Multilayer Thin-film Interference Filter Advanced Optics

Dr. Ribal Georges Sabat

Jimmy Zhan

November, 2013

Abstract The theory and background of multilayer thin-film interference filters was developed, via optical

analysis and formalisms of electromagnetic theory. A complete model on transmission spectra,

resonance peaks, and peak FWHM of the filter as functions of various parameters were

developed, and plotted on MATLAB, which was later compared to previous experimental

evidence. Results show that the theoretical models on transmission characteristics to be

accurate.

Table of Contents Abstract ......................................................................................................................................................... 1

Introduction .................................................................................................................................................. 3

Theory ........................................................................................................................................................... 5

Electromagnetic Wave Propagation ......................................................................................................... 5

Planar Optical Boundary between Two Absorption Free Media .............................................................. 6

Planar Optical Boundary between Absorption-free Incident Medium and a Metal ................................ 7

Thin Film Optics......................................................................................................................................... 7

Background ................................................................................................................................................... 9

Coherence ................................................................................................................................................. 9

Interference .............................................................................................................................................. 9

Model .......................................................................................................................................................... 11

Plane-Parallel Optical Cavity ................................................................................................................... 11

Thin Film Fabry-Perot Interference Filter on Glass Substrate ................................................................ 16

Determination of Ф𝒂𝒃 (Phase Shifts at Metal-Dielectric Interface) ...................................................... 19

Summary ................................................................................................................................................. 22

Results ......................................................................................................................................................... 23

References .................................................................................................................................................. 24

Introduction

Monochromatic light sources are often needed at wavelengths where laser sources do not

exist, or are too costly to use. In these cases, a broadband source in junction with a filter can be

used. However, most optical filters have rather broad pass bands. One simple example is the

Wratten filter, which involves dyed gelatin sheets.

A multilayer interference filter involves a transparent dielectric layer sandwiched between two

thin film layers of metals [4]. These filters can be designed with various optical properties, such

as band pass or band stop. The arrangement of two thin metal films enclosing a dielectric layer

is similar to an optical resonating cavity, in which the two metal films are semitransparent and

act as half mirrors. The light is repeatedly reflected between the two half mirrors, and

interferes with itself, greatly amplifying specific wavelengths (resonant frequencies,

constructive interference) while attenuating other wavelengths (destructive interference). This

results in very narrow transmission bandwidths [2]. A particular design of the filter consisting of

Magnesium Fluoride dielectric layer, enclosed between thin films of Aluminum, will be the

focus of this report.

The earliest form of interference filter is called the Fabry-Perot etalon, invented by Charles

Fabry, and Albert Perot. This filter consists of a mechanically variable air gap (acting as the

dielectric), with two planar parallel glass half mirrors as end plates [1, 3]. A later form of the

filter utilizes solid dielectric medium bound on each side by a thin metal semitransparent film,

and it was called a Metal Dielectric Metal (MDM) interferometer. An even later form utilizes

stacks of thin films for both the metal and the dielectric. This final form is called metal dielectric

thin film Fabry Perot interference filter. See Figure 1 for a diagram of a basic multilayer

interference filter.

The optical models will be developed by solving Maxwell’s Equation for the propagation of

electromagnetic waves, and their interactions with optical media (absorption, transmission, and

reflection) and at various interfaces.

Figure 1: A multilayer interference filter. In this case, the two metal films are aluminum, and the

dielectric layer is composed of Magnesium Fluoride. The stack is mounted on a glass substrate

(Watt, 2010).

Theory

Electromagnetic Wave Propagation The light is assumed to be a plane polarized harmonic electromagnetic wave, propagating in a

linear isotropic homogeneous medium. The wave can be characterized by several scalar

parameters along with Maxwell’s Equations.

Since absorption is being studied, the wavenumber, permittivity, and the refractive index are

allowed to be complex numbers. In the dielectric, these numbers will be purely real, and in

metals they can be complex.

The complex refractive index is defined as follows,

𝑁 = 𝑛 − 𝑖𝑘

(1)

Where the real part 𝑛 pertains to the ratio of the speed of light in the medium to that of

freespace, and the imaginary part 𝑘 is the extinction coefficient, a measure of absorption in the

medium (note the similarity with the imaginary part of the complex permittivity and complex

wavenumber).

Another material parameter, the characteristic impedance 𝑦, is defined as the ratio of the

amplitude of the H vector to the E vector.

𝐻 = 𝑦𝐸

(2)

Inside the material, and rewriting equation (2) using the Poynting vector,

𝑁

𝑐𝜇(�̂� x �⃗� ) = �⃗⃗�

�̂� x �⃗⃗� = −𝑁

𝑐𝜇�⃗�

𝑦 =𝑁

𝑐𝜇= 𝑁𝑌0 = 𝑁√휀0𝜇0

(3)

Where 𝑌0is the characteristic impedance, in this case, of freespace. (Note, this is the reciprocal

of the usual definition for wave impedance).

The irradiance, also known as the intensity, is the power flux density of the electromagnetic

wave. It can be computed by taking the average of the Poynting vector.

