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 ⃗ (𝑠̂ x 𝐸⃗ ) = 𝐻 𝑐𝜇 ⃗ =− 𝑠̂ x 𝐻 𝑦=

𝑁 𝐸⃗ 𝑐𝜇

𝑁 = 𝑁𝑌0 = 𝑁√𝜀0 𝜇0 𝑐𝜇 (3)

Where 𝑌0 is 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 1 1 1 ⃗ ) = 𝑅𝑒(𝐸𝐻 ∗ ) = 𝑦𝐸(𝑥)2 = 𝑛𝑌0 𝐸(𝑥)2 𝑅𝑒(𝐸⃗ x 𝐻 2 2 2 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). 𝑛

𝑠𝑖𝑛𝜃

3) Snell’s law of refraction. 𝑛1 = 𝑠𝑖𝑛𝜃2. 2

1

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 = (

2 𝐸𝑟 ) 𝐸𝑖 𝑇𝐴𝑁

2 𝐸𝑡 𝑇 = 𝜏(𝑦1 , 𝑦2 , 𝜃1 )2 = ( ) 𝐸𝑖 𝑇𝐴𝑁

𝑅+𝑇 =1 (5) Where 𝑅, 𝑇, 𝜌, 𝜏 are the reflectance, transmittance, reflection coefficient, and transmission 𝐼

𝐼

coefficient, respectively. 𝑅 = ( 𝐼𝑟 ) , 𝑇 = (𝐼𝑡). This applies only to light energy flow normal to the 𝑖

𝑖

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 𝑛1 − 𝑛2 = 𝑦1 + 𝑦2 𝑛1 + 𝑛2

𝜏1−2 =

2𝑦1 2𝑛1 = 𝑦1 + 𝑦2 𝑛1 + 𝑛2

𝑅1−2 = (𝜌1−2 )2 = ( 𝑇1−2 =

𝑛1 − 𝑛2 2 ) 𝑛1 + 𝑛2

𝑦2 4𝑦1 𝑦2 4𝑛1 𝑛2 (𝜏1−2 )2 = = (𝑦1 + 𝑦2 )2 (𝑛1 + 𝑛2 )2 𝑦1 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) + 𝐴22 (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. 𝜆𝑖 is 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) = 2𝑛𝑐 𝑑𝑐𝑜𝑠(𝜃𝑐 ) 𝛿=

2𝜋 4𝜋𝑛𝑐 𝑑 4𝜋𝑑 ∆𝐿 = cos(𝜃𝑐 ) = cos(𝜃𝑐 ) 𝜆0 𝜆0 𝜆𝑐 𝛿 = 2𝜋

∆𝐿 𝜆0

𝑛𝑖 sin(𝜃𝑖 ) = 𝑛𝑐 sin(𝜃𝑐 ) = 𝑛𝑒 sin(𝜃𝑒 ) 𝜆0 = 𝑛𝑖 𝜆𝑖 = 𝑛𝑐 𝜆𝑐 = 𝑛𝑒 𝜆𝑒 (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 . 2𝜋∆𝐿 𝛿𝑟𝑒𝑠𝑜𝑛𝑎𝑛𝑐𝑒 = = 2𝜋𝑚 𝜆0 (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=

𝐼𝑒 𝐼𝑖

1 𝐼𝑖 = 𝑛𝑖 𝑌0 𝐸𝑖 𝐸𝑖∗ 2 𝐼𝑒 =

1 𝑛𝑒 𝑌0 𝐸𝑒 𝐸𝑒∗ 2 (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 (−

𝑖𝑘𝑐 𝑑 ) [1 + ∑(𝜌𝑏+ 𝜌𝑎− )𝑛 exp(−𝑖𝑛𝛿)] cos(𝜃𝑐 ) 𝑛

𝐸𝑒 =

𝑖𝑘𝑐 𝑑 ) cos(𝜃𝑐 ) 1 − (𝜌𝑏+ 𝜌𝑎− ) exp(−𝑖𝛿)

