Study Of Factors Affecting Degradation In Performance Of Spin-Polarized Atomic Gyroscopes

Study of Factors Affecting Performance of Spin-Polarized Atomic Gyroscopes

Uyen Nguyen Huynh Department of Electrical Engineering University of California, Berkeley

Faculty Mentor: Professor Andrei M. Shkel Graduate Student: Max Perez Department of Mechanical and Aerospace Engineering University of California, Irvine

- 1 - Abstract:

The atomic gyroscope designed in Professor Shkel’s lab operates based on nuclear magnetic resonance (NMR) phenomena in which the intensity of circularly polarized light is modulated at the Larmor frequency (specified by NMR gas). The atomic gyroscope uses the folded optics architecture, in which the emitted light enters a micro-cell and bounces several times from the cell wall before being reflected back to the plane of the emitted light. The modulated light is detected by photodiodes after multiple reflections inside a NMR cell. The

NMR cell is a Silicon substrate edged to a pyramidal-shaped hole where its four sides function as micro-reflectors. It is important to maintain the polarization state and a high intensity of reflected light to achieve high gyroscope performance. Unfortunately, bulk Silicon reflectance is only as high as 30%. To improve the cell’s reflectivity, a high reflectance coating method is employed by depositing alternating layers of high and low indices of refraction on the reflector sides of the

NMR cell. MATLAB models were developed to analyze effects of a multilayer thin-film reflector on the properties of circularly polarized light after its first and second reflections.

Experiments also were carried out to study reflected circularly polarized light at various incident angles. The analytic and experimental results show that multilayer thin-film can improve the reflectivity of the NMR cell up to 99% while maintaining the polarization state of incident light.

They also show that fabrication imperfection can degrade the mentioned properties of multilayer reflectors but still maintain a reflectance two times higher than that of the bulk Silicon reflector.

Key Terms:

Multilayer thin-film reflector, nuclear magnetic resonance.

Introduction:

1. Nuclear Magnetic Resonance Gyroscope (NMRG):

- 2 - Since the late 1900s, macro-gyroscopes have been used as inertial sensors to measure angular rates for many navigation applications. However, as the demand for smaller and cheaper devices has increased, the traditional macro-gyroscope appears too complex, too large and thus, too expensive. Researchers have tried to employ the original concept of the macro-gyroscope to find new ways to design smaller and cheaper micro ones while maintaining the macro-gyroscope performance.

B cos(ω t) 1 a

Static Field B o

Figure 1: Cross section of assembled NMRG (Illustration by Jesper Eklund)

In Professor Shkel’s lab, the micro-atomic gyroscope studied is the nuclear magnetic resonance gyroscope (NMRG) (Figure 1) which applies the NMR phenomena. Its basic components are a base-plate supporting a VCSEL (vertical cavity surface emitting laser) and photodiodes, a quarter-wave plate and an NMR cell. It functions by sending a linearly polarized light, which changes to circularly polarized light as it passes through a quarter-wave plate placed above the VCSEL. This light travels to the NMR cell where Rubidium and Xenon vapor (NMR gas) trapped between two glass plates allows NMR phenomena to take place. Along with the

- 3 - existence of the static magnetic field Bo and the pulsing magnetic field B1 (shown in figure 1), circularly polarized light intensity is modified at Larmor frequency ωL of the NMR gas (Stejskal and Memory 4). The four sides of the NMR cell acts as micro-reflectors to guide the modified light back to the photodiodes. When the gyroscope rotates at frequency ωR (figure 2), the photodiodes observe an intensity frequency ωobserved. The intensity frequency extracted from the photodiodes and the known Larmor frequency of the NMR gas are used to determine the angular rate of the gyroscope: ωR = ωL – ωobserved.

ω R ω observed

Figure 2: When NMRG is rotating at frequency ωR, the photodiodes observe the modified light intensity frequency of ωobserved

2. NMR Cell:

m 1.8 mm

1.8 mm

Figure 3: Cross section of NMR cell Figure 4: NMR cell with filled NMR gas (Picture taken and cell fabrication by Max Perez)

- 4 - The NMR cell is an important part of the NMRG. It is where the NMR phenomena take place. It uses the folded optics architecture to guide emitted light back to photodiodes after being reflected twice inside the cell, as well as a means to trap NMR gas between two glass plates

(figure 4). The cell is made of Silicon substrate, which is edged to a pyramidal-shaped hole. Each side of the cell forms a 54.7o angle with the horizontal plane (figure 3). They are also coated with high reflectance coating to improve bulk-Silicon reflectance from 30% to as high as 99%. This method potentially reduces approximately 90% loss in optical power due to bare Silicon reflectors after two reflections inside the cell to less than 10% loss due to the high reflectance coating reflectors. In the context of the NMRG, the high reflectance coating is designed for light at 795 nm wavelength. The thin-film thicknesses are also designed such that the high reflectance band (see the Multilayer Thin-Film Design section) extends for a wide range of incident angles that the NMRG operates in. Since the VCSEL emits light at an angle α from 0 o to 20o, the first

o o o reflection angle θ1 varies from 34.7 to 54.7 , and the second reflection angle θ2 varies from 15.9 to 35.9o (figure 5).