𝐼(𝑥) = |𝐼 | =1

2𝑅𝑒(�⃗� x �⃗⃗� ) =

1

2𝑅𝑒(𝐸𝐻∗) =

1

2𝑦𝐸(𝑥)2 =

1

2𝑛𝑌0𝐸(𝑥)2

(4)

Where 𝐸(𝑥) = 𝑅𝑒(𝑬(𝑥)) is the scalar amplitude of the electric field.

Planar Optical Boundary between Two Absorption Free Media At the boundary of two absorption free media (dielectrics) with designated optical impedance

𝑦1, 𝑦2, at some incident angle 𝜃1, some well-known relationships between the incident,

reflected, and transmitted waves can be summarized.

1) The incident, reflected, and transmitted wave vectors form a plane (called the plane of

incidence), which also includes the normal to the surface.

2) Angle of incidence is equal to the angle of reflection. 𝜃1 = 𝜃1′ . (Law of reflection).

3) Snell’s law of refraction. 𝑛1

𝑛2=

𝑠𝑖𝑛𝜃2

𝑠𝑖𝑛𝜃1.

4) Invariance of frequency.

5) Energy conservation at the boundary.

Using the last rule, the following equations are derived.

𝑅 = 𝜌(𝑦1, 𝑦2, 𝜃1)2 = (

𝐸𝑟

𝐸𝑖)𝑇𝐴𝑁

2

𝑇 = 𝜏(𝑦1, 𝑦2, 𝜃1)2 = (

𝐸𝑡

𝐸𝑖)

𝑇𝐴𝑁

2

𝑅 + 𝑇 = 1

(5)

Where 𝑅, 𝑇, 𝜌, 𝜏 are the reflectance, transmittance, reflection coefficient, and transmission

coefficient, respectively. 𝑅 = (𝐼𝑟

𝐼𝑖) , 𝑇 = (

𝐼𝑡

𝐼𝑖). This applies only to light energy flow normal to the

interface, so only the tangential component (parallel to the boundary) of the electric field is

involved, regardless of incident angle 𝜃1. These restrictions allow the modelling of all dielectric

interfaces for collimated beams of normal and oblique incidences for a particular polarization.

There is generally no phase shift between incident and transmitted waves while the reflected

wave is shifted by 𝜋 radians if 𝑛1 < 𝑛2.

For normal incidences, it can be shown that,

𝜌1−2 =𝑦1 − 𝑦2

𝑦1 + 𝑦2=

𝑛1 − 𝑛2

𝑛1 + 𝑛2

𝜏1−2 =2𝑦1

𝑦1 + 𝑦2=

2𝑛1

𝑛1 + 𝑛2

𝑅1−2 = (𝜌1−2)2 = (

𝑛1 − 𝑛2

𝑛1 + 𝑛2)2

𝑇1−2 =𝑦2

𝑦1

(𝜏1−2)2 =

4𝑦1𝑦2

(𝑦1 + 𝑦2)2=

4𝑛1𝑛2

(𝑛1 + 𝑛2)2

𝑅1−2 + 𝑇1−2 = 1

(6)

If the wave is not at normal incidence, then both the reflectance and transmittance (as well as

the coefficients of reflection and transmission) will vary as a function of the angle of incidence.

Planar Optical Boundary between Absorption-free Incident Medium and a Metal Examples of such boundaries include air-metal, or any dielectric-metal interfaces. In this case 𝑅

𝑇 do not add up to 1 because of absorption (they remain real however). The coefficients 𝜌 and

𝜏 are in general complex numbers, embodying phase differences. There is a finite phase shift

occurring for both reflected and transmitted waves at the boundary, for any angle of incidence

[2].

In general,

𝑅 + 𝑇 + 𝐴 = 1

(7)

Where A is the absorbance.

Thin Film Optics A thin film deposited on a substrate presents three optical boundaries, air-film, film-substrate,

and substrate-air. Multiple reflections will occur within the media, and interference effects are

possible. A quantitative classification of an optical medium as to whether it is “thick” or “thin” is

that when illuminated by a monochromatic light, whether the path length difference between

the multiple reflections at different boundaries are less than the coherence length of the light

source or not. If the optical path length differences are less than the coherence length, then the

media is “optically thin”, and interference can occur.

Most films are “optically thin”, and most substrates are “optically thick”. However, in reality

thin films have significant amount of surface and bulk textures, and thus scatterings tend to

suppress some interference. This effect is even more pronounced in multilayered films. Thus, in

an experimental setup we should expect to see deviations in transmission characteristics from

the predicted models.

Background

Coherence Coherence is an ideal property of waves that enables stationary (temporally and spatially

constant) interference. It contains several distinct but related concepts, such as temporal

coherence, spatial coherence, and spectral coherence. In reality, no waves can be truly

coherent. A measure of coherence is the coherence time, and coherence length.

The relationship between the property of coherence and the phenomenon of interference is

that interference results from the addition of waves of different relative phases. See next

section. Two waves are said to be coherent if they have constant relative phases. A single wave

is said to be coherent if it has a constant relative phase with itself after a time delay, or after a

spatial delay. Thus in principle, coherence is a measure of how quickly a wave’s phase drifts in

time or space. In practice, the interference visibility can be used to measure the coherence of a

wave.

