## Abstract

We use calculation and simulation to characterize an all-reflective monolithic gyroscopic structure that supports 3 sets of orthogonal, spatially dense and continuous helical optical paths. This gyroscope differs from current fiber optic and ring laser gyroscopes primarily in the free space multi-turn nature of the optical path. The design also creates opportunities for introducing gain while minimizing spontaneous emission noise from those gain regions. The achievable angular measurement precision for each axis, given ideal components and no gain, is calculated to be ~0.001°/hr for a structure of ~6.5 cm diameter, ~1 watt average optical power, and a wavelength of 0.5 µm. For fixed power, the uncertainty scales as the reciprocal cube of the diameter of the structure. While the fabrication and implementation requirements are challenging, the needed reflectivities and optical surface quality have been demonstrated in more conventional optics. In particular, the low mass, compact character, and all reflective optics offer advantages for applications in space.

©2000 Optical Society of America

## 1. Introduction

We describe a strategy for using reflective optics to produce many spatially dense turns of an optical beam. The beam traverses an approximately helical path. This strategy offers advantages in the form of a higher spatial density of turns in small structures or greater precision, for a given instrument mass, in large structures. The elimination of optical fiber reduces mass and facilitates the nesting of all three axes plus the signal processing elements in a single volume. This approach requires very high quality reflecting elements and/or a strategy for including discretized gain, as well as a monolithic optical structure with very specifically structured optical surfaces.

#### 1.1 The Sagnac interferometer as the sensing element

The instrument design is based on the Sagnac interferometer. As in conventional fiber gyroscopes, the interferometer senses differential phase shifts of optical signals counterpropagating along a common, approximately helical path. In general, increasing the product of the pathlength of the light and the area circumscribed by the optical path increases the precision of the measurement. This requires that the net loss remain small. We explore here this all-reflective strategy for achieving such long, low net loss optical paths with maximum circumscribed area in minimum volume.

#### 1.2 Objective

Our primary objective is to maximize measurement precision while minimizing mass and volume. We do this under the constraint that the system support low loss propagation of a minimum diameter Gaussian beam through the reflective optical system. This requires a periodic refocusing and redirecting of the Gaussian beams. The beams are as densely packed as allowed by the need to maintain low beam loss and low distortion [2]. We recognize that the task of fabricating the surfaces so identified and maintaining low loss is demanding. We note, however, that advances in mirror coating [3] and optical fabrication support consideration of design strategies such as those advanced here.

## 2. Measurement precision

The minimum uncertainty in the measured rotation rate *δ*Ω for an optical gyroscope given an optical pathlength *L*, light at wavelength *λ*_{o}
in microns, a photon flux of *n*_{ph}
photons/second at the detector, a detector quantum efficiency of *η*, and an integration time *τ* is

We have approximated *L* as produced by *N* circular turns of diameter *D*. This yields the relationship 4*NA*=4*Nπ*(*d*/2)^{2}=*N*(*πd*)*d*=*N*·(*circumference*)·*d*=*Ld*. *A* is to be regarded as the average area circumscribed by each loop in the helical path and *d* is the average diameter of those loops. Typical parameters for a good navigational grade gyroscope are, *L*=1 km, *d*=10 cm, *λ*_{o}
=1 µm, *η*=0.3 and *τ*=1 sec. For fiber, a typical value for photon flux is *n*_{ph}
=3×10^{15} photons/sec, corresponding to 1 mW of power. These parameters yield an uncertainty in rotation rate *δ*Ω of ~5×10^{-8} radians/sec, or 0.01°/hr [1]. The number of turns *N* for this example, given the assumed circular path, is 3,183. In general, a complete gyroscope requires sensing of motion about three orthogonal axes. There will typically be a need for three orthogonal structures having the above properties, or 9,549 turns for this example. We develop a comparison below of the performance expected for such conventional gyroscopes and our all-reflective multi-turn gyroscope.

