# Kingdom of multilayer mirrors

Today we will get acquainted with multilayer mirrors, find out why they are needed and how they are modeled using the method of transfer matrices.

An ordinary bathroom mirror (and its better counterparts) is nothing more than a thin smooth metal film. When reflected from it, about five percent of the light is lost. Sometimes this is critical - say, in telecom (the less signal we lose, the less we put intermediate amplifiers) or in complex optics such as periscopes (if you lose 5% on each mirror, very little will reach the observer).

On the other hand, the absorption of light on a mirror causes it to heat up. And if 5% of the laser pointer is something ephemeral, then 5% of the industrial laser for cutting is about a watt, which can noticeably heat a thin film. It is still more interesting with pulsed lasers, where the peak power is sufficient to melt the mirror. Something like this:

Probably, many have heard about the enlightenment of optics. This is a tricky coating of lenses, which allows to reduce the reflection from their front surface to almost zero. That is, the light will not be wasted on reflection, but will go completely into the optical system. Physically, this occurs due to

The same idea can be used in the opposite direction: so that the reflection does not weaken, but rather intensifies. We will need a layer cake from two different materials, each a quarter wavelength thick. At each of the junctions of the two materials, part of the light is reflected back. If all reflections that come out will have the same phase,

Such a “layer cake” is called a

Let's take a closer look at the reflection scheme. An incident beam is reflected back at each of the boundaries of two media. True, each reflected ray is also reflected on each of the boundaries that it passes along the way back - now in the other direction. Each of these rays is reflected once again ... hell, how do you count them all?

The

Generally speaking, there are two waves at each point: one propagates to the right (let's call it

For any two points, the vectors will be connected by a certain linear expression that takes into account the propagation of light through the medium and through the boundaries of two media. This expression can be written using matrices. Basically, we need two matrices. The first (let's call it M

Combining M

That is, we simply multiply the matrices in a row for everything that the light passes through (it goes from left to right):

- the boundary of air and 1 layer (entrance to the mirror)

- distribution in the first layer

- the border of 1-2 layers

- the distribution in the second layer

- the border of 2-3 layers

and so on until you exit the mirror. And we get one 2x2 matrix that describes the entire mirror at once!

It was not by chance that I chose points on both sides of the mirror. At point A, the upper component of the vector (which flies to the right) is the wave that we send to the mirror. The bottom component is the reflection that we want to calculate. At point B (immediately behind the mirror), the top component is the transmission of the mirror, and the bottom is zero because nothing falls onto the mirror from the back. The result is an elementary matrix equation

where r and t are the reflection and transmission, respectively. From here, not forgetting that the intensity is the square of the amplitude, we get the reflection

In conclusion, we note that for other wavelengths, the layers will have a thickness different from a quarter of the wavelength, so the reflection coefficient may change. To find out in what spectral range the mirror will reflect, it is necessary to repeat the calculation for different wavelengths.

The code is straightforward. For each wavelength, you need to calculate M

1. M

A problem arises if, starting from a certain wavelength, the material begins to absorb light (say, semiconductors behave this way). Usually, for the wavelength at which absorption begins, there is no good tabulated data at all; and there is a gap between the values to the left and to the right of it. In this case, the regions to the left and right of the “bad” point are interpolated separately, and the point itself is ignored.

2. If the mirror consists of N pairs of layers, then instead of calculating M

1.~~not at~~ all a mirror, but just a useless piece of glass. At the maximum, the reflection is approximately equal to 99.97%.

2.

3. The

4.

Of course, first of all, the transfer matrix method is needed by the corresponding R&D. Laser diodes, fiber lasers, mirrors (ranging from terahertz to soft X-rays), narrow-band filters and even optical illumination are considered to be just them.

Further, we saw that interference is very sensitive to the thickness of the layers. In principle, if we approximately know the composition of a certain layered structure, we can determine the thickness of its layers with an accuracy of the order of a nanometer. To do this, measure its reflection spectrum and adjust it by varying the layer thicknesses in the algorithm. It turns out such a system for reverse engineering of layered structures. Moreover, its cost is orders of magnitude less than the cost of an electron microscope with the same resolution.

