Optical Basic Concepts

Optical thin films are primarily the study of interference between the multiple reflections of light from optical interfaces between two different media. Some principles and terminology are drawn from the basics of geometrical optics.

Index of Refraction \( n \):

\[ n \equiv \frac{c}{v}, \quad v = \frac{c}{n}, \quad c = 2.99792458 \times 10^8 \text{ m/s} \] where \( c \) is the speed of light in vacuum, and \( v \) is the speed of light in a medium.

Wavelength (λ) and frequency (ν):

\[ \lambda = \frac{v}{\nu}, \text{in vacuum:} \lambda = \frac{c}{\nu} \]

Wave Number (\(\sigma\)):

The wave number (\(\sigma\)) is defined as the number of wavelengths per centimeter:

\[
\sigma = \frac{1}{\lambda}
\]

where \(\sigma\) has units of cm\(^{-1}\).

Optical Path Difference (OPD)

The optical path difference (OPD) is proportional to the time required for light to travel between two points. In a homogeneous medium, it is given by:

\[
OPD = n d
\]

where:
– \(n\) is the refractive index of the medium,
– \(d\) is the physical distance between the two points.

Snell’s Law of Refraction

Snell’s law describes the relationship between the angles of incidence and refraction when light passes between two media with different refractive indices:

\[
n_1 \sin \theta_1 = n_2 \sin \theta_2
\]

where:
– \(n_1\), \(n_2\) are the refractive indices of the two media,
– \(θ_1\), \(θ_2\) are the angles of incidence and refraction, respectively.

The incident ray, refracted ray, and the surface normal all lie in the same plane. Additionally, when light propagates through a series of parallel interfaces, the quantity \(n\sin θ\) remains conserved.

Internal Angles in Thin Films

In a stack of plane parallel thin-film surfaces of various indices, the angle of a ray within a given film can be calculated using Snell’s law. Because the law applies at each interface and follows from the previous interface, the internal angle in the film is independent of the order of the films or where that film is within the stack:

\[n_1\sin\theta_1=n_2\sin\theta_2=n_3\sin\theta_3=n_4\sin\theta_4\]

The emmerging ray from the stack refracts back to an index of 1.0 and is parallel to the entering ray because the index is the same.

Optical coatings on lenses are not strictly plane parallel thin-film surfaces, but on the scale of the thickness of these films(~ 1 μm), the approximation is valid.

On a more microscopic scale (~ 1 nm), the surfaces of real coatings are not usually very smooth or flat. However, at wavelengths of one or two orders of magnitude greater than that, the approximation of “flat” is still valid.

In the deep ultraviolet (UV) or x-ray region, these things can be of concern because of the shorter wavelengths.

Reflection

Law of reflection:

\[\theta_1=-\theta_2\]

The incident ray, the reflected ray, and the surface normal are coplanar.

Total internal reflection (TIR) occurs when the angle of incidence of a ray propagating from a higher-index medium to a lower-index medium exceeds the critical angle :

\[\sin\theta_\text{c}=n_2/n_1\]

At the critical angle, the angle of refraction \(\theta_2\) equals 90°.

The reflectance amplitude \(r\) of an interface between \(n_1\) and \(n_2\) at normal incidence (\(\theta_1\) = 0), with no absorption, is given by the Fresnel reflection equation:

\[r=(n_1-n_2)/(n_1+n_2)\]

When absorption is included:

\[N_1=n_1-ik_1\]

where

\[r(\text{complex})=(N_1-N_2)/(N_1+N_2)\]

When the angle of incidence (AOI) \(\theta_1\) is not \(0^\circ\), the \(s-\) and \(p-\)polarizations have different effective indices:

\[N_S=n\times\cos\theta\]

\[N_P=n/\cos\theta\]

In these cases

\[r_S=(N_{1S}-N_{2S})/(N_{1S}+N_{2S})\]

and

\[r_P=(N_{1P}-N_{2P})/(N_{1P}+N_{2P})\]

Normally, thin-film calculation/design software takes care of all of these details.

