbitrary incident angle )

TRANSMISSION AND REFLECTION COEFFICIENTS

(Impedance approach for arbitrary incident angle )

Remember that, at normal incidence,

Define an impedance out of certain components of E and H chosen such that their Poynting vectors point normal to the interface.

S_x

Æ called characteristic wave impedances referred to the x direction

Replacing η₁ and η₂ by η_1x and η_2x above,

and then converting η to n, gives the same results:

TRANSMITTANCE AND REFLECTANCE

(AT AN ARBITRARY INCIDENT ANGLE)

TRANSMITTANCE AND REFLECTANCE

(AT AN ARBITRARY INCIDENT ANGLE)

Power per unit area is Poynting vector, s = E × H In applying conservation of energy,

Note that the medium with the greater impedance has the larger E field, since

Let A be assumed to be a unit area,

x

BREWSTER’S ANGLE BREWSTER’S ANGLE

Figure 2.5 Reflection and transmission coefficients as a function of incident angle for air to glass interface.

Brewster angle only for TM pol.

= 0 ; when θ₁ + θ₂ = 90^o

The direction of reflection if it were to exist (dashed line) coincides with the direction of polarization of the transmitted wave.

From a microscopic viewpoint, the reflected wave is generated by the oscillation of electric dipoles in the transmission medium.

The oscillating dipole does not radiate in the oscillation direction Æ Brewster’s condition Brewster’s condition is a consequence of impedance

matching with reference to the vertical direction Æ

From Snell’s law : n₁sinθ₁ = n₂ sinθ₂ →η₂ sinθ₁ =η₁sinθ₂

x

x 2

1

η

η =

Phase diagram

WAVE EXPRESSIONS OF LIGHT WAVE EXPRESSIONS OF LIGHT

In media 1 and 2, the plane waves are expressed as

Inserting E₁ and E₂ into the wave equation,

The wavelength matching condition (phase matching condition) on the boundary for the z direction is,

EVANESCENT WAVES EVANESCENT WAVES

When the total internal reflection takes place,

does the field abruptly become zero on the boundary?

Because of the boundary condition of continuity, the field in the less dense medium cannot

abruptly become zero.

So called evanescent wave (surface wave) exists!

when n = n₂/n₁ < sinθ₁

The effective depth h of the penetration of the evanescent wave, which is defined as the depth where the amplitude decays to 1/e of that on the boundary, is 1/γ.

n₁ > n₂

Form a standing wave in x direction

Transmission and Reflection Coefficients for Total Internal Reflection (TIR)

Transmission and Reflection Coefficients for Total Internal Reflection (TIR)

TIR

when n = n₂/n₁ < sinθ₁

Since the magnitudes of the denominator and numerator are identical, the absolute value of the reflection coefficient is unity.

The phase shift difference can be used to make waveplates.

For example, a phase difference of π/2 results in a quarter-waveplate, which is quite useful because it is independent of wavelength.

Phase diagram

where

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛ × k n

k n

1 1

For Total Internal Reflection (TIR) For Total Internal Reflection (TIR)

and represent the ratio of momenta in medium 1 and medium 2.

tan δ

tan δ

For Total Internal Reflection (TIR) For Total Internal Reflection (TIR)

r I

t = +

The evanescent wave is a result of light energy that goes into the less dense medium only for a short distance and then comes back again into the dense medium.

The transmission coefficient describes the magnitude on the boundary but just a short distance inside the less dense medium.

For Total Internal Reflection (TIR) For Total Internal Reflection (TIR)

||

||

||

||

2

||

2

||

2 cos

) 1

1 (

δ ^j

δ

j

e

e n t n

e

r =

⇒ = +

=

⊥

⊥

⊥ −

⊥

−

⊥

−

⊥

= e

⇒ t = + e

= δ e

r

1

2 cos

With an increase in cosθ₁, the magnitude of the transmission coefficient into the second medium increases, also the momentum perpendicular to the boundary (kn₁cosθ₁) increase.

The transmission coefficients reach their maxima at the critical angle(n = sinθ₁).

The amount of phase shift, however, reaches zero at the critical angle.

This fact rejectsthe simple explanation that the phase delay is the time needed for the evanescent wave to go out into the less dense medium and come back into the dense medium. It cannot safely be said that the phase delay is due to the round-trip time.

Anyway, the phase shift is needed to match the boundary condition.

Æ Goos and Hanchen clarify these matters.

This is as if the photon in the optically dense medium is being pushed out into the less dense medium by this momentum.

In 1947, Goos and Hanchen looked at the difference between reflection from a silver surface and total internal reflection at a glass–air interface.

The reflected light from the silver surface takes the direct path SQ₁P₁

The light that has undergone total internal reflection takes the route SQ₁Q₂P₂,

D

Near the critical angle, where the δ’s are at their minimum, the D’s are at their maximum.

Evanescent Field and Its Adjacent Fields Evanescent Field and Its Adjacent Fields

In the dense medium, a standing wave E_s1 is formed by the incident and reflected waves.

n₁ > n₂

Form a standing wave in x direction The position of maximum intensity is shifted downward by δ radians.

Because of the continuity condition,

the amplitude |E₂| of the evanescent wave

has to be identical to that of the standing wave E_s1 evaluated on the boundary;

E-field expressions near the boundary.

With a decrease in the incident angle θ₁from 90° approaching the critical angle θ_c, both δ and γ decrease and h(= γ⁻¹) increases

Evanescent Field and Its Adjacent Fields Evanescent Field and Its Adjacent Fields

This means that with a decrease in θ₁, the maxima shift toward the interface and the evanescent wave shifts into the optically less dense medium.

In practice, with the wavelengths 0.85~1.55 μm and n₁~ 1.55, n₂ ~ 1.54, h = 1~10 μm.

k-diagrams (phase diagrams) for the Graphical Solution of the Evanescent Wave

k-diagrams (phase diagrams) for the Graphical Solution of the Evanescent Wave

by eliminating

When total internal reflection exists, k_2x is imaginary :

where

β

K γ

K–β diagram :

K–γ diagram :

Radius = ^{2 2}

1 2

k n −n

Radius = kn₁

large θ₁ largeβ small K

large γ fast decay

θ₁Æ θ_c γ Æ 0

very slow decay

θ₁< θ_c no evanescent

WHAT GENERATES THE EVANESCENT WAVES?

WHAT GENERATES THE EVANESCENT WAVES?

(Corrugated metal surface) (Array of metal pins)

To satisfy the boundary condition in z-direction,

When d < λ/2 (one-half of the free-space wavelength), all component waves become evanescent waves.

( Assume k_y = 0.)

Example 2.5 Example 2.5

A lossy glass is bordered by air in the x = 0 plane.

The propagation constant in the z direction on the border is β + jα.

Find the expression for the evanescent wave in the air near the boundary.

Propagation constants in air,

Phase matching along the z direction

Finally, the expressions for the evanescent field become

not acceptable since it goes infinitely as x increases.