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We describe the behavior of optical trajectories in multipass rotationally asymmetric cavities (RACs) using a phase-space motivated approach. Emphasis is placed on generating long optical paths. A trajectory with an optical path length of 18 m is generated within a 68 cm3 volume. This path length to volume ratio (26.6 cm−2) is large compared to current state of the art multipass cells such as the cylindrical multipass cell (6.6 cm−2) and astigmatic Herriott cell (9 cm−2). Additionally, the effect of small changes to the input conditions on the path length is studied and compared to the astigmatic Herriott cell. This work simplifies the process of designing RACs with long optical path lengths and could lead to broader implementation of these multipass cells.
© 2016 Optical Society of America
1. Introduction
The detection of trace amounts of molecules in the gas phase is paramount to applications in defense [1–9], atmospheric studies [10–12], industrial process control [13], and medical diagnostics [14]. Optical absorption spectroscopy (OAS) measures wavelength-dependent changes in the intensity of light as it interacts with a sample under test and is useful for specific and quantitative detection of trace gases [15]. As an optical measurement technique, OAS has the benefit of being nondestructive, inexpensive, and fast. One strategy to improve the sensor detection limits, and therefore detect gases at low concentrations, is to increase the optical path length (OPL), the interaction length of the optical beam and the sample under test. The measured change in intensity of the optical source, described by the Beer-Lambert Law, is exponentially sensitive to the OPL [16]. Increasing the OPL typically improves the signal-to-noise ratio in OAS.
Multipass cells increase the OPL using mirrors to reflect light through the same airspace many times. Long OPLs can be achieved by increasing the inter-mirror distance and/or the number of reflections inside of the cell. Increasing the inter-mirror distance will also increase the cell volume, resulting in a longer measurement response time. Small cell volumes that support many reflections are favorable for applications that require fast response times or place limits on the footprint and weight of the cell.
State-of-the-art multipass cells such as the astigmatic Herriott cell support OPLs greater than 200 m in a 2200 cm3 volume [10]. The OPL to cell volume ratio is 9.1 cm−2 and is useful for comparing different cells. The sensitivity to alignment and large size of these cells can limit their use in applications requiring compact and/or mobile sensors. Because the OPL in a Herriott cell is controlled by adjusting the spacing between the two mirrors, and in the astigmatic case, the mirror twist, Herriott cells are difficult to miniaturize while maintaining a large OPL. Cylindrical cells can require a less demanding mirror system, but these cells exhibit a smaller OPL to volume ratio of 6.6 cm−2 (16 m in a 250 cm3 volume) [17], compared to 9.1 cm−2 in the Herriott cell. Further increases in the OPL to volume ratio have been achieved in Herriott cells by constricting the volume of the cell such that the optical mode occupies a larger fraction of the usable cell volume; in this way Herriott cells have demonstrated OPL to volume ratios of 28 cm−1 [11]. Hollow core fibers used for OAS can provide very large OPL to volume ratios, but are unsuitable for certain applications because of the long time constants associated with filling the fiber with the test gas [18].
Recently, cells based on a single reflective surface formed by the boundary of a sphere with quadrupole deformations were demonstrated. These so-called rotationally asymmetric cavities (RACs) supported OPLs of up to 15.5 meters in simulation [19,20] and 6 meters in experiment [21]. The volume of these cells is small (68 cm3), making them useful for applications requiring fast gas refresh times and compact sensors. The OPL can be controlled via the design of the cell and the input conditions for the optical beam, giving RACs with OPL to volume ratios that exceed unconstricted astigmatic Herriott cells by a factor of about 3: a ratio of 26.6 cm−2 for an 18 m OPL in a 68 cm3 cell. If the volume is reduced in a technique similar to that employed for the Herriott cell, the ratio could be increased to 36.2 cm−2 (assuming the total cell volume is constricted to a cylinder with a diameter 10% larger than the largest dimension in the lateral direction of the mode volume).