Temporal coherence describes the correlation between waves observed at different moments

in time. Temporal coherence tells us how monochromatic a light source is, so in general, the

broader the frequency spectrum of a source, the smaller the coherence time.

𝜏𝑐∆𝑓 ≅ 1

(8)

Thus wave packets or white light, which have broad range of frequencies, tend to have smaller

coherence times, than for example, a monochromatic laser turned on for all times.

Spatial coherence describes the correlation between waves at different points in space (the

ability that at two points in space for a wave to interfere). If a wave has only one value of

amplitude over an infinite length, then it is perfectly spatially coherent. A light bulb filament,

for example, consists of point sources that emit light independently without a fixed phase

relationship, and is thus non-coherent. Thus, only a point source of light would be truly spatially

coherent.

Interference Interference is a phenomenon that can be observed from nearly all waves, including optical

waves, acoustic waves, and water waves. Two waves superimpose on each other and form

nodes of constructive and destructive interferences, resulting in waves of greater and lower

amplitudes. The interfering waves must have the nearly the same frequencies and be coherent,

and have phase differences of integers of 𝜋 or 2𝜋.

Let the displacement of the two waves as a function of position and time be represented by,

𝑈1(𝒓, 𝑡) = 𝐴1(𝒓) exp{𝑖(𝜙1(𝒓) − 𝜔𝑡)}

𝑈2(𝒓, 𝑡) = 𝐴2(𝒓) exp{𝑖(𝜙2(𝒓) − 𝜔𝑡)}

(9)

Where 𝐴 is the amplitudes, 𝜙 is the phase, and 𝜔 is the angular frequency.

The displacement sum of the two waves is,

𝑈(𝒓, 𝑡) = 𝐴1(𝒓) exp{𝑖(𝜙1(𝒓) − 𝜔𝑡)} + 𝐴2(𝒓) exp{𝑖(𝜙2(𝒓) − 𝜔𝑡)}

(10)

The intensity/irradiance of the light at 𝒓 is given by,

𝐼(𝒓) = ∫ 𝑈(𝒓, 𝑡)𝑈∗(𝒓, 𝑡)𝑑𝑡 ∝ 𝐴12(𝒓) + 𝐴2

2(𝒓) + 2𝐴1(𝒓)𝐴2(𝒓) cos(𝜙1(𝒓) − 𝜙2(𝒓))

(11)

Which can be expressed in terms of the intensity of the individual waves,

𝐼(𝒓) = 𝐼1(𝒓) + 𝐼2(𝒓) + 2√𝐼1(𝒓)𝐼2(𝒓) cos(𝜙1(𝒓) − 𝜙2(𝒓))

(12)

Thus, the interference pattern maps out the difference in phase between the two waves, with

maxima occurring when the phase difference is a multiple of 2𝜋. If the two beams are of equal

intensity, the maxima are four times as bright as the individual beams, and the minima have

zero intensity.

The two waves must have the same polarization to give rise to interference fringes since it is

not possible for waves of different polarizations to cancel one another out or add together.

Instead, when waves of different polarization are added together, they give rise to a wave of a

different polarization state.

Model

Plane-Parallel Optical Cavity

Figure 2: A plane-parallel optical cavity. Shown here with a non-zero angle of incidence for

generality (Watt, 2010).

As illustrated in Figure 2, the structure of a plane-parallel optical cavity (in the simplest case

that it is “optically thin” with thickness 𝑑), including the dielectric medium of refractive index

𝑛𝑐, bound on both the incident and emergent sides by media of refractive indices 𝑛𝑖, and 𝑛𝑒,

through optical boundaries designated 𝑎 and 𝑏.

𝜆𝑖is the incident plane wave (its direction shown by the arrow). It is incident on the surface at

an angle 𝜃𝑖 to keep the generality. Multiple reflections produce a series of transmitted beams

𝜆𝑒 on the emergent (right) side.

From the way this diagram is shown, one may falsely believe that interference will not be

possible unless light is incident at a normal angle, because the optical paths of the transmitted

beams do not overlap. (Unless a lens is used to focus all the transmitted beams to a focal point,

commonly used in a Fabry-Perot interferometer). However, these lines represent only the

wavevectors (directions of the wavefronts) of the incident and transmitted beams. So they only

show the directions of propagations of the waves. The actual wavefronts are much wider than

the thickness of the cavity.

Using the models of section 2 and using the ray optics model for phase shifts and path lengths,

the following observations are made.

1) Phase shift is zero across both interfaces for transmitted waves.

2) Reflections within the cavity introduces phase shifts of 𝜙𝑎 and 𝜙𝑏, both of which are

zero because the cavity refractive index is greater than incident and emergent media

refractive indices.

3) The only phase shift between two consecutive transmitted beams is therefore 𝛿,

resulting from the optical path length difference ∆𝐿.