𝜏𝑏+ (𝜏𝑎+ 𝐸𝑖 ) exp (−

(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, 𝐸𝑒 𝐸𝑒∗ 𝐸𝑒 𝐸𝑒∗

=

(𝜏𝑏+ 𝜏𝑎+ )(𝜏𝑏+ 𝜏𝑎+ )∗ 𝐸𝑖 𝐸𝑖∗ = (1 − 𝜌𝑏+ 𝜌𝑎− exp(−𝑖𝛿))(1 − 𝜌𝑏+ 𝜌𝑎− exp(−𝑖𝛿))∗ |𝜏𝑎+ |2 |𝜏𝑏+ |2 𝐸𝑖 𝐸𝑖∗

[1 − |𝜌𝑏+ ||𝜌𝑎− | exp(𝑖(𝜙𝑎 + 𝜙𝑏 − 𝛿))][1 − |𝜌𝑏+ ||𝜌𝑎− | exp(−𝑖(𝜙𝑎 + 𝜙𝑏 − 𝛿))] 𝐸𝑒 𝐸𝑒∗ = 𝐸𝑒 𝐸𝑒∗

=

|𝜏𝑎+ |2 |𝜏𝑏+ |2 𝐸𝑖 𝐸𝑖∗ 2

1 + |𝜌𝑎− |2 |𝜌𝑏+ | − 2|𝜌𝑎− ||𝜌𝑏+ | cos(𝜙𝑎 + 𝜙𝑏 − 𝛿) |𝜏𝑎+ |2 |𝜏𝑏+ |2 𝐸𝑖 𝐸𝑖∗ 2

1 + |𝜌𝑎− |2 |𝜌𝑏+ | − 2|𝜌𝑎− ||𝜌𝑏+ | [1 + cos(𝜙𝑎 + 𝜙𝑏 − 𝛿) − 1] |𝜏𝑎+ |2 |𝜏𝑏+ |2 𝐸𝑖 𝐸𝑖∗

𝐸𝑒 𝐸𝑒∗ =

4|𝜌𝑎− ||𝜌𝑏+ |

2

(1 − |𝜌𝑎− ||𝜌𝑏+ |) [1 +

(1 − |𝜌𝑎− ||𝜌𝑏+ |)

2 sin

2 𝜙𝑎

+ 𝜙𝑏 − 𝛿 ] 2 (18)

Then the total left to right transmittance is, 𝑛𝑒 𝑇=( ) 𝑛𝑖

|𝜏𝑎+ |2 |𝜏𝑏+ |2 2

(1 − |𝜌𝑎− ||𝜌𝑏+ |) [1 +

4|𝜌𝑎− ||𝜌𝑏+ | (1 − |𝜌𝑎− ||𝜌𝑏+ |)

2 sin

2 𝜙𝑎

+ 𝜙𝑏 − 𝛿 ] 2

𝑛𝑒 𝑇𝑎+ 𝑇𝑏+ 4√(𝑅𝑎− 𝑅𝑏+ ) 𝜙 + 𝜙𝑏 − 𝛿 −1 2 𝑎 𝑇=( ) [1 + ] 2 sin 𝑛𝑖 (1 − √(𝑅 − 𝑅+ ))2 2 (1 − √(𝑅 − 𝑅 + )) 𝑎

𝑏

𝑎

𝑏

(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, 𝑛𝑒 𝑇𝑎+ 𝑇𝑏+ 4√(𝑅𝑎− 𝑅𝑏+ ) 𝛿 −1 𝑇𝑚𝑎𝑥 2 𝑇=( ) [1 + sin ] = 2 2 𝛿 𝑛𝑖 (1 − √(𝑅 − 𝑅 + )) 2 (1 − √(𝑅𝑎− 𝑅𝑏+ )) 1 + 𝐹 sin2 2 𝑎 𝑏

