**RoboMirror and One-way/Two-way Mirror** Nkiruka Uzuegbunam Studing Transmission of Light through a One-way (Two-way) Mirror =============================================================================== This post investigates vision through a partially hazing in indoor scenes. In particular, as part of my thesis, I am studying the transmission of light through one-way/two-way mirrors. Please comment, send corrections, ask questions, or as always, cite appropriately. [Click here](../index.html) to navigate back to home page. What are One-way/Two-way Mirrors? ------------------------------------------------------------------------------- One-way/Two-way mirrors (TWMs) are partially reflective and partially transparent constructions. Typically, TWMs are composed of a transparent medium, and a reflective medium. Glass, acrylic or polycarbonate subtrate is used as the transparent medium. The reflective medium is usually a metal such as aluminium or silver. What are One-way/Two-way Mirrors Used for? ------------------------------------------------------------------------------- One-way (often called Two-way) mirrors are used for security, observation, teleprompter, windows, e.t.c. They are designed to provide privacy in various situations. How do One-way/Two-way Mirrors work? ------------------------------------------------------------------------------- A normal mirror, here I assume a plane mirror, is most commonly, essentially a plate of glass, coated with a reflective coating, most often silver or aluminium. The metal coat allows the mirror to reflect most of the light that hits it, creating the reflection we see in the mirror. ![Figure [mirror1]: Metal coating on a mirror reflects most of the light hitting the mirror, creating reflections. Picture obtained from Wikipedia, https://upload.wikimedia.org/wikipedia/commons/5/52/Mirror.jpg](../assets/Mirror.jpg) TWMs are generally designed to allow light to pass equally in both directions. This is by coating a transparent medium with a very thin, almost transparent layer of metal. The sparse distribution of the metal molecules ensures that some light is reflected, while the rest pass through the TWM. TWM will appear like a plane mirror when one side of the mirror is *much more* brightly lit than the other side. Most TWM vendors state that the reflective side (i.e. the metal coated side) should face the brightly lit room, while the transparent layer faces the darker room. The reason for this can be seen in the figure below. Assume a 2-room scenerio. Room A is brightly lit. Room B is dark, with very little light. The TWM is mounted in a gap on the wall connecting Room A to Room B. The reflective side of the TWM faces into room A. The transparent side of the TWM faces into room B. ![Figure [mirror2]: Essential operation of a TWM. Picture obtained from TWM Vendor site: http://www.twowaymirrors.com/](../assets/how-does-a-two-way-mirror-work.jpg) Recall, in physics class that when light hits an object, it is either reflected, refracted (i.e. transmitted), absorbed or does some combination of the three actions. As such, when light from room A that hits the reflective side of the TWM, most of it is reflected back into room A creating a reflection in the mirror. A large portion of the remaining light is refracted and tranmitted through the TWM into room B, to the user facing the transparent medium in room B. The light in room B is *much much less* than the light in room A. As such, the transmitted light into room B, from room A, is also much greater than any amount of light sent from room B to room A. Recall in image processing class, that we are able to see objects only because light reflects off the objects into our eyes. Hence, a user in room B, can see clearly into room A, without being distracted by any light coming from room B. Math of One-way/Two-way Mirrors for a Few scenarios ------------------------------------------------------------------------------- As stated, research into vendor design of TWMs shows that most are designed with transparent medium and a reflective medium. The transparent medium is most commonly glass or acrylic or polycarbonate subtrate. The reflective medium is most commonly a semi-transparent, very thin, layer of metal like aluminium (Al) or silver (Ag). The refractive indices for the materials are shown in the table below. Refractive index of a material is dimensionless, given by the ratio of the speed of light in a vacuum to the speed of light in the material: $ n = $ refractive index $ = \frac{c}{v} $ where c = speed of light in vacuum $\approx$ 3.0 x $10^8$ m/s, v = speed of light in medium. The refractive index is used in the calculation of the reflection coefficient, and hence the reflectance of the materials. Reflectance is the ratio of reflected light power to incident/incoming light power. For non-magnetic media, at normal incidence (light hits perpendicular to the medium, 0 degrees from the normal) it is defined as: $ R = $ reflectance $ = \left(\frac{n_{1} - n_{2}}{n_{1} + n_{2}}\right)^2 $ where $n_i$ is the refractive index of medium i. Material | Refractive Index | ---------| ----------------- Glass | 1.50 Acrylic | 1.490 Polycarbonate | 1.585 Silver | 0.151 Aluminium| 1.097 Air | 1.0 [Table [refractiveIndex]: Refractive indices of Common TWM materials. Numbers obtained from http://refractiveindex.info/. Any general optics text should have similar tables] Using the reflectance, we can verify the optical properties of the TWM. Consider the scenerios