The design space for RACs is very large and selecting appropriate cell parameters and input conditions is difficult. Here, we describe beam trajectories inside of RACs and demonstrate how an understanding of these trajectories in phase space can enable the design and selection of trajectories with long OPLs (> 18 m) in compact geometries. We also analyze and discuss the sensitivity of the trajectories to changes in the input conditions and demonstrate that RACs are comparable to the astigmatic Herriott cell.
2. Methods
The trajectory of an optical beam inside of a cell can be determined numerically by simulating propagation and reflection in the interior of the cell. The interior geometries of the RACs in this work are deformed spheres defined by two quadrupole deformations [19–21]. The boundary of the cell geometry is described by the following equations:
where ϵxy and ϵyz are the magnitudes of the quadrupole deformations in the xy-plane and yz-plane, respectively, and may range from 0 to 1, though only small deformations are considered here (ϵxy, ϵyz ≤ 0.15); φ and θ are as defined in Fig. 1(a). As either of the deformation parameters increases from 0 to 1, the cross section of the cell in the corresponding plane changes from a circle with radius R0 to a quadrupole. Fig. 1(b) depicts the cross-sectional geometry of a cell in the xy-plane with deformation parameters of (ϵxy, ϵyz) = (0.01, 0.02).
Fig. 1 (a) Depiction of the coordinate system used for defining the cell shape and ray position and momentum. The azimuthal angle φ is defined as increasing counterclockwise from the x-axis, and the vertical angle θ is defined as zero along the z-axis. (b) A schematic of the envelope of the light trajectories in multipass cells under different deformations. Trajectories using cell deformation parameters of (ϵxy, ϵyz) = (0.01, 0.02), (0.02, 0.04), and (0.05, 0.1) are shown by the red, gold, and green shaded regions, respectively. Cells with deformations of (ϵxy, ϵyz) = (0.05, 0.1) and (0.1, 0.2) are shown in dashed lines; the innermost dashed curve has deformation parameters of (0.1, 0.2). (c) Solid model of one half of a RAC to be used as a multipass gas cell with deformation parameters of (ϵxy, ϵyz) = (0.0021, 0.0076).
To use the RACs as a multipass cell, a hole is fabricated into the cell wall for the entrance and exit of the input laser beam, and gas input and output ports are added at positions far from the intended optical path [19]. These additions are shown in a CAD drawing of one half of a cell in Fig. 1(c). The gas ports are sufficiently far from the mode volume to ignore in the numerical models. Light is coupled into the RAC through the entrance hole, and then reflects off of the interior surface a large number of times. The trajectory that the light follows can be controlled by altering the geometry of the RAC and the input conditions. The projection of the volume occupied by the trajectory onto the xy-plane is shown for three different deformations in Fig. 1(b); a single cell boundary is used for the three projections of the mode volume.
We utilize a custom ray tracing program for calculating the trajectory of rays of light inside the cell. An optical beam is simulated by calculating the trajectories of 13 individual rays chosen to model the envelope of the optical beam. The ray bundle comprises a central ray surrounded by 12 rays equidistant from the central ray and spaced angularly by 30 degrees as shown in the inset of Fig. 2(b). The 12 surrounding rays have position and input angle offsets from the center ray that can be selected to model optical beams of various diameters and behavior, such as collimated, focusing, or diverging. In our simulations the beam width is 200 μm entering the RAC, and the surrounding rays are focused such that they cross in the center of the cell.
Fig. 2 (a) Real space trajectory inside a RAC with deformation ϵxy = 0.01 and ϵyz = 0.03. The input position is given by (φ0, θ0) = (0, 0.5π), and the beam is input at an angle of (φp0, θp0) = (0.99π, 0.485π). (b) A plot of beam size as a function of reflection number for the trajectory in (a). The global focus time is indicated by an arrow. The inset shows a cross section of the beam, which is modeled by a center ray and 12 exterior rays.