2θ 2θ 2 1

Photodiode VCSEL

Figure 5: Light path inside NMR cell

3. Multilayer Thin-Film Design:

- 5 - The high reflectance coating adapted in this design consists of a stack of alternating layers of high and low indices of refraction and of equal optical thickness (figure 6). For the wavelength λo, for which the optical layer thicknesses are λo/4, all the multiply-reflected beams are in phase at the normal incident angle, which results in a constructive interference. The coating leads to a high reflectance R at λo

n f – n R = ( S o )2 (1) 2N n f + n S o where N is the number of high and low refractive index pairs, no is the refractive index of the

2N incident medium, nS is that of the substrate and f = (nH/nL) (Heavens 78). Notice that the more layers and/or the larger difference between nH and nL, the higher the reflectance R. The advantage of this multilayer system is a high reflectance band for a finite wavelength spectrum center at λo. The reflectance band of a quarter-wave stack at normal incident angle is shown in figure 7 (Heavens 79). The multilayer system improves approximately 99% reflectance for a spectrum of wavelengths (Heavens 79), from 700 nm to 900 nm in this plot, and low reflectance elsewhere. The high reflectance band will shift as the incident angle θi changes. It is accounted for by a factor of cos(θr) (figure 8), where θr is the refractive angle by Snell’s law, in the observed thickness of each layer by the incident light. If the incident angle is significant, light at wavelength λo can fall out of the high reflectance band.

L L H L

Incident Light

Substrate n n n n n n n n o H L H L H L S

- 6 - Figure 6: Typical quarter-wave high-reflecting stack with 6 layers of high (nH) and low (nL) refractive indices with thicknesses LH and LL, respectively, that satisfy nH* LH = nL* LL = λo/4 (Sharma 313)

Figure 7: Sample graph showing the reflectance band of a quarter-wave stack at normal incident angle with respect to optical wavelength

λ /4 o

λ /4 * cos(θ ) o r θ …………. θ r i

Incident Light

Figure 8: Incident light at an angle θi observed a thickness of λo/4 * cos(θr)

Besides the difficulty of maintaining high reflectance at all incident angles, a multilayer system also introduces a phase delay between the multiply-reflected beams (figure 9a) (Macleod

- 7 - 17). Moreover, the reflectance of the s- and p-waves—where s-waves are the perpendicular components and p-wave are the parallel components of light with respect to the incident plane

(figure 9b)—depends on both the mediums and incident angle; each follows its own Fresnel’s reflection curve (Sharma 31 and 33), as shown in figure 9c. Both of the properties of thin-film mentioned above alter the polarization state of light. It can change the incident circularly polarized light, where the s- and p- components have equal magnitude and a 90o phase delay, to elliptical polarized light, where the phase difference and relative magnitudes are arbitrary

(Collett, 1993).

Incident light Reflected light

θ Air i Si

Refracted light a) b)

Figure 9: Properties of reflection light

a) A thin-film layer introduces phase delay ∆δ = 2πL/λ between R1 and R2 beams. b) s and p waves at surface of reflection. c) Fresnel’s reflectance curves

- 8 - 3. Project Objective:

To achieve high gyroscope performance, it is important to maintain high optical intensity to overcome noise introduced by photodiodes. Another important aspect is the ability to preserve the circular polarization state of light inside the NMR cell where the NMR phenomena occur; the performance of NMR cell reflectors directly affects the performance of the gyroscope. This paper reports the effect of multilayer thin-film reflectors on the polarization state of light inside the NMR cell and the effect of fabrication imperfection on both reflectance and polarization state of light.

Materials:

The analytic and experimental results are reported for the following samples:

- Bare Si cell (figure 10a)

- 6 Si_SiO2 layer cell (6L) (figure 10a)

- 6 Si_SiO2 layer wafer (6LW) (figure 10b)

Test Area

Figure 10: Test samples. a) 6L or bare Si cell b) 6LW. (Picture taken and cell fabrication by Max Perez)

All samples were prepared by Max Perez. A Bare Si cell is an NMR cell without any coating. The 6L sample has multilayer coating on the four incline walls of the cell, whereas the

6LW sample has the layers deposited on a planar Silicon wafer.