Another application follows from here - feedback on the production of mirrors. The reflection spectrum of the manufactured mirror is easy to measure and compare with the theoretical. If the differences are significant, modeling can show what exactly went wrong in the process. Moreover, feedback can be obtained in real time during the manufacturing process: a light shines on the sprayed mirror, and the measured reflection spectrum is displayed on the screen.

The formulas above describe normal reflection (i.e., perpendicular to the mirror). In reality, you often need mirrors that reflect at an angle. For such calculations, the algorithm is a little complicated: you have to add another cycle for different values of the angles.

It is slightly more difficult to calculate a concave or convex mirror: it is necessary to separately consider different parts of the surface. Usually in such a situation one has to sacrifice something: the spectrum, the angles of reflection, the polarization properties. This task is often trusted in a genetic algorithm tuned to maximize the required parameters. Let's say you can make a mirror that reflects light from all angles, but the spectrum and quality will not be the best. Or make a mirror with a reflection of about 99.999% - but only for one wavelength and at one angle.

A small plus is that it is not necessary to use a periodic structure: the thickness of the layers can vary as you like (such a mirror is called

M. Bourne, E. Wolf, "Fundamentals of Optics."

The dxdy forum has a good post about Chebyshev polynomials.

Pictures: KDPV , 1 , 2 , 3 .

### What is wrong with ordinary mirrors?

An ordinary bathroom mirror (and its better counterparts) is nothing more than a thin smooth metal film. When reflected from it, about five percent of the light is lost. Sometimes this is critical - say, in telecom (the less signal we lose, the less we put intermediate amplifiers) or in complex optics such as periscopes (if you lose 5% on each mirror, very little will reach the observer).

On the other hand, the absorption of light on a mirror causes it to heat up. And if 5% of the laser pointer is something ephemeral, then 5% of the industrial laser for cutting is about a watt, which can noticeably heat a thin film. It is still more interesting with pulsed lasers, where the peak power is sufficient to melt the mirror. Something like this:

### A bit of physics

Probably, many have heard about the enlightenment of optics. This is a tricky coating of lenses, which allows to reduce the reflection from their front surface to almost zero. That is, the light will not be wasted on reflection, but will go completely into the optical system. Physically, this occurs due to

**destructive interference**from different layers of the antireflection coating.*Anti-reflective binoculars.*The same idea can be used in the opposite direction: so that the reflection does not weaken, but rather intensifies. We will need a layer cake from two different materials, each a quarter wavelength thick. At each of the junctions of the two materials, part of the light is reflected back. If all reflections that come out will have the same phase,

**constructive interference**will occur.and the reflected signal will have the highest possible intensity.Such a “layer cake” is called a

**dielectric mirror**, a**multilayer mirror**, or a**distributed Bragg reflector**(in English**distributed Bragg reflector**,**DBR**). It is “distributed” because the reflection does not occur on one surface, but on several at once. The reflection coefficient can easily reach 99.99%, which means that compared with metal mirrors, losses are reduced by 2-3 orders of magnitude.### A bit of math

Let's take a closer look at the reflection scheme. An incident beam is reflected back at each of the boundaries of two media. True, each reflected ray is also reflected on each of the boundaries that it passes along the way back - now in the other direction. Each of these rays is reflected once again ... hell, how do you count them all?

The

**transfer matrix**method comes to the rescue . Its essence is that we stop monitoring each ray separately, and look only at the result of their addition. That is, we define a vector that describes the wave at each point.Generally speaking, there are two waves at each point: one propagates to the right (let's call it

*E*), the second to the left (^{R}*E*) Then our vector has two elements, which are complex: after all, we are interested in both the amplitude of the light (the^{L}__absolute__value of the number) and its phase (respectively, the__phase of the__number).For any two points, the vectors will be connected by a certain linear expression that takes into account the propagation of light through the medium and through the boundaries of two media. This expression can be written using matrices. Basically, we need two matrices. The first (let's call it M

_{1}) connects the vectors to the left and to the right of the interface. The second (M_{2}) describes the propagation of a wave in a homogeneous medium (between interfaces).**How it all comes out**

Let's start with M

where n

Combining the expressions, we get the

Matrix M

where L is the layer thickness, lambda is the wavelength.

