1. Dielectric mirror manufacture High quality dielectric mirrors manufacture is based on the following theoretical considerations. On the surface of the glass (for example, quartz), with refractive index n s, transparent (without absorption) dielectric layer with a refractive index n> n s is deposited. The layer thickness d is chosen so that its optical thickness n d is equal to 0 /4, i.e., a quarter wavelength in a vacuum. Here 0 is a selected (fixed) wavelength. This increases the reflective index of the surface. Really, the waves reflected from the front and back borders of the layer are in phase, as the phase lag of the second wave in π radians, accumulated during its propagation inside the film back and forth, changing the phase offset of the first wave by π radians when reflected from optically more dense medium. In other words, in the front surface of the layer the wave phase jump occurs and the same π radians phase change gives the optical path difference of rays. So, reflective and refractive waves are superimposed amplified each other. Theoretical consideration 2 n0n0 nBnB nSnS 1 3 Fig. 5.1. Let the light falls from the air (medium 1 in Fig. 5.1) along the normal to the surface of the glass (medium 3) with deposited thin film (medium 2) on it. By adding the wave amplitude reflected from the outer surface of the film with wave amplitudes coming from the film after multiple reflections, we get the formula for the total amplitude reflection coefficient of RI:

2. Here, R 12 and R 23 - the amplitude reflection coefficients on the borders of the media 1-2 and 2-3, respectively, γ - phase shift, taking into account the phase jumps at the border. Fig. 5.2 shows a graph of the total (5.2)(5.2) power reflection coefficient of the "air (n 0 = 1) - film (n B ) - glass (n s )» system for radiation with a wavelength λ depending on the optical film thickness n B ⋅ d. For quarter wavelength thickness film (n B ⋅ d=mλ/4, m=1,3, 5…) expression for R is simplified: 0.1 0.2 0.3 R /4 /2 3 /4 nBdnBd n B =1.5 n B =1.7 n B =2.5 Fig. 5.2 The higher the refractive index of the film n B, the higher the maximum value of the system reflection coefficient. From this follows the possibility of increasing the reflection from the surface by depositing to it a quarter wavelength layer. When value n B is close to the n s (for example, n B ≅ 1.5), then the border is absent and it is impossible to distinguish film from the glass substrate. (5.1)(5.1)

3. The system reflection coefficient R = 0.4 (Fig. 5.2) is the reflection coefficient of the border, “air - glass" and does not depend on the thickness of the film. The same value of R is reached for n B  d = mλ / 2 (m = 0, 1, 2...) for all values of n B. In practice, the high reflection coefficient are making through the use of multilayer coatings with alternating high (n B ) and low (n H ) refractive indexes. If the optical thickness of all layers are equal to λ 0 / 4, the reflected by their boundaries waves are in the same phase and in the result of interference they reinforce each other. Such multi-layer dielectric coating provide high reflectivity only in a limited range of wavelengths near the value for which the optical thickness of the layers is λ 0 / 4. Thus, the amplitude reflection coefficient of a multilayer mirror is calculated by adding up the amplitudes of the rays reflected from each layer, taking into account the multiple reflections within each layer and the phase jumps at the layer boundaries. At exact equality of the optical thickness of the layers to λ 0 / 4 multilayer mirror reflection coefficient at normal incidence angle of light in air is calculated from the following simple formulas for the even (2N) and odd (2N +1) number of layers, respectively: (5.3)(5.3)

4. Multilayer mirror manufacture technology After complete the polishing process with surface quality up to 14-th class of accuracy, the substrate is covered by dielectric layers. To obtain reflective coefficient up to 0.99 for 0.63  m wavelength the number of layers reaches 22. There are many different technologies to deposit layer on the substrate, some of them are: thermal evaporation in vacuum, electron-beam method, ion-beam method, chemical vapor deposition, molecular beam epitaxy and others. Evaporation involves two basic processes: a hot source material evaporates and condenses on the substrate. Evaporation takes place in a vacuum, i.e. vapors other than the source material are almost entirely removed before the process begins. In high vacuum (with a long mean free path), evaporated particles can travel directly to the deposition target (substrate) without colliding with the background gas. At a typical pressure of 10 -4 Pa, 0.4-nm particle has a mean free path of about 60 m. Hot objects in the evaporation chamber, such as heating filaments, produce unwanted vapors that limit the quality of the vacuum. Fig. 5.3 shows ion-beam deposition schematic. Current integrator A controls the deposition speed. Magnetic field and diaphragms 1 and 2 guides the ion beam to the substrate. Fig.5.3. Ion beam deposition. 1 2