## 3. Open multi-turn configuration for implementing an optical gyroscope

We develop here the design parameters for a series of free space, approximately helical paths packed at maximum density as illustrated in Fig.s 1a and 1b. In addition to the freedom to nest the three basic systems, the material needed for guiding the optical beams in our model is restricted to a thin shell. The individual segments of the optical path approximate the modes of a confocal resonator. The mirror facets in this case are, however, micro off-axis parabolas oriented so as to guide the beam through a series of optimally packed, nearly 90 degree turns. The geometry we use for this example is that of three families of beams formed in a Cartesian coordinate system with one family sensing rotation about the x-axis, one sensing rotation about the y-axis and one sensing rotation about the z-axis, see Fig. 1b. The routing mirror facets for any given family are confined to the two opposing spherical sections, as seen in Fig. 1a.

## 4. Limiting precision for a given characteristic dimension

We calculate the minimum uncertainty achievable for a given characteristic dimension of this gyroscope assuming ideal components. We seek the maximum number of turns that can be localized within an approximately spherical volume subject to the constraint that the diffraction ripple introduced by any given aperture be less than 1%. This requires each mirror facet diameter *D*≅4.6√2·*ω*_{i}
[2]. Here *ω*_{i}
is the radius of the beam waist for the i^{th} path segment. The relationship between pathlength *2Z*_{i}
, beam waist *ω*_{i}
, and wavelength *λ*_{o}
is 2*Z*_{i}
=2${\pi \omega}_{i}^{2}$/*λ* [2].

We illustrate the relationship between the minimum uncertainty in measured rate of rotation, the diameter of the including sphere, and the number of mirror facets that can be included in a row. We use as an example the particular structure shown in Fig. 1. There the number of mirror facets per row is 21. For a sphere of radius *R*, the length of a single side of a cube inscribed in that sphere is *a*=2*R*√3. The worst case confocal parameter is 2Z_{i}=2R. Substituting this into the above equations using a wavelength of *λ*=0.5µm yields *ω*_{o}
=18.8µm. Here *Ω*_{o}
is the upper limit on beam radius. A worst case estimate of the diameter of a sphere that supports this 21-turn-per-row configuration is small, i.e., 2*R*=2*Z*_{o}
=2${\pi \omega}_{o}^{2}$/*λ*=0.446*cm*.

We calculate the limiting precision of this particular free-space optical gyroscope illustrated in Fig. 1 for ideal components by finding the actual total area enclosed, *A*_{TOTAL}
=*NA*, in equation 1. Our Advanced Systems Analysis Program (ASAP) simulation gives this area as 0.00192 m^{2}. For photon flux we take *n*_{ph}
=3×10^{18} photons/sec which corresponds to 1 Watt. This is a higher flux than typically used for fiber. We regard this as accessible given the use of free space optics. Using *η*=0.5 and *τ*=1 sec, Eq. (1) yields *δ*Ω=7.99×10^{-6}
*rad*/sec=1.65°/*hr*. This is a modest precision, however, the diameter of the gyroscope, <5 mm, is small compared to most conventional gyroscopes offering this degree of precision. Scattering from the multiple surfaces must be addressed as a potential source of error. See Sections 8 and 9 below.

As regards scaling the gyroscope size, the limiting uncertainty for ideal components decreases approximately as the reciprocal cube of the characteristic dimension. The beam area at the mirror facets also increases with the characteristic dimension so that higher average powers can be tolerated. In our preferred configuration the optics are contained within the surrounding shell of the structure. This has the consequence that the mass of the structure increases slowly while the limiting precision improves rapidly with increasing dimensions. For example, a gyroscope having the approximate size of a tennis ball, *2R*=6.5 cm, has an average enclosed area *A*_{TOTAL}
=3.36 m^{2}. For this total area with all other values the same, the limiting angular resolution is *δΩ*=0.001°/hr. This number is an order of magnitude better precision than the typical fiber gyro for a structure 2/3 the size. We note that the reduction in uncertainty is due in significant part to the higher photon flux.

For a sphere of approximately 18 cm diameter, or about the size of a volleyball, the total enclosed area is 68.35 m^{2}. This yields an uncertainty of *δΩ*=0.00005°/hr. There is thus reason to believe that precision beyond the values now characteristic of current optical gyroscopes might be realized.