∆𝐿 = 𝑛𝑐(1 → 2 + 2 → 3) − 𝑛𝑒(1 → 𝑥) = 2𝑛𝑐𝑑𝑐𝑜𝑠(𝜃𝑐)

𝛿 =2𝜋

𝜆0∆𝐿 =

4𝜋𝑛𝑐𝑑

𝜆0cos(𝜃𝑐) =

4𝜋𝑑

𝜆𝑐cos (𝜃𝑐)

𝛿 = 2𝜋∆𝐿

𝜆0

𝑛𝑖 sin(𝜃𝑖) = 𝑛𝑐 sin(𝜃𝑐) = 𝑛𝑒sin (𝜃𝑒)

𝜆0 = 𝑛𝑖𝜆𝑖 = 𝑛𝑐𝜆𝑐 = 𝑛𝑒𝜆𝑒

(13)

4) The phase shift 𝛿 therefore depends on the physical thickness of the cavity, the

refractive indices of the incident and cavity media, the vacuum wavelength, and the

incident angle.

5) Resonant condition is when the phase shift is a multiple of 2𝜋. This occurs when the

optical path length difference ∆𝐿 is an integer multiple 𝑚 of the vacuum wavelength 𝜆0.

𝛿𝑟𝑒𝑠𝑜𝑛𝑎𝑛𝑐𝑒 =2𝜋∆𝐿

𝜆0= 2𝜋𝑚

(14)

6) Without meeting this condition, the phase shift between consecutive transmitted

beams will cause a cumulative decoherence (scattering), and transmission will be weak.

7) There are multiple possible resonant wavelengths 𝜆0.

8) Similar results can be obtained for resonant reflections. (I.e. reflection and transmission

analysis are symmetrical).

Let 𝑇, 𝑅, 𝜏, 𝜌 be the irradiance/intensity transmittance, irradiance reflectance, (amplitude)

transmission coefficient, and reflection coefficient, respectively, at the two boundaries 𝑎, and

𝑏. Let the superscript +, and – indicate whether the wave is right or left going at the interfaces.

Then, the left to right transmittance is,

𝑇 =𝐼𝑒𝐼𝑖

𝐼𝑖 =1

2𝑛𝑖𝑌0𝐸𝑖𝐸𝑖

∗

𝐼𝑒 =1

2𝑛𝑒𝑌0𝐸𝑒𝐸𝑒

∗

(15)

Where the 𝐸 is the electric field amplitudes of the left to right transmitted waves at interface 𝑎

and 𝑏, and they are,

𝐸𝑎 = 𝜏𝑎+𝐸𝑖

𝐸𝑒 = 𝜏𝑏+𝐸𝑏 = 𝜏𝑏

+𝐸𝑎 exp (−𝑖𝑘𝑐𝑑

cos(𝜃𝑐)) [1 + ∑(𝜌𝑏

+𝜌𝑎−

𝑛

)𝑛 exp(−𝑖𝑛𝛿)]

𝐸𝑒 =𝜏𝑏+(𝜏𝑎

+𝐸𝑖) exp (−𝑖𝑘𝑐𝑑

cos(𝜃𝑐))

1 − (𝜌𝑏+𝜌𝑎

−) exp(−𝑖𝛿)

(16)

Phase shifts and amplitude changes at the interfaces can be accounted for via complex

coefficients,

𝜏𝑏+ = |𝜏𝑏

+|exp (𝑖𝜙𝑏′ )

𝜏𝑎+ = |𝜏𝑎

+|exp (𝑖𝜙𝑎′ )

𝜌𝑏+ = |𝜌𝑏

+|exp (𝑖𝜙𝑏)

𝜌𝑎− = |𝜌𝑎

−|exp (𝑖𝜙𝑎)

(17)

The amplitude squared (irradiance) of the emergent wave is calculated thusly,

𝐸𝑒𝐸𝑒∗ =

(𝜏𝑏+𝜏𝑎

+)(𝜏𝑏+𝜏𝑎

+)∗𝐸𝑖𝐸𝑖∗

(1 − 𝜌𝑏+𝜌𝑎

− exp(−𝑖𝛿))(1 − 𝜌𝑏+𝜌𝑎

− exp(−𝑖𝛿))∗

𝐸𝑒𝐸𝑒∗ =

|𝜏𝑎+|2|𝜏𝑏

+|2𝐸𝑖𝐸𝑖∗

[1 − |𝜌𝑏+||𝜌𝑎

−| exp(𝑖(𝜙𝑎 + 𝜙𝑏 − 𝛿))][1 − |𝜌𝑏+||𝜌𝑎

−| exp(−𝑖(𝜙𝑎 + 𝜙𝑏 − 𝛿))]

𝐸𝑒𝐸𝑒∗ =

|𝜏𝑎+|2|𝜏𝑏

+|2𝐸𝑖𝐸𝑖∗

1 + |𝜌𝑎−|2|𝜌𝑏

+|2− 2|𝜌𝑎

−||𝜌𝑏+| cos(𝜙𝑎 + 𝜙𝑏 − 𝛿)

𝐸𝑒𝐸𝑒∗ =

|𝜏𝑎+|2|𝜏𝑏

+|2𝐸𝑖𝐸𝑖∗

1 + |𝜌𝑎−|2|𝜌𝑏

+|2− 2|𝜌𝑎

−||𝜌𝑏+| [1 + cos(𝜙𝑎 + 𝜙𝑏 − 𝛿) − 1]