_{1}. Imagine that light is incident on the interface between two media on both sides. For each of the rays, we can calculate the percentage of reflection and refraction, and then add the results. Expressions are derived from Fresnel formulas. If light fell only on the left (E_{1 }^{R}), then reflection and refraction would be given by the formulaswhere n

_{1}and n_{2}are the refractive indices of two media. If the light fell only on the right (E_{2 }^{L}), we would have accordinglyCombining the expressions, we get the

Matrix M

_{2}It looks much simpler. Since the rays flying left and right do not interact, the matrix is diagonal. The amplitude of the light does not change - that means the modulus of the matrix elements is 1. Only the phase changes: it increases for a ray flying to the right, and decreases for a flying to the left (minus sign in exponential). It turns outwhere L is the layer thickness, lambda is the wavelength.

Combining M

_{1}and M_{2}, we can connect two vectors at any point in space. For example, between the points A and B in the figure below, the dependence will be as follows:That is, we simply multiply the matrices in a row for everything that the light passes through (it goes from left to right):

- the boundary of air and 1 layer (entrance to the mirror)

- distribution in the first layer

- the border of 1-2 layers

- the distribution in the second layer

- the border of 2-3 layers

and so on until you exit the mirror. And we get one 2x2 matrix that describes the entire mirror at once!

It was not by chance that I chose points on both sides of the mirror. At point A, the upper component of the vector (which flies to the right) is the wave that we send to the mirror. The bottom component is the reflection that we want to calculate. At point B (immediately behind the mirror), the top component is the transmission of the mirror, and the bottom is zero because nothing falls onto the mirror from the back. The result is an elementary matrix equation

where r and t are the reflection and transmission, respectively. From here, not forgetting that the intensity is the square of the amplitude, we get the reflection

In conclusion, we note that for other wavelengths, the layers will have a thickness different from a quarter of the wavelength, so the reflection coefficient may change. To find out in what spectral range the mirror will reflect, it is necessary to repeat the calculation for different wavelengths.

### Code and a bit of optimization

The code is straightforward. For each wavelength, you need to calculate M

_{1}and M_{2}, and then multiply them the desired number of times. Since the results for different wavelengths are independent, the calculations parallel well on multi-core processors. The code on which the examples below were considered is written in MATLAB. I will mention a few subtleties.1. M

_{1}and M_{2}for different wavelengths are different due to different refractive indices (this is called the word**dispersion**). Typically, the refractive index values are tabulated and change quite smoothly, so they are well interpolated by the polynomial.A problem arises if, starting from a certain wavelength, the material begins to absorb light (say, semiconductors behave this way). Usually, for the wavelength at which absorption begins, there is no good tabulated data at all; and there is a gap between the values to the left and to the right of it. In this case, the regions to the left and right of the “bad” point are interpolated separately, and the point itself is ignored.

2. If the mirror consists of N pairs of layers, then instead of calculating M

_{1}and M_{2}for each layer, you can calculate them once for a pair of layers, and then multiply them N times in the desired order. In other words, first calculate the transfer matrix for a pair of layers, and then raise it to the Nth power.**Chebyshev polynomials**can greatly help in this .**Power raising using Chebyshev polynomials.**

Let M be a 2x2 unimodular matrix (i.e., its determinant is 1 or -1). Then,

where

,

a

are the Chebyshev polynomials of the second kind.

where

,

a

are the Chebyshev polynomials of the second kind.

### Interesting results

1.