## 5. Routing of beams in the open multi-turn gyroscope

We discuss here one strategy for developing a beam pattern of the type we have described. One chooses a particular location as a starting point and then uses four successive reflections, each at, approximately, a right angle. This forms a nearly square loop. The path brings the beam back to the starting location except displaced by 4.6 times the radius of the Gaussian beam at the mirror facet. This pattern is repeated until the beam has walked the available distance, ance, ~*a*=2*R*/√3. Fig. 1a and the associated animation illustrate one family of mirror facets and a portion of the beam path produced by the mirror facet array

The array shown in Fig. 1a comprises four quadrants, each of which has 10 rows. Each row contains 21 facets. One loop of the beam reflects from one mirror facet in each of the four quadrants. The beam is routed to the adjacent facet in the starting row upon the completion of one loop and, in this manner, “walks” along a row of mirror facets parallel to the axis about which it senses motion. When the beam reaches the end of a row, it is routed to the next row where it proceeds to walk back along the axis in the opposite direction. The animation that accompanies Fig. 1 illustrates the placement of each of the four quadrants of mirrors on a sphere and provides a rotating perspective of the family of mirror facets. There may be reason in some systems to interleave paths in such a way as to minimize thermal effects as is done in fiber gyros. We did not seek that particular goal here.

We have analyzed the polarization effects of the system using ASAP. Tracing a *p* polarized beam through our gyroscope with R=1cm, or 377 turns, the resulting beam is 95.6% polarized. Polarizing elements can be introduced to maintain the polarization closer to 100%.

The complete set of possible paths is formed by utilizing the full area available on a given pair of opposing sections of the sphere for mapping the family of facets. We do not show a maximized number of paths here since the packing density is so high as to render visualization difficult. Fig. 1b shows how the three different families of mirror facets can be placed on the same sphere to sense motion in each of the three orthogonal directions. Each family of mirror facets is shown in a different color. The beams are omitted in Fig. 1b to facilitate viewing the facet arrays.

## 6. Effects of paraboloidal optics on a Gaussian beam

The accumulated effect of the paraboloidal optics on a beam propagating through this large number of reflections is a concern. While we do not address the total cumulative effect here, we do analyze the properties of a single sequence of 4 reflections. We show that the most serious beam distortions are self compensating in any given loop of the approximately helical path. Since the centers of the segmented mirrors in Fig. 1 align to a paraboloidal shape, we analyze the optical characteristics of a pair of identical, facing paraboloids using the ASAP optical modeling and analysis code.

In the model, shown in Fig. 2, a collimated grid of rays is simulated as beginning to the right of the lower vertical line and traveling in a direction parallel to the optical axis of the first paraboloid and towards an off-axis portion of that first paraboloid. This ray grid intersects and is reflected towards a geometrically perfect focus, passes through focus and continues through its second intersection and reflection with the first paraboloid. It is again a collimated grid of rays, but directed towards the second paraboloid where it will be reflected through perfect geometrical focus, recollimated, and directed back to the first paraboloid.

In the analysis which follows, the rays are evaluated in the vertical plane located at the mid-point between the two paraboloids. (It is stressed here that the analysis is purely geometrical in nature; no diffraction has been considered in this effort.) The distribution of rays is different for those reflecting from the first parabolic surface and the rays reflecting from the second parabolic surface. We illustrate this using spot diagrams in Fig.s 3a, 3c and 4a. In Fig. 3a, we see that the ray grid is a uniform, regularly spaced pattern. After the rays have completed the reflection off the first parabolic reflector, the pattern has changed dramatically, as shown in Fig. 3c. The overall size of the pattern increases by almost 10%, and the ray density becomes non-uniform. The rays are still perfectly collimated, but the intensity distribution of the beam is altered.

When this distorted grid pattern completes the reflection from the second parabolic surface, the intensity redistribution is compensated. The final intensity distribution is shown by the spot diagram in Fig. 4a. Thus, the full path causes the beam to undergo an intensity redistribution, and then the inverse of that redistribution back to its original pattern. We simulated a flat top intensity profile beam following the complete path as a means of better exhibiting this redistribution process. The initial flat top profile is shown in Fig. 3b. The resulting intensity shift is shown in the redistributed profile (Fig. 3d) that occurs after completing the circuit through the first paraboloid. After the second paraboloid, the intensity pattern is the flat top profile again, as shown in Fig. 4b.