𝐸𝑒𝐸𝑒∗ =

|𝜏𝑎+|2|𝜏𝑏

+|2𝐸𝑖𝐸𝑖∗

(1 − |𝜌𝑎−||𝜌𝑏

+|)2[1 +

4|𝜌𝑎−||𝜌𝑏

+|

(1 − |𝜌𝑎−||𝜌𝑏

+|)2 sin2 𝜙𝑎 + 𝜙𝑏 − 𝛿

2 ]

(18)

Then the total left to right transmittance is,

𝑇 = (𝑛𝑒

𝑛𝑖)

|𝜏𝑎+|2|𝜏𝑏

+|2

(1 − |𝜌𝑎−||𝜌𝑏

+|)2[1 +

4|𝜌𝑎−||𝜌𝑏

+|

(1 − |𝜌𝑎−||𝜌𝑏

+|)2 sin2 𝜙𝑎 + 𝜙𝑏 − 𝛿

2 ]

𝑇 = (𝑛𝑒

𝑛𝑖)

𝑇𝑎+𝑇𝑏

+

(1 − √(𝑅𝑎−𝑅𝑏

+))2 [1 +

4√(𝑅𝑎−𝑅𝑏

+)

(1 − √(𝑅𝑎−𝑅𝑏

+))2 sin2

𝜙𝑎 + 𝜙𝑏 − 𝛿

2]−1

(19)

However, we have made the assumption that transmission and reflections within the cavity

introduces zero phase shifts, so that 𝜙𝑎′ = 𝜙𝑏

′ = 𝜙𝑎 = 𝜙𝑏 = 0. Then the above expression for

transmittance simplifies to,

𝑇 = (𝑛𝑒

𝑛𝑖)

𝑇𝑎+𝑇𝑏

+

(1 − √(𝑅𝑎−𝑅𝑏

+))2 [1 +

4√(𝑅𝑎−𝑅𝑏

+)

(1 − √(𝑅𝑎−𝑅𝑏

+))2 sin2

𝛿

2]−1

=𝑇𝑚𝑎𝑥

1 + 𝐹 sin2 𝛿2

Where 𝑇𝑚𝑎𝑥 is the maximum transmittance, and 𝐹 is the finesse factor,

𝑇𝑚𝑎𝑥 =𝑛𝑒

𝑛𝑖

𝑇𝑎+𝑇𝑏

+

(1 − √(𝑅𝑎−𝑅𝑏

+))2

𝐹 =4√(𝑅𝑎

−𝑅𝑏+)

(1 − √(𝑅𝑎−𝑅𝑏

+))2

(20)

A local maximum in the transmittance occurs at each cavity resonant wavelength 𝜆𝐶𝑝, 𝜆0𝑝,

satisfying the following conditions,

sin21

2𝛿(𝜆𝐶𝑝) = sin2

1

2𝛿(𝜆0𝑝) = 0

1

2𝛿(𝜆𝐶𝑝) =

1

2𝛿(𝜆0𝑝) =

4𝜋𝑛𝐶𝑑

2𝜆0𝑝,𝑚cos(𝜃𝐶) =

4𝜋𝑑

2𝜆𝐶𝑝,𝑚cos(θC) = mπ

(21)

For 𝑚 = 0, 1, 2, 3…

In which case,

𝑇 = 𝑇𝑚𝑎𝑥

The integer 𝑚 is called the order number of the interference. (Note, some people call 𝑚 + 1

the order number).

The transmittance is therefore dependent on the refractive indices, the internal reflectance and

transmittance, and the angle of incidence, which is non-zero (not normal), will shift the

resonance to a shorter wavelength.

The finesse factor is a measure of how sharp the resonance is. It depends solely on the internal

reflectance. The higher the internal reflectance, the sharper the resonance curve.

The FWHM (bandwidth) of the 𝑚𝑡ℎ order resonance transmission can be shown to be,

𝑇12,𝑚

=𝑇𝑚𝑎𝑥,𝑚

2=

𝑇𝑚𝑎𝑥,𝑚

1 + 𝐹 sin2

𝛿 (𝜆12,𝑚

)

2

𝛿 (𝜆12,𝑚

)

2=

4𝜋𝑛𝐶𝑑

2𝜆0,1/2 ,𝑚cos(𝜃𝐶) = 𝑚𝜋 + sin−1 (±√

1

𝐹) = 𝑚𝜋 ± sin−1 (√

1

𝐹)

𝜆0,1/2,𝑚 =2𝜋𝑛𝐶𝑑 cos(𝜃𝐶)

𝑚𝜋 ± sin−1 (1

√𝐹)

𝐹𝑊𝐻𝑀𝜆0,𝑚 = |𝜆0,12

,+− 𝜆0,1

2,−|𝑚

= 2𝜋𝑛𝐶𝑑 cos(𝜃𝐶) |1

𝑚𝜋 + sin−1 (1

√𝐹)−

1

𝑚𝜋 − sin−1 (1

√𝐹)|

𝐹𝑊𝐻𝑀𝜆0,𝑚 = 4𝜋𝑛𝐶𝑑 cos(𝜃𝐶) ||sin−1 (

1

√𝐹)

𝑚2𝜋2 − (sin−1 (1

√𝐹))

2||

(22)

Thin Film Fabry-Perot Interference Filter on Glass Substrate

In the previous section we have discussed the transmission characteristics of a model involving

non-absorbing (all dielectric, no metal) optical cavity. In this section, we will (by several

considerations) extend the previous model to a model that includes multiple metal films, and a

glass substrate. This model will give a realistic analysis of an actual multilayer interference filter.