**Typical reflection spectrum.**A simple example is a mirror of alternating layers of TiO_{2}and SiO_{2}, each 10 times. Reflection reaches a maximum in a certain range: in the picture - from 420 to 600 nm, that is, our mirror works in the blue-green region. Outside the operating range, reflection jumps from zero to small values; in these areas, the mirror is2.

**More layers - better reflection.**By the way, it is customary to consider not layers, but their pairs. In the picture below, the red spectrum is 5 pairs of TiO_{2}/ SiO_{2}blue - 10 pairs In practice, do not use too many pairs, because it increases the production time. Approximate numbers - 5-7 pairs for mirrors in conventional laser diodes and fiber lasers; 20-30 for very specific applications such as quantum optics.3. The

**contrast**or difference in refractive indices of the two materials. The larger it is, the less pairs are needed for a mirror of the same quality. In the picture below, the spectra of mirrors of 10 pairs of TiO_{2}/ SiO_{2}(blue) and ZrO / SiO_{2}(lilac). The latter has a smaller refractive index difference, therefore, the maximum reflection is 99.24% (versus 99.97% for TiO_{2}/ SiO_{2}) - in other words, the losses in the ZrO / SiO_{2}mirror are 25 times greater.4.

**Precision of manufacturing.**The layers are extremely thin (0.1-0.2 microns), and small deviations noticeably affect the quality. For reproducibility of the spectrum, it is critical to track not the thickness of each layer, but the thickness of the pair. Let's see what happens with our mirror of 10 pairs of TiO_{2}/ SiO_{2}(blue spectrum) for various manufacturing errors. If all layers of one material will be 10% thicker, and the second - 10% thinner (green), then the thickness of the pair will remain unchanged and the quality of the mirror will change slightly. At the same time, the deviation of only one layer by 5% changes the thickness of the pair and noticeably shifts the spectrum (red curve).### Where is it needed

Of course, first of all, the transfer matrix method is needed by the corresponding R&D. Laser diodes, fiber lasers, mirrors (ranging from terahertz to soft X-rays), narrow-band filters and even optical illumination are considered to be just them.

Further, we saw that interference is very sensitive to the thickness of the layers. In principle, if we approximately know the composition of a certain layered structure, we can determine the thickness of its layers with an accuracy of the order of a nanometer. To do this, measure its reflection spectrum and adjust it by varying the layer thicknesses in the algorithm. It turns out such a system for reverse engineering of layered structures. Moreover, its cost is orders of magnitude less than the cost of an electron microscope with the same resolution.

Another application follows from here - feedback on the production of mirrors. The reflection spectrum of the manufactured mirror is easy to measure and compare with the theoretical. If the differences are significant, modeling can show what exactly went wrong in the process. Moreover, feedback can be obtained in real time during the manufacturing process: a light shines on the sprayed mirror, and the measured reflection spectrum is displayed on the screen.

### What's next

The formulas above describe normal reflection (i.e., perpendicular to the mirror). In reality, you often need mirrors that reflect at an angle. For such calculations, the algorithm is a little complicated: you have to add another cycle for different values of the angles.

It is slightly more difficult to calculate a concave or convex mirror: it is necessary to separately consider different parts of the surface. Usually in such a situation one has to sacrifice something: the spectrum, the angles of reflection, the polarization properties. This task is often trusted in a genetic algorithm tuned to maximize the required parameters. Let's say you can make a mirror that reflects light from all angles, but the spectrum and quality will not be the best. Or make a mirror with a reflection of about 99.999% - but only for one wavelength and at one angle.

A small plus is that it is not necessary to use a periodic structure: the thickness of the layers can vary as you like (such a mirror is called

**aperiodic**). You can immediately vary a dozen thicknesses - what a vastness for the genetic algorithm! This is how mirrors for x-ray lithography, which is used in modern technological processes in microelectronics, are calculated.### Sources

M. Bourne, E. Wolf, "Fundamentals of Optics."

The dxdy forum has a good post about Chebyshev polynomials.

Pictures: KDPV , 1 , 2 , 3 .