## 7. Acceptable reflection losses

Given the large number of reflections characteristic of this approach there is a critical need to keep each reflection loss small. A key challenge is that of providing high reflectivity for the many mirror facets with each element having a slightly different orientation. Reflecting surfaces can be made that have reflectivities of 99.99% [3]. For 10,000 reflections at this reflectivity the loss is -4.3 dB. Another option is that of including small incremental gain at discrete locations as discussed in Sections 8 and 9 below.

## 8. Detection

The detection of the differential phase shift between optical signals counterpropagating in the two possible directions can be performed in a manner similar to that in a conventional gyroscope. Obviously care must be taken to minimize any path differences between the two counterpropagating signals other than that due to the small nonreciprocal Sagnac phase shift caused by rotation of the gyroscope. A detection method, such as that using an intentional nonreciprocal phase modulation and phase sensitive detection for a minimum configuration gyroscope [1] would serve this detection purpose.

The new difficulty encountered in detection using this open multipass scheme is most likely to occur in reducing nonreciprocal variations in the counterpropagating optical paths to a level small compared to the phase noise. This will require ensuring that the two counterpropagating paths established by the signal injection and by the guiding action of the multifaceted reflector have a sufficiently small nonreciprocity to avoid signal degradation.

One possible solution is to operate the open multipass system as a multi-turn ring laser oscillator by closing the optical path as in a conventional single turn ring laser oscillator, but closing the path so as to include the entire multi-turn path. This will provide a stabilization of the two counterpropagating optical beams by essentially the same mechanism as in the single turn ring oscillator.

## 9. Noise

As in conventional gyroscopes the principal unavoidable noise is the shot noise inherent in the optical signal. A difference between this open multipass optical system and conventional gyroscopes in noise minimization options occurs because of the different nature of the optical path. The optical path in this open multipass geometry has a long rod-like geometry that utilizes the minimum possible aperture at each turning point in the system. This provides an inherent spatial filtering that minimizes spontaneous emission noise from gain regions included within the system more effectively than could be accomplished in fiber systems, e.g..

Assuming isotropic spontaneous emission from the gain regions and an aperture diameter of 4.6√2·*ω*_{o}
[2] for the separation between the active reflecting regions of ~2*R*, the fraction of the spontaneous emission that is collected at the next optical element is ~10${\omega}_{o}^{2}$/*R*
^{2}. For our approximately confocal geometry this is ~*πλ/R*, or for our representative example where *R*=3.25 cm, the fraction of spontaneous emission coupled into the propagating beam is ~5×10^{-5}.

This suggests including discrete gain regions within the system, as in the form of active mirrors. Other options also exist for reducing the spontaneous emission further as by temporal gating of signals formed as short pulses. Nonlinear methods of noise reduction, such as use of saturable absorber mechanisms can also be considered. Scattering from the mirror facets will also be a potential source of noise and means for reduction of that noise will need to be examined.

## 10. Conclusions

In conclusion, we have presented an all-reflective optical gyroscope design as a possible alternative to conventional fiber optic and ring laser gyros. The class of gyroscopes so identified offers a higher spatial density of turns in smaller structures or reduced uncertainty for a given mass of material in large structures. While the task of fabricating the complex high quality mirror facets that are needed is a challenging barrier, the potential advances deserve examination. Work remains in the form of articulating optimal noise reduction strategies.

## 11. Acknowledgments

We thank Alan Shapiro for discussion concerning the reflectivity of dielectric coatings, Paul Ashley for discussion on gyroscope technology, and Darryl Engelhaupt and Bruce Peters for discussions concerning optical fabrication.

## References and links

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**2. **Anthony E. Siegman, *Lasers*, (University Science Books., Mill Valley, CA1986).

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**4. **Richard L. Fork, Spencer T. Cole, Lisa J. Gamble, William M. Diffey, and Andrew S. Keys, “Optical amplifier for space applications,” Optics Express **5**, 292–301 (1999). http://www.opticsexpress.org/oearchive/source/14181.htm. [CrossRef] [PubMed]

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