- 9 - Methods:

Figure 11 is the experimental setup that uses Thorlabs’s equipment to characterize the optical properties of the samples. Tests were done on both the full NMR cell to observe the optical power after two reflections inside the cell, and the half NMR cell to characterize the optical power, power split ratio, and phase difference at various incident angles. The power split ratio and phase difference are used to determine the polarization state of light. It can also be observed visually by the Poincare Sphere shown in figure 11.

Poincare Sphere Sensor Sample ¼ Wave-Plate ½ Wave-Plate VCSEL

λ = 795nm

Full Cell ½ Cell

Figure 11: Experimental setup

The two poles represent circular polarization state, the equator is linear polarization state, and elliptical polarization state is everywhere else on the sphere. Power split ratio mentioned measures the relative existence of the s- and p- waves in the detected light. It fluctuates from 0 to

1, where 0.5 represents equal existence of both components. Phase difference measures a phase delay between the s- and p- waves. It varies from -180o to 180o. A special case is when power

- 10 - split ratio at 0.5 and phase difference at ±90o, circularly polarized light is detected. The reading of the power from the sensor is in dBm and it is related mW through the following relationship:

A dBm = 10log10(P/1mW) (2) where A is the optical power in dBm and P is the power in mW. This data is used to determine the reflectance R = PR/Po of reflected light; PR is the reflected light power and Po is the initial light power. All the samples’ surfaces were taped to prevent the sensor from collecting reflected light from other sources than the NMR cells.

Light emitted from the VCSEL is linearly polarized light and is tuned to wavelength λ at

795nm. After going through both the half-wave plate and quarter-wave plate, the emitted light is changed into circularly polarized light. It is directed to the mounted sample and the reflected light is collected at the sensor. From the sensor, optical properties such as power split ratio, phase difference, and power are recorded. The experimental data was plotted and compared to the analysis results that were generated by MATLAB using Sophocles J. Orfanidis’s MATLAB functions derived by for electromagnetic waves. Since the experimental initial light studied is in circular polarization state, the initial light in the analysis is also circular polarized light in which the phase differential is 90o and the power split ratio is 0.5. In addition to comparing analytic and experimental data, the result from the 6L sample is compared with the 6LW sample to study the effect of fabrication limitations on the reflectance as well as polarization state.

Results:

The descriptions for the graphs in figure 12, 13 and 14 are explained as follows. The solid lines with the abbreviation T represent the analysis results generated by MATLAB using

Orfanidis’s functions. The blue and red dotted lines with the abbreviation E are experimental results and the magenta dotted lines P are the interpolated result of the second reflection based on

- 11 - the first-reflection experimental data. The independent variable α is the angle of light from the

VCSEL seen in figure 5.

1. Bare Silicon Cell:

For both graphs in figure 12, the analytic and experimental results are consistent. They are as expected from the optical properties of bare Silicon reflector. Figure 12b shows that the phase difference changes 180o after each reflection, so reflected light still maintains ±90o phase delay. However, figure 12a shows the deviation of power split ratio from the desired value 0.5.

These results show that initial circularly polarized light will change into elliptical polarized light after the first and second reflection from the Silicon reflectors. The consistency of these analytic and experimental results also confirms the trustworthy of the Orfanidis’s functions.

- 12 - b) Figure 12: Analytic and experimental results of the bare Silicon cell

2. 6 Si_SiO2 Layer Cell:

- 13 - b)

Figure 13: Analytic and experimental results of the 6 Si_SiO2 layer cell

The analytical graphs of figures 13a and b demonstrate promising results for the 6L sample. After the first and second reflection, the power split ratio is still at 0.5 and the phase difference is at either ±90o. That means the polarization state of light inside the cell will be maintained as circularly polarized, which is desired. However, the experimental data is not consistent with the analysis results. The power split ratio approaches 0.4 at low α and 0.6 at high

α. The phase graph follows the same trend where there is approximately 40o difference between

T and E lines at low α after the first reflection.

3. 6 Si_SiO2 Layer Wafer:

To eliminate part of the fabrication effect that may cause the inconsistence in the 6L results seen in figure 13, a 6LW sample, for which the layers were deposited on a Silicon substrate planar instead of at an incline, was tested. Its results in figure 14 are encouraging. The agreement between analysis and experimental results confirm that the 6 Si_SiO2 multilayer design has a potential of maintaining the circular polarization state of light after the both the first and second reflections.