The 13 individual ray trajectories are calculated in MATLAB using custom code that simulates traversal across the RAC and specular reflection off of the RAC wall Dataset 1, [22]. The input rays are initially propagated across the cell interior, and the position where the ray intersects the surface of the cell is calculated using a quasi-Newton method with line-backtracking [23]. The algorithm consists of estimating the intersection of the ray and cell boundary, then applying Newton’s method to converge on a point. The accuracy of the result is evaluated by comparing the momentum calculated by connecting the ray-cell intersection point with the previous intersection point to the momentum calculated assuming specular reflection. If the error is larger than the set bound, typically 10−10, a new approximation is selected and the process is repeated. Once the error is below the specified level, the process repeats for the specified number of reflections.
We describe the intersection of a ray with the cell geometry using angular coordinates (φn, θn) and the momentum as the angle a ray makes with the origin (φpn, θpn). For both position and momentum, φ and θ give the azimuthal and vertical angular position, Fig. 1(a); the numbered subscript, n, denotes the reflection number; and a subscript “p” denotes coordinates corresponding to momentum. The initial conditions are indicated by n = 0. Figure 2(a) shows the real space trajectory for the central ray that enters the RAC at a point on the surface with angular coordinates (φ0, θ0) = (0.03π, 0.47π), at an angle of (φp0, θp0) = (1.03π, 0.56π) for a RAC with geometry parameters ϵxy = 0.02, ϵyz = 0.07, and R0 = 2.54 cm. These input conditions result in a two-bounce mode in this RAC, where the beam travels back and forth along the x-axis with each reflection and crosses the yz-plane near the center of the RAC. We restrict our study to these modes because the surface is left free of reflections near the yz-plane, which allows the RAC to be milled in two parts without the seam interfering with normal operation. Similarly, the gas flow ports may be placed along this plane. Two-bounce modes also give a long OPL per reflection, as the length of the RAC is traversed with each reflection. Additionally, the distortion of the beam is relatively low, as the beam is incident nearly normal to the surface; a large angle of incidence would lead to elongation of the beam spot, limiting the number of reflections, and is therefore undesirable.
The OPL, spot size, and shape of the ray bundle are characterized by collectively analyzing the trajectories of the 13 rays. The exit condition for the simulated beam is determined by identifying the first occurrence of spatial overlap between the entrance/exit hole and any of the 12 perimeter rays. The OPL is calculated by summing the length of the central ray for each traversal of the RAC up to the calculated exit. The spot size of the simulated optical beam is the area of the 12-sided polygon that the rays form at each reflection. The shape of the beam is also calculated as a function of reflection using the polygon; for each reflection, the ray farthest from the center ray is marked as the major axis of the beam and is used as a measure of the deformation of the beam. While the magnitude of each reflection depends on the polarization of the incoming beam, the total effect of polarization is small (0.01%), and is not considered further.
3. Identifying trajectories with long OPLs
Ray trajectories depend on a large number of variables, including cell geometry and input position and momentum, making the identification of trajectories with long OPLs challenging. In addition to a long OPL, trajectories useful for OAS should maintain a small spot size, preventing a partial exit before the designed OPL has been achieved and avoiding spatial overlap of beam spots on the cell surface for continuous wave operation. Also, the beam should preserve a relatively circular cross-section or a significant portion may not exit.
Figure 2(a) depicts a trajectory in real space that is characteristic of the trajectories examined in this work. These two-bounce, axial trajectories can temporarily focus the beam. The calculated spot size versus reflection number for a beam with a central ray that follows the trajectory in Fig. 2(a) is shown in Fig. 2(b). The minimum in the spot size area, which occurs after 125 reflections for this trajectory, is referred to as the global focusing time (GFT) [19]. Up to the GFT, the spot size decreases and after the GFT, the spot size increases. To keep the beam smaller than the exit hole for many reflections, a large GFT should be used.