Reflections

Reflectance intensity \(R\) is what is measured with a photometer. It is the product of the reflectance amplitude \(r\) and its complex conjugate \(r^*\):

\[R=rr^*\]

Each interface between two media of differing index has Fresnel reflection. As seen in the figure, a ray falling on the first surface has \(r_1\) reflected from it. The transmitted part has \(r_2\) reflected when it falls on the second surface of the shaded medium. Part of this second reflection is reflected when it falls back on the first surface. Part of that reflection is reflected when it falls again on the second surface, etc.

When all of the reflections from the first and second surfaces of a thin film are considered, the resulting reflectance \(r\) is rigorously given by

\[r=(r_1+r_2e^{-i\varphi})/(1+r_1r_2e^{-i\varphi})\]

Here, \(e^{-i\varphi}\) is the complex phase factor that accounts for the phase difference between \(r_1\) and \(r_2\) caused by the optical thickness of the film:

\[e^{-i\varphi}=\cos\varphi-i\sin\varphi\]

These equations properly account for the multiple internal reflections in the medium.

Example Reflection Calculations

A soap buble might have an index of reflection of 1.4. The reflections in air (\(n=1.0\) from the first and second surfaces would be

\[r_1=(1-1.4)/(1+1.4)=-1/6\]

\[r_2=(1.4-1)/(1.4+1)=+1/6\]

If the bubble is infinitesimally thin, the two reflections cancel each other:

\[r_1+r_2=-1/6+1/6=0.0\]

where \(\varphi=0\), so that

\[r=(r_1+r_2*1)/(1+r_1r_2*1)=0.0\]

If the thin film of the bubble were one quarter-wave optical thickness (QWOT=\(\lambda/4)\)=\(nd\)) at the wavelength under consideration, the path of the ray from the first surface to the second and back to the first would be one half wavelength. In this case \(\varphi=180^\circ\). Here, the two reflections would add to each other for maximum effect:

\[r=[r_1+r_2*(-1)]/[1+r_1r_2*(-1)]=-0.3243\]

\[r\approx{r_1+r_2}\approx{-1/6+(-)1/6}\approx{-1/3}\]

The reflectance \(R\) would be \(rr^*\) or ~\(1/9\), which is ~\(11\%\). The appearance of the relfection when the bubble has a thickness of one QWOT would be white. As the bubble became thinner, its appearance would change toward no reflectance or “black.”

In the normal case, a bubble has a thickness of many QWOTs. For a given physical thickness, there are more QWOTs in blue light (short wavelength) than in the red (long wavelength). Therefore, the blue and red “rays” are at different phases and therefore have different reflectance. This causes the rainbow that we see in soap bubbles.

Reflectance as Vector Addition

The case of the soap bubble is illustrated from the viewpoint of vectors.

When \(\varphi=0\), \(r_1=-1/6\) and \(r_2=+1/6\).

When \(\varphi=90^\circ\):

When \(\varphi=180^\circ\) (one QWOT):

When \(\varphi=360^\circ\) (two QWOTs or one half-wave OT):

Reflectance Amplitude Diagram

The reflectance amplitude diagram , often referred to as a circle diagram, follows from the foregoing vector diagrams.

The outermost circle represents \(r=1.0\) and also \(R=rr^*=1.0\) or \(R=100\%\) reflectance. The origin of the real and imaginary axes is where \(r=0.0\) or zero reflectance.

Any new layer starts from the point of the reflectance amplitude of whatever lies beneath it, whether that is a substrate or a stack of coating layers on a substrate. This point is represented by point \(A\) in the figure, where the starting reflectance amplitude is \(r_A\) and its phase is \(\varphi_A\).

After the addition of a given physical thickness (PT=\(d\)) and optical thickness (OT) of a layer of given index, the resulting reflectance reaches point \(B\). The new reflectance amplitude is \(r_B\) and its phase is \(\varphi_B\). Point B would then be the starting reflectance for the next layer.