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

𝑇𝑚𝑎𝑥

𝑛𝑒 𝑇𝑎+ 𝑇𝑏+ = 𝑛𝑖 (1 − √(𝑅 − 𝑅 + ))2 𝑎

𝐹=

𝑏

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, 1 1 sin2 𝛿(𝜆𝐶𝑝 ) = sin2 𝛿(𝜆0𝑝 ) = 0 2 2 1 1 4𝜋𝑛𝐶 𝑑 4𝜋𝑑 𝛿(𝜆𝐶𝑝 ) = 𝛿(𝜆0𝑝 ) = cos(𝜃𝐶 ) = cos(θC ) = mπ 2 2 2𝜆0𝑝,𝑚 2𝜆𝐶𝑝,𝑚 (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, 𝑇1,𝑚 = 2

𝑇𝑚𝑎𝑥,𝑚 = 2

𝑇𝑚𝑎𝑥,𝑚 1+

𝐹 sin2

𝛿 (𝜆1,𝑚 ) 2

2

𝛿 (𝜆1,𝑚 ) 2

2

=

4𝜋𝑛𝐶 𝑑 1 1 cos(𝜃𝐶 ) = 𝑚𝜋 + sin−1 (±√ ) = 𝑚𝜋 ± sin−1 (√ ) 2𝜆0,1/2 ,𝑚 𝐹 𝐹 𝜆0,1/2,𝑚 =

2𝜋𝑛𝐶 𝑑 cos(𝜃𝐶 ) 1 𝑚𝜋 ± sin−1 ( ) √𝐹

𝐹𝑊𝐻𝑀𝜆0 ,𝑚 = |𝜆0,1,+ − 𝜆0,1,− | = 2𝜋𝑛𝐶 𝑑 cos(𝜃𝐶 ) | 2

2

𝑚

𝐹𝑊𝐻𝑀𝜆0 ,𝑚 = 4𝜋𝑛𝐶 𝑑 cos(𝜃𝐶 ) ||

1 1 𝑚𝜋 + sin−1 ( ) √𝐹 1 sin−1 ( ) √𝐹

−

1

| 1 𝑚𝜋 − sin−1 ( ) √𝐹

1 2 𝑚2 𝜋 2 − (sin−1 ( )) √𝐹

||

(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, 4𝑛𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 𝑛𝑎𝑖𝑟 4𝑛𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 ⏊ = = 1 − 𝑅𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒−𝑎𝑖𝑟 <1 2 (𝑛𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 + 𝑛𝑎𝑖𝑟 ) (𝑛𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 + 1)2 𝑛𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 − 𝑛𝑎𝑖𝑟 2 𝑛𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 − 1 2 ⏊ 𝑅𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒−𝑎𝑖𝑟 =( ) =( ) <1 𝑛𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 + 𝑛𝑎𝑖𝑟 𝑛𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 + 1 (23)

⏊ 𝑇𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒−𝑎𝑖𝑟 =

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/metalbulk/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 𝑇𝑎+ 𝑇𝑏+ 4√(𝑅𝑎− 𝑅𝑏+ ) 𝜙 + 𝜙𝑏 − 𝛿 −1 2 𝑎 = [1 + ] 2 sin 𝑛𝑎𝑖𝑟 (1 − √(𝑅 − 𝑅 + ))2 2 + )) − (𝑅 (1 − √ 𝑎 𝑅𝑏 𝑎 𝑏 𝑇𝑀𝐷𝑀𝑐𝑎𝑣𝑖𝑡𝑦