The first consideration is that we now have a substrate before the emergent medium, and it is

not semi-infinite.

1) Substrate Cavity Resonance. Most substrates are orders of magnitude thicker than the

films, and are generally optically too thick to exhibit any interference effects. The path

length difference for the substrate cavity is much larger than that for the dielectric film

cavity, and thus any interference resonance produced by the substrate cavity will be

very spectrally distinct (longer wavelength) from those produced by our dielectric film

cavity. However in reality, it is nevertheless possible that the substrate cavity

interference resonances will produce some artifacts in experimental results.

2) Exit Medium. Another issue with the substrate is that it is an additional layer before the

exit medium (which is air). Since both the substrate (glass) and the exiting medium (air)

are non-absorbing and all dielectric, we can use the results from section 2 between two

absorption-free media, such that no phase shifts occur for glass to air transmissions.

Also, using the assumption from the previous consideration, we will ignore multiple

reflections in the glass substrate (do not consider it as an optical cavity for the

wavelengths we are interested in). Therefore, for substrate to air interface, there is only

a finite attenuation due to a single internal reflection in the glass. For the case of normal

incidence,

𝑇𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒−𝑎𝑖𝑟⏊ =

4𝑛𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒𝑛𝑎𝑖𝑟

(𝑛𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 + 𝑛𝑎𝑖𝑟)2=

4𝑛𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒

(𝑛𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 + 1)2= 1 − 𝑅𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒−𝑎𝑖𝑟

⏊ < 1

𝑅𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒−𝑎𝑖𝑟⏊ = (

𝑛𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 − 𝑛𝑎𝑖𝑟

𝑛𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 + 𝑛𝑎𝑖𝑟)2

= (𝑛𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 − 1

𝑛𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 + 1 )2

< 1

(23)

The second consideration concerns how to treat the two additional metal layers to the all

dielectric core cavity structure. In principle, multiple reflections within each individual metal

film can occur (such that the metal film itself is an optical cavity). However, a finite phase shift

will be introduced at each reflection internal or external to the metal layer. In this case,

resonance in the metal film will not be achieved, and the coefficients of reflection and

transmission will in general by complex quantities with angle of incidence dependencies.

1) The two metal films involved in transmission filters will generally have a very narrow

range of thickness (around few tens of nanometers). Thicker films will block all light,

while thinner films will have low reflectivity (and thus Finesse will be too low).

2) Metal films with above range of thickness tend to have columnar grain structure and

porosity that are highly dependent on the vacuum vapour deposition process.

Scatterings due to these grains in the structure will quickly decohere light beams. Thus,

with multiple reflections, no resonance can form within the metal films. The metal films

have only an absorption effect on the light.

3) Certain metals (such as aluminum) form oxides when exposed to air, and thus must be

protected with another coating layer. This further complicates the model.

4) These considerations imply that full analytical model of the entire device is nearly

impossible, and thus an empirical model should be used. This empirical model treats

each dielectric-metal-dielectric structure as a single interface, with the appropriate

coefficients of transmission and reflection, and a finite phase shift for transmission. This

is similar to the previously derived all-dielectric model, except now 𝜙𝑎−, 𝜙𝑏

′ , 𝜙𝑎 , 𝜙𝑏 are

non-zero. Any absorption due to the metal will be accounted for in the magnitude of the

complex coefficient of transmission.

5) In effect, we have reduced the model from air/coating/metal/metal-

bulk/metal/dielectric-bulk/metal/metal-bulk/metal/substrate/air, to air/DMD-DMD/air.

With these considerations in mind, the filter transmission, resonant transmission

conditions, and bandwidth for the thin film metal-dielectric-metal-substrate filter

(MDM) are given by,

𝑇𝑀𝐷𝑀𝑐𝑎𝑣𝑖𝑡𝑦 =𝑛𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒

𝑛𝑎𝑖𝑟

𝑇𝑎+𝑇𝑏

+

(1 − √(𝑅𝑎−𝑅𝑏

+))2 [1 +

4√(𝑅𝑎−𝑅𝑏

+)

(1 − √(𝑅𝑎−𝑅𝑏

+))2 sin2

𝜙𝑎 + 𝜙𝑏 − 𝛿

2]−1

𝑇𝑀𝐷𝑀𝑐𝑎𝑣𝑖𝑡𝑦 = 𝑇𝑚𝑎𝑥 [1 + 𝐹 sin2𝜙𝑎 + 𝜙𝑏 − 𝛿

2]−1

(24)

Where the subscript 𝑎 and 𝑏 designate the two interfaces (air-metal-dielectric, and

dielectric-metal-substrate).