5. In the thermal method, metal wire is fed onto heated ceramic evaporators known as "boats" due to their shape. A pool of melted metal forms in the boat cavity and evaporates into a cloud above the source. Alternatively the source material is placed in a crucible, which is radiatively heated by an electric filament, or the source material may be hung from the filament itself (filament evaporation). In the electron-beam method, the source is heated by an electron beam with an energy up to 15 keV. In the ion-beam deposition the ions are accelerated, focused or deflected using high voltages or magnetic fields. Optional deceleration at the substrate can be employed to define the deposition energy. This energy usually ranges from a few eV up to a few keV. At low energy molecular ion beams are deposited intact (soft landing) At a high deposition energy a molecular ion fragments, while atomic ions penetrate further into the material, a process known as ion implantation. The following materials are used for deposition: magnesium fluoride MgF 2, silicone dioxide SiO 2, zinc sulfide ZnS (n=2.32). Evaporated materials deposit nonuniformly if the substrate has a rough surface. Because the evaporated material attacks the substrate mostly from a single direction, protruding features block the evaporated material from some areas. This phenomenon is called "shadowing".

6. Deposition of dielectric layers on a multilayer mirror substrate is made using vacuum equipment VU-1A (ВУ-1А). External view of this equipment is shown in fig.5.4. It contains a vacuum chamber Fig.5.4. Vacuum equipment VU-1A (ВУ-1А) to manufacture multilayer mirrors. (in the center) with heating element and two control stands (left and right). Fig.5.6. Thermal deposition equipment of Bendix company. hexafluoroaluminate (cryolite Na 3 AlF 6 ) allowing fabrication of dielectric mirrors. This front view of the Bendix high-vacuum thermal evaporator system shows the major components including the vacuum chamber, controls stack (with gauges), and vacuum valves on the front panel. Fig.5.6 shows thermal deposition unit of Bendix company (USA). It is configured for four sources and is primarily intended for optical thin-film coatings. The filaments would normally be loaded with both metals (such as aluminum) and dielectric materials such as sodium

7. FIBER-OPTIC GYROSCOPE Fiber optic gyro (FOG) was firstly proposed by V. Vali and Shortheel in 1976. They forecasted FOG sensitivity at the level of 0.001 deg/h. The sensitivity is limited by photodiode’s shot noise. However they demonstrated the device with sensitivity of 2 deg/s. From that moment development 4 1 3 2 Fig. 5.7. FOG simplified block-diagram 1 - Light source, 2 – directional coupler, 3 – fiber coil, 4 – photodiode. Output signal b1 b2 of fiber and other FOG components and also scientists’ effort over the world resulted in the creation of FOG and at present time they are in serial production in many countries. Modern FOG has accuracy sufficient to use them in miniaturized strapdown INS and they can compete with RLG by many parameters including accuracy, price, size reliability etc. Fiber-optic gyro like RLG are different sensor to measure Sagnac phase. In case of FOG this phase is defined by expression (2.8, lecture 2). Here sensitivity increasing is reached for the account of multiwinding coil, so that production NS w is sufficiently high. Now, it is widely used interferometric FOG. Optical scheme to measure Sagnac phase is diagramically shown in fig.5.7. Light beam from the source1 is split on two counter propagated beams b1 and b2 in directional coupler 2 and after passing the fiber coil 3 in opposite direction they are provided to photodiode 4. Intensity of the two waves superposition is defined as:

8. (5.4) where A 1, A 2 – the amplitudes of b1 and b2 electrical fields. Suppose that A 1 =A 2 and there are no losses in the coil, we have: (5.5) IdId  s a b Fig.5.8. FOG Output characteristic. IdId  s I 0 – the light source intensity. The current on the output of the photodiode is depicted in fig.5.8a. As can be seen from the figure the sensitivity of the device for small phase tends to zero, i.e. (dI d /d  s )  0 for small  s, where I d – photodiode current and it is impossible to define direction of rotation because for + and – angle rate we have the same  s. In order to increase sensitivity, initial  /2 phase shift is used. Fig. 5.8.b shows good sensitivity. In this case photodiode current is: I d = 0.5DI o sin  s  0.5DI o  s ; (5.6) D =  q/h.  - photodiode quantum efficiency, q – electron charge, h – Planck constant, - light frequency.

9. FOG output characteristic is linear in the small region around zero, because it is sine function: 1% linearity corresponds to less than 0.1 rad. in Sagnac phase. It means that for fiber length L=500 m, coil radius R = 10 cm, we have FOG dynamic range  3 deg/h. Initial Phase Biasing There are four techniques of FOG initial phase shifting: 1.Static technique; 2. Dynamic technique; 3. Phase nulling technique and 4. Heterodyne technique. Static technique is not used in practice, 4 1 3 2 Fig.5.9. Dynamic technique of biasing in FOG. 1 – light source, 2 – directional coupling, 3 – fiber coil, 4 –photodiode, 5 – phase modulator, 6 – generator, 7 – synchronous demodulator. 7 5 6 V heterodyne technique is difficult to realize due to introduction of undesired nonreciprocity in counter propagated waves and is not used in practice, too. So, dynamic technique and phase nulling technique are often used in practice and we’ll discussed them now in detail. FOG schematic using dynamic technique is diagramically depicted in fig.4.3. The essence of this technique is that phase deference between counter propagated waves is modulated by frequency  m =2  f m =2  /2T с, where T с – the time it takes the light to pass around coil (fiber length L). Using sinusoidal modulation, variable component of the photodiode current is:

10.  m – modulation amplitude. Bessel function series decomposition yields: (5.7) (5.8) This amplitude is maximum when  m =1.8 rad. This method does not improve nonlinearity of FOG output characteristic. FOG with dynamic biasing has high scale factor nonlinearity when measuring large angle rate. Sagnac phase nulling technique is that the controlled frequency shifting p is given for two counter propagated waves. In this case phase deference  r is: Demodulation at frequency  m results in amplitude.  r =2  nL p / -  s = 2  L(n p / - 2R  /C)/. Fig.5.10. Phase nulling technique in FOG. 1 – light source, 2 – directional coupling, 3 – fiber coil, 4 – photodiode, 5 – phase modulator, 6 – voltage control generator, 7 – information processing unit. 4 1 3 2 7 5 6 V p (5.9) n – refraction coefficient of the fiber. The aim is: phase deference drive to null by controlling deference frequency p. As a result we have: p = 2  R/ ( n). (5.10)

11. Here, FOG become equivalent to RLG. Fig. 5.10 shows block diagram FOG using phase nulling technique. The controlled deference frequency p is proportional to angle rate measured. Scale factor for such a FOG is: K = 2R/( n). (5.11) Basic FOG Errors Any nonreciprocal phase effect of counter propagated waves in FOG, except rotation, results in errors. As in laser gyro, slow changes in phase nonreciprocity define FOG drift, and fast changes – random walk. Fast changes are due to frequency fluctuation, shot noise of the photodiode, coherent and non coherent back scattering. Slow changes are due to thermal effects in fiber, polarization coupling between the modes, magnetic field, thermal changes in refraction coefficient etc. Besides, intensity, polarization and frequency changes result in phase nonreciprocity because of nonlinear optical effects. FOG drift and random walk result from the following main nonreciprocal effects: Drift Counteraction between counter propagated waves Modulation polarization Beams intensity and deference intensity changes Temperature gradients in fiber coil Changes in fiber refraction coefficient