𝜙𝑎 + 𝜙𝑏 − 𝛿 −1 = 𝑇𝑚𝑎𝑥 [1 + 𝐹 sin ] 2 2

(24) Where the subscript 𝑎 and 𝑏 designate the two interfaces (air-metal-dielectric, and dielectric-metal-substrate). Thus, the resonant condition becomes, 𝜙𝑎 + 𝜙𝑏 − 𝛿 =0 2 𝜙𝑎 + 𝜙𝑏 − 𝛿 Ф𝑎𝑏 − 𝛿 | |≡| | = 𝑚𝜋 2 2 4𝜋𝑛𝑐 𝑑 4𝜋𝑑 𝛿 = 𝛿(𝜆𝐶 , 𝜃𝐶 ) = 𝛿(𝜆0 , 𝜃𝐶 ) = cos(𝜃𝐶 ) = cos(𝜃𝐶 ) 𝜆0 𝜆𝐶 𝜙𝑎 = 𝜙𝑎 (𝜆0 , 𝑛𝐶 , 𝜃𝑖 ) 𝜙𝑏 = 𝜙𝑏 (𝜆0 , 𝑛𝐶 , 𝑛𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 , 𝜃𝐶 (𝑛𝐶 , 𝜃𝑖 )) sin2

(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 𝜙𝑎 + 𝜙𝑏 2𝜙𝑎 ≡ = = 𝜙𝑎 2 2 2 tan(𝜙𝑎 ) =

𝑛𝐶2

2𝑛𝑐 𝑘𝑚 2 − 𝑘2 − 𝑛𝑚 𝑚 (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 𝑛𝐶 𝑑. Given 𝜆, we still need to find 𝑛𝐶 𝑑. | |

Ф𝑎𝑏 − 𝛿(𝜆0𝑝,𝑚 ) Ф𝑎𝑏 − 𝛿(𝜆𝐶𝑝,𝑚 ) |=| | = 𝑚𝜋 2 2

Ф𝑎𝑏 − 𝛿(𝜆0𝑝,𝑚+1 ) Ф𝑎𝑏 − 𝛿(𝜆𝐶𝑝,𝑚+1 ) |=| | = (𝑚 + 1)𝜋 2 2 𝛿 = 𝛿(𝜆𝐶𝑝,𝑚 ) = 𝛿(𝜆0𝑝,𝑚 ) = |

1 𝜆0𝑝,𝑚

−

1 𝜆0𝑝,𝑚+1

4𝜋𝑛𝐶 𝑑 4𝜋𝑑 = 𝜆0𝑝,𝑚 𝜆𝐶𝑝,𝑚

|=

1 2𝑛𝐶 𝑑 (28)

Thus solving for Ф𝑎𝑏 comes down to the task of finding the product 𝑛𝐶 𝑑. 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 𝑛𝐶 𝑑 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 sin2 sin2

𝜙𝑎 + 𝜙𝑏 − 𝛿 −1 ] 2

𝜙𝑎 + 𝜙𝑏 − 𝛿 Ф𝑎𝑏 − 𝛿 = sin2 ( )≡0 2 2

Ф𝑎𝑏 − 𝛿 1 4𝜋𝑛𝐶 𝑑 1 4𝜋𝑛𝐶 𝑑 (1 ∓ ∆𝑚 )]| | | ≅ 𝑚𝜋 = | [Ф𝑎𝑏 − ]| ≅ | [Ф𝑎𝑏 − 2 2 𝜆0𝑝,𝑚 (1 ± ∆𝑚 ) 2 𝜆0𝑝,𝑚 Where in the last step I used the Binomial Approximation, since ∆𝑚 ≪ 1. Then, Ф𝑎𝑏 − 𝛿 2𝜋𝑛𝐶 𝑑 | | ≅ 𝑚𝜋 ± ∆ 2 𝜆0𝑝,𝑚 𝑚 Ф𝑎𝑏 − 𝛿 Ф𝑎𝑏 − 𝛿 2𝜋𝑛𝐶 𝑑 2𝜋𝑛𝐶 𝑑 sin2 ( ) = sin2 | | ≅ sin2 (𝑚𝜋 ± ∆𝑚 ) ≅ sin2 ( ∆ ) 2 2 𝜆0𝑝,𝑚 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 𝑛𝐶 𝑑, 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.