In addition to a small spot size to ensure that the beam exits the cavity, the beam area should also be small compared to the area of the mode volume projected onto the surface of the RAC. This reduces the possibility of overlap between reflections and can also increase the OPL. Figure 3(a) shows the beam area after 400 reflections compared to the total surface area on the boundary of the RAC to which the trajectory is confined versus the cavity deformation parameters. The same nearly-axial input trajectory is used to calculate all points. This input condition is chosen because nearly axial inputs along the x-axis reliably result in two-bounce modes for all of the RAC deformations considered here. Points in the lower left half of the space of Fig. 3(a) are not plotted because these conditions result in trajectories that quickly degenerate into chaotic orbits, and the spot size of a beam input along one of these chaotic trajectories grows large quickly, making these trajectories inappropriate for use in a multipass cell. Small ϵyz and ϵxy are preferable to maintain a small spot size relative to the surface area occupied by the mode.
Fig. 3 (a) Color plot of the sum of calculated beam size at 400 reflections divided by the area of the RAC surface covered by reflections for various deformation parameters. The lower left half is left blank, as deformations of this type lead to highly chaotic trajectories. (b) Color plot for global focus time versus cell deformation parameters. Trajectories are calculated using the input position (φ0, θ0) = (0, 0.5π), and an input momentum of (φp0, θp0) = (0.98π, 0.48π). Points with GFTs > 150 are shown in bright yellow.
Figure 3(b) shows the GFT calculated for the same trajectories used in Fig. 3(a). A large GFT is desirable because the beam spot size decreases over more reflections, reducing spatial overlap between reflections on the RAC surface and also the probability of partial or early exit of the beam. The region with the largest GFT is where ϵyz is large and ϵxy is small. However, when designing the cell geometry, the spot size and mode volume must also be considered (Fig. 3(a)). Since the GFT is more strongly influenced by the choice of input conditions, the spot area should be prioritized when selecting the cell deformation.
Surface of section plots are useful for analyzing the behavior of many modes or trajectories in these cells [23]. We plot the surface of section by fixing the input position (φ0, θ0) and varying the momentum (φp0, θp0) of the input beam. Figure 4(a) depicts sin(χ) versus the azimuthal angular position for each reflection, where χ is the angle of reflection, for a cell with deformation parameters (0.07, 0.00). This map shows a subset of the trajectories that may exist in the RAC with the specified input conditions. Chaotic trajectories form scattered regions of points on this plot, while predictable quasiperiodic trajectories form curves [23]. The two-bounce modes emphasized in this manuscript manifest as the curves centered on φ = π and split over φ = 0 and φ = 2π. The outermost curves contain a large spread of angular positions, while the inner curves correspond to trajectories with small angular spread. For a given number of reflections, the individual points that make up the outer curves will be more widely spaced in angular position than the same number of reflections on an inner curve. This makes the outer trajectories preferable for use in OAS, as additional space between reflections will reduce the possibility of spot overlap and premature exits.
Fig. 4 (a) The surface of section for a quadrupole deformed circle. Chaotic trajectories produce scattered points on this plot, while some trajectories are confined to sets of curves. The two bounce trajectories considered here are confined to curves centered on φ = π and split over φ = 0 and φ = 2π. (b) A schematic showing the angle of incidence χ.
The calculated trajectories comprise a time-ordered set of points that describe the location of reflections off of the boundaries of the RAC. While an initial position and momentum were specified for the calculations, any position-momentum pair in the calculated trajectory can be selected as the input condition for the cell—resulting in the same trajectory—by fabricating the input/output aperture at the selected position and aligning the incident beam to give the specified momentum. Because the beam and input/output aperture have finite dimensions, the location of the aperture is nontrivial. The angular position of the reflections on one side of the yz-plane for a particular trajectory is shown in Fig. 5(a); there is a similar shape representing reflections on the other side of the RAC. The density of reflections (reflections/solid angle) is lower in the interior of the map as shown in Fig. 5(b). Since the aperture and beam occupy finite areas in this map, locating the aperture in a low density region increases the likelihood of selecting a configuration that supports long OPLs.