Thus, the resonant condition becomes,

sin2𝜙𝑎 + 𝜙𝑏 − 𝛿

2= 0

|𝜙𝑎 + 𝜙𝑏 − 𝛿

2| ≡ |

Ф𝑎𝑏 − 𝛿

2| = 𝑚𝜋

𝛿 = 𝛿(𝜆𝐶 , 𝜃𝐶) = 𝛿(𝜆0, 𝜃𝐶) =4𝜋𝑛𝑐𝑑

𝜆0cos(𝜃𝐶) =

4𝜋𝑑

𝜆𝐶cos (𝜃𝐶)

𝜙𝑎 = 𝜙𝑎(𝜆0, 𝑛𝐶 , 𝜃𝑖)

𝜙𝑏 = 𝜙𝑏(𝜆0, 𝑛𝐶 , 𝑛𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 , 𝜃𝐶(𝑛𝐶 , 𝜃𝑖))

(25)

Here the phase shift between two consecutive transmitted beams due to cavity path lengths

difference is 𝛿, and is shown to be dependent on vacuum wavelength, cavity thickness 𝑑,

refractive index 𝑛𝐶 , and the wave propagation angle in the cavity 𝜃𝐶 . The two phase shifts due

to DMD interfaces, 𝜙𝑎, and 𝜙𝑏, are shown to be additionally dependent on the angle of

incidence 𝜃𝑖.

𝜃𝑖 and 𝜃𝐶 are not related by Snell’s Law due to the presence of a metal film. However, 𝜃𝐶 is a

function of 𝜃𝑖 as shown. Therefore, the phase shift 𝛿 is also dependent on the incidence angle

𝜃𝑖, through 𝜃𝐶 . In the case of normal incidence, 𝜃𝑖 = 0, the model simplifies greatly.

The value of 𝛿 is predictable from the cavity thickness and refractive index (if given normal

incidence). However, 𝜙𝑎 and 𝜙𝑏 (involving metal thin films) are highly sensitive to the vacuum

deposition process and conditions. Thus, it is best to determine 𝜙𝑎 and 𝜙𝑏 empirically for a

given thin film filter.

However, it is still possible to model 𝜙𝑎 and 𝜙𝑏 analytically, in terms of the metal and dielectric

refractive indices 𝑁𝑚 and 𝑛𝐶 . For the Al-MgF2-Al MDM filters, the two phase shifts at the DMD

interfaces 𝜙𝑎 and 𝜙𝑏 are assumed to be equal, and independent of aluminum thickness.

Reported results show that for aluminum half mirrors of thickness on the order of 20 → 30 𝑛𝑚,

the phase shifts at the DMD interface is about 30 𝑑𝑒𝑔𝑟𝑒𝑒𝑠. [6, 7, 8, 9]

𝑁𝑚 = 𝑛𝑚 − 𝑖𝑘𝑚

Ф𝑎𝑏

2≡

𝜙𝑎 + 𝜙𝑏

2=

2𝜙𝑎

2= 𝜙𝑎

tan(𝜙𝑎) =2𝑛𝑐𝑘𝑚

𝑛𝐶2 − 𝑛𝑚

2 − 𝑘𝑚2

(26)

Note that this model, and the referenced value of 𝜙𝑎 = 30 𝑑𝑒𝑔𝑟𝑒𝑒𝑠 do not necessarily apply to

all experiments. Ultimately the value is sensitive to deposition conditions. The most accurate

value for the phase shift is through empirical determination.

Determination of Ф𝒂𝒃 (Phase Shifts at Metal-Dielectric Interface) The method to determine Ф𝑎𝑏 experimentally is as follows. Firstly, we must have normal

incidence. Secondly, we must know the values of peak wavelengths (for various orders of 𝑚).

Since Ф𝑎𝑏 is a function of 𝛿, which is a function of 𝜆 and 𝑛𝐶𝑑. Given 𝜆, we still need to find 𝑛𝐶𝑑.

|Ф𝑎𝑏 − 𝛿(𝜆0𝑝,𝑚)

2| = |

Ф𝑎𝑏 − 𝛿(𝜆𝐶𝑝,𝑚)

2| = 𝑚𝜋

|Ф𝑎𝑏 − 𝛿(𝜆0𝑝,𝑚+1)

2| = |

Ф𝑎𝑏 − 𝛿(𝜆𝐶𝑝,𝑚+1)

2| = (𝑚 + 1)𝜋

𝛿 = 𝛿(𝜆𝐶𝑝,𝑚) = 𝛿(𝜆0𝑝,𝑚) =4𝜋𝑛𝐶𝑑

𝜆0𝑝,𝑚=

4𝜋𝑑

𝜆𝐶𝑝,𝑚

|1

𝜆0𝑝,𝑚−

1

𝜆0𝑝,𝑚+1| =

1

2𝑛𝐶𝑑

(28)

Thus solving for Ф𝑎𝑏 comes down to the task of finding the product 𝑛𝐶𝑑. Two methods are

shown.

1) Since the cavity dielectric is deposited in between two layers of metal films, in situ

monitoring and post process determination of the thickness and refractive index will be

difficult. However, one can pre-calibrate the deposition thickness monitor reading

(usually a quartz resonator gauge [10]), by depositing a single layer of test film at an

identical or geometrically related position/angle, and measuring the thickness of that

test layer with an ellipsometer. The ellipsometer will give both the refractive index and

the thickness of the dielectric.