depicted below. Label, as indicated, the light entering the TWM, passing from air to metal, metal to glass, then back to glass. ![Figure [mirror3]: General Scenario. Light from the bulb hits the book. A portion of this light is reflected to the TWM, at an angle $\theta_{1}$ to the normal N. The TWM transmits a portion of this light. Clip art of table, books, and bulb obtained from openclipart.org](../assets/opticsMirror1.png) ![Figure [mirror4]: Optics at boundary of TWM. Input light power from book of radiance L is transmitted through the metal. The transmitted radiance is decreased to $L_{1}$. Passage from the metal to glass further decreases radiance to $L_{2}$. Passage from glass back to air results in final radiance $L_{3}$](../assets/opticsMirror2.png) Let us calculate the transmitted radiance in Scenario 1 (S.1) shown in Figure [mirror4]. Assume normal incidence, i.e. light strikes along normal N meaning $\theta_{1}$ shown in Figure [mirror3] is 0. Also, assume the reflective metal coating is silver, Ag, while the transparent medium is glass.. Using the refractive indices in Table [refractiveIndex], we can calculate the reflectance as follows: $R_{air}^{Ag} = \left(\frac{n_{air} - n_{Ag}}{n_{air} + n_{Ag}}\right)^2 = \left(\frac{1 - 0.151}{1 + 0.151}\right)^2 \approx 0.74 \Rightarrow 74$% of light L is reflected, leaving $\approx$ 26% to pass through. Hence, $L_{1} = 0.26L$. Repeating this for the remaining scenarios, we have: $R_{Ag}^{glass} = \left(\frac{n_{Ag} - n_{glass}}{n_{Ag} + n_{glass}}\right)^2 = \left(\frac{0.151 - 1.5}{0.151 + 1.5}\right)^2 \approx 0.67$. This means that $\Rightarrow 67$% of $L_{1}$ is reflected. Thus, $L_{2}= (1-0.67) * 0.26 * L \approx 0.0858L$ is transmitted. $R_{glass}^{air} = \left(\frac{n_{glass} - n_{air}}{n_{glass} + n_{air}}\right)^2 = \left(\frac{1.5 - 1.0}{1.5 + 1.0}\right)^2 \approx 0.04$. This means that $\Rightarrow 4$% of $L_{2}$ is reflected. Thus, $L_{3}= (1-0.04) * 0.0858 * L \approx 0.0824L$ is transmitted. Hence, only about 8% of input radiance is transmitted. Then, assuming pure reflection at each boundary (no loss of light from absorption, or other scattering), this means that $\approx$ 92% of the input light is reflected by the TWM, with the bulk of the reflection occuring at the boundary of air and metal. *Note: I noticed the same result happens if we go the other direction. That is, light moves from air to glass to metal to air. Some vendors claim that the TMW must be setup in a particular direction for effective privacy. This may be a result of their production process or for aesthetics or other subjective viewing purposes. This may also be due to the metal providing the most direct reflection. As light does not need to be bent as it passes from metal to glass to air, we get the most undistorted reflection occuring at the boundary of metal and air.* **Privacy via TWM** The TWM achieves privacy when the amount of light on the reflecting side is *much much* larger than the light on the transparent side. Using our calculations above, assume a result of 91% reflectance, with 9% transmission. I did the calculations a little different in my notes but the method is the same. With the figure below, assume the light $L^{A} \gg L^{B}$ such that $L^{A} > 100*L_{B}$. Two things happen in this setup: * The light reaching room B, composed of reflection from room B, and transmittance from room A is essentially $L_{B} = 0.09*L_{A} + 0.91*L_{B} = 0.09*100*L_{B} + 0.91 * L_{B} \approx 9*L_{B}$. The light transmitted from A to room B is much larger than reflections from B. Hence, light transmitted from A dominates, and person in room B can observe what is room A. * The light reaching room A, composed of reflection from room B, and transmittance form B is essentially $L_{A} = 0.91*L_{A} + 0.09*L_{B} \approx 0.91*L_{A}$. The reflection from room A back into room A is much larger than transmittance from room B. Hence, reflections from A dominate, and the person in room A can only see their reflections in the TWM in room A. ![Figure [mirror5]: Reflection and transmittance of radiance L on both sides of the TWM](../assets/opticsMirror3_1.png) Application of One-way/Two-way Mirrors for Therapy ------------------------------------------------------------------------------- In my project, RoboMirror, a setup for Augmented/Mixed Reality designed for face-focused interactions, we use a TWM and a camera mounted on a robotic arm (see figure below). The robotic arm follows the user in front of the TWM, and the image from the camera is piped back to the user, and displayed on the TWM. Hence, the user sees a modified reflection of themselves rather than thier pure reflection. This is designed to be used for therapeutic activities, mixed or augmented reality since the camera image can be modified any which way before being sent to the TWM for display. Summary ------------------------------------------------------------------------------- In this work, we have explained how a TWM works. A TWM, designed for privacy, reflects most of the light hitting it, while allowing others to pass through. It is composed of a transparent medium, often glass, acrylic or some transparent plastic, coated with a very thin layer of a reflective medium, often silver or aluminium. A TWM acts like a mirror when the reflective side is much more brightly lit than the transparent side. In my next post, I will further explain the robotic mirror, my intended usage of the TWM. [Click here](../index.html) to navigate back to home page.