- 14 - a)

Figure 14: Analytic and experimental results of 6 Si_SiO2 layer wafer

4. Reflectance results of full cells and ½ cells:

The reflectance results of bare Silicon cell, 6L, and 6LW samples are plotted in figure 15.

The reflectance of Silicon is at about 0.3 and above 0.9 for the 6LW sample, as expected. For the

6L sample, the reflectance is only as high as 0.6 at low α and 0.7 at high α.

- 15 - Figure 15: Reflectance after the first reflection

Figure 16: Reflectance after the second reflection

Figure 16 shows the reflectance results of the unfilled 6 Si_SiO2 layer cell (figure 10a),

NMR vapor filled cell (figure 4), and unfilled bare Silicon cell. It is plotted against the VCSEL drive current, which is proportional to emitted light wavelength. The peak seen in the filled

- 16 - NMR red curve is due to the absorption of the NMR gas. Once again, the reflectance of Silicon is about 10% after the second reflection, as expected. The unfilled cell performance is as high as

80%. And the filled cell performance is only 40%.

Discussion:

The Results section shows the deviation of the 6L experimental results from both the analysis results and the 6LW experimental results. This can be explained by the following two possible causes. First is due to the emitted light spot size that is larger than the surface of the cell’s reflectors (demonstrated by figure 17). This may lead to the difference in 6L and 6LW reflectance results (figure 15). The second cause is the non-uniform thicknesses of thin-film layers deposited on the cell walls. This effect can be seen in figure 17b. The color spectrum represents different coating thicknesses. Each color corresponds to a thickness that yields a high reflection band at that color wavelength. When red light at 795 nm reflected from other color regions (figure 17b), both the reflectance and polarization state of reflected light will be degraded and deviated from the analysis results. The non-uniform thickness effect is confirmed by the analysis results in figure 18. The graphs show the behaviors of 795 nm light when it is reflected from various thickness regions of the cell reflector. The λs in the legend give implicit thicknesses of the color regions seen in figure 17b. VCSEL

Figure 17: Possible sources cause variation in 6L and 6LW results

- 17 - At reflector thicknesses corresponding to λ = 700 and 800 nm (red light regions), the power split ratio and phase difference are still compatible to the 6L analysis results (Figure 18).

However, at thicknesses corresponding to λs in orange, yellow, green, and blue, the graphs deviate from the desired values (0.5 for power split ratio, and ±90o for phase difference) and the

6L analysis results.

- 18 - Figure 18: Analysis results of 6L cell for different thin-film thicknesses.

Another concern addressed here is from figure 16, where a 40% difference in reflectance results between the filled and unfilled cells is observed. One hypothesis is that part of the optical power is lost when light passes through the glass plates of the filled cell four times. As a rule of thumb, each time it goes through an air-glass boundary, 4% of the optical power is lost. It results in a total of 16% optical power loss due to the glass plate. The remaining loss is assumed to be caused by other factors such as light scattering of the NMR gas, and/or loss in experimental procedure.

Conclusion:

From the analysis and experimental results, the the NMR cell’s high reflectance coating can potentially improve gyroscope performance. The 6 Si_SiO2 multilayer system can maintain both high reflectance and the circular polarization state of light inside the NMR cell.

Imperfection in fabrication such as non-uniform thicknesses of thin-film layers inside the cell will change the polarization state of light and reduce the reflectance of the cell reflectors.

However, the overall reflectance of cell reflectors is still two times higher than bare Silicon at the first reflection and four times higher at the second reflection for the filled cell. The high reflectance coating method promises an improvement in gyroscope performance when the fabrication imperfection effect is further reduced.

Acknowledgements:

I would like to thank Professor Andrei M. Shkel for welcoming me into his phenomenal

MicroSystems Lab team and opening my eyes to a demanding but rewarding research life. I owe many thanks to Max Perez for his guidance throughout my summer research despite his busy

- 19 - schedule. I thank Jesper Eklund, who patiently explained the NMRG concept many times, as well as everyone in the MicroSystems Lab for a welcoming environment.

Works Cited:

Collett. Polarized Light. New York: Marcel Dekker, Inc., 1993.

Heavens. Thin Film Physics. London: T. & A. Constable Ltd, 1970.

Macleod. Thin-Film Optical Filters. New York: American Elsevier Publishing Company, Inc., 1969.

Memory, Stejskal. High Resolution NMR in the Solid State. NewYork: Oxford University Press, Inc., 1994.

Orfanidis. Electromagnetic Waves and Antennas. 21 June 2004. .

Sharma. Optics. Amsterdam: Elsevier, Inc., 2006.

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