2) If the dielectric is thick enough as to support two distinct peaks of resonant wavelengths

(i.e. 𝑚,𝑚 + 1 orders), then the product 𝑛𝐶𝑑 can be determined quite easily as shown in

the last of the above equations.

However, the order numbers 𝑚 of the observed interference peaks are generally not known,

although the total phase shifts between two consecutive peaks is always expected to be 2𝜋. If

one has a thick enough dielectric as to exhibit many peak wavelengths, then a multi-parameter

fit incorporating all the peaks would yield a value for Ф𝑎𝑏. In practice, scatterings at the thin

film interfaces limit the maximum observable order number to roughly 𝑚 = 3. Thus, a

theoretical model should be used in tandem with the experimental method, to help guide data

analysis. For example, the data in Figure 3 shows two peaks, in which case one needs to only

attempt the orders 𝑚 = 0,1, 𝑚 = 1, 2, 𝑚 = 2, 3 (keeping to low orders as discussed), when

trying to determine Ф𝑎𝑏.

Figure 3: Sample data of transmission peaks of Al-MgF2-Al on glass filter (Watt, 2010).

Figure 4: Another sample data, showing a high symmetry peak (Watt, 2010).

Figure 4 shows sample data for a filter with one peak, fitted to a Lorentzian lineshape, not to

the filter model. Nevertheless, high symmetry of the peak wavelength is one indication of

conformance to the ideal model conditions, since the transmittance is an even function of the

fractional wavelength deviation ∆𝑚 for the 𝑚𝑡ℎ order peak, for small deviations in either

direction.

𝑇𝑀𝐷𝑀𝑐𝑎𝑣𝑖𝑡𝑦 = 𝑇𝑚𝑎𝑥 [1 + 𝐹 sin2𝜙𝑎 + 𝜙𝑏 − 𝛿

2]−1

sin2𝜙𝑎 + 𝜙𝑏 − 𝛿

2= sin2 (

Ф𝑎𝑏 − 𝛿

2) ≡ 0

|Ф𝑎𝑏 − 𝛿

2| ≅ 𝑚𝜋 = |

1

2[Ф𝑎𝑏 −

4𝜋𝑛𝐶𝑑

𝜆0𝑝,𝑚(1 ± ∆𝑚)]| ≅ |

1

2[Ф𝑎𝑏 −

4𝜋𝑛𝐶𝑑

𝜆0𝑝,𝑚

(1 ∓ ∆𝑚)]|

Where in the last step I used the Binomial Approximation, since ∆𝑚≪ 1. Then,

|Ф𝑎𝑏 − 𝛿

2| ≅ 𝑚𝜋 ±

2𝜋𝑛𝐶𝑑

𝜆0𝑝,𝑚∆𝑚

sin2 (Ф𝑎𝑏 − 𝛿

2) = sin2 |

Ф𝑎𝑏 − 𝛿

2| ≅ sin2 (𝑚𝜋 ±

2𝜋𝑛𝐶𝑑

𝜆0𝑝,𝑚∆𝑚) ≅ sin2 (

2𝜋𝑛𝐶𝑑

𝜆0𝑝,𝑚∆𝑚)

(29)

Summary According to the theoretical model developed above,

1) Peak wavelength for a given interference order 𝑚 is determined by both the optical

thickness 𝑛𝐶𝑑, and total phase shift due to both metal-dielectric interfaces Ф𝑎𝑏.

2) The FWHM/Bandwidth/Finesse is dependent on the geometrical mean of the

reflectances at the two metal-dielectric interfaces, which in turn depend on the

thickness of the metal films. In general, the thicker the films, the higher the reflectance,

and thus the higher the Finesse.

3) Peak symmetry is dependent on both the metal and dielectric film uniformity, and near

constant phase shifts at the two metal-dielectric interfaces.

Results

Several Aluminum-MagnesiumFluoride-Aluminum MDM transmission filters were produced in

the labs (March, 2013). The filters were constructed by sequentially depositing thin films on

pre-cleaned glass microscope slides acting as the substrate. The basic schematic is shown

below.

Figure 5: Basic construction of a metal-dielectric-metal interference filter on a glass substrate

(Howe-Patterson, Warner, Zhan, 2013).

References

1. Fabry C and Perot A 1899, Theorie et applications d’une nouvelle method de

spectroscopie interferentielle Ann. Chim Phys. Paris 16, 115-44. 2. Jeong Justin, Pike Jessica, Stephen Gord, "Multi-Layer Interference Filter", Queen's University,

2013.

3. Howe-Patterson Matt, Warner Alex, Zhan Jimmy, “Multilayer Interference Filter”, Queen’s

University, 2013.

4. Watts, M., "Design of a Multi-Layer Interference Filter", lab manual, Queen's University, 2010.

5. Thin-Film Optical Filters, H. A. Macleod, 3rd Edition, Taylor & Francis 2001

6. Metal-Dielectric Multi-layers, John Macdonald, Elsevier Monographs on Applied Optics.

No.4, Elsevier, NY 1977.

7. M. N. Polyanskiy. Refractive index database, http://refractiveindex.info. Accessed Nov,

2013.

8. Calculate Spectral Reflectance of Thin Film Stacks,

http://www.filmetrics.com/reflectance-calculator. Acessed Nov, 2013.