Fig. 5 (a) Angular position for each reflection on the hemisphere of the cell along the negative x-direction (Fig. 1(a)). Each point corresponds to a reflection on a trajectory with an OPL of 18 m. (b) Color histogram of the number of reflections per bin for 105 reflections on the given trajectory versus the angular position of the reflections.
Using the tools and strategies described above, we have isolated trajectories with characteristics such as long OPL, ease of coupling, and minimal spot overlap. These simulated trajectories may reach as many as 359 reflections before exiting the RAC, corresponding to an OPL of 18.1 meters in a cell with a volume of only 68 cm3, and an OPL to volume ratio of 26.6 cm−2. This 18 m trajectory may be coupled to in a RAC with deformations ϵxy = 0.0021 and ϵyz = 0.0076 at an input position of (φ0, θ0) = (6.25, 1.694) and an input angle of (φp0, θp0) = (−3.134, 1.375). The spot pattern of this trajectory is shown in Fig. 5.
Changing the average radius of the RAC while maintaining the other parameters, such as the angular input conditions, the beam width entering the RAC, and the size of the entrance/exit hole, has a superlinear effect on the OPL achievable by the RAC. For example, for the longest trajectory we simulated in this manuscript, doubling R0 results in an OPL of 104 m, a factor of 5.7 greater than the original OPL. In cases where longer OPLs are necessary, increasing the size of the RAC is a viable option. However, the volume is proportional to the cube of the radius of the cell, so the OPL to volume ratio could decrease if this is done. Increases in the OPL can also be realized by reducing the width of the initial beam or the radius of the entrance/exit hole.
4. Trajectory stability
In any device with potential applications in the field, consideration must be given to how the device will behave when conditions are not ideal. The approach presented here allows for deviations in the input conditions, such as would result from misalignment of the source with the RAC entrance/exit hole. Because input conditions are chosen such that the coupling point is in a region of the RAC with a low density of reflections, a small change in the input momentum or position will result in a small change to the overall trajectory, and the entrance/exit will remain in a low density region. Small changes in the input alignment will generally result in OPLs comparable to the designed OPL; however, it is possible that a change in the input results in a shorter OPL due to an early full or partial exit of the beam.
Many trajectories with very long OPLs exist in RACs with deformations similar to those shown here. Light along these optical trajectories may travel for a very large number (thousands) of reflections without returning to its exact original position, but most trajectories return to a point near the entrance within only 100 to 200 reflections. The finite size of the entrance hole therefore becomes a limiting factor for long OPLs. Given two nearby trajectories, one would expect nearly equal OPLs, but the difference between the two may result in one trajectory exiting the cell before the other, resulting in different OPLs for the trajectories and discontinuities in the calculated OPL despite smoothly varying input and cell parameters. The effect of small changes to input conditions on the overall OPL is demonstrated in Fig. 6 using deformation parameters of ϵxy = 0.0021 and ϵyz = 0.0076. The area of the region in this phase space characterizes the stability of the trajectory; regions with larger areas are less sensitive to changes in the input conditions. This effect is to be expected in multipass cells with long OPLs, and has been reported elsewhere [24]. Compared to the astigmatic Herriott cell at similar number of reflections, the RAC has an increased dependence on these small changes by a factor of 0.75 in one dimension and a decreased dependence by a factor of 2.5 along the perpendicular direction, which combined increase the solid angle of admittance by a factor of 1.9. For a 183 reflection orbit in an RAC with a one inch radius, this allows a window of acceptable misalignment with an area of 0.01 mm2 at the cell surface. When the OPL is increased to 18 m (359 reflections), the acceptable misalignment decreases by 70% to 0.003 mm2 at the surface of the cell. This is larger than the solid angle of acceptance for a Herriott cell with the same number of passes by a factor of 2. This decrease in sensitivity to input conditions indicates that the RAC should be no more difficult to align than a Herriott cell for a given number of reflections. Fortunately, other beam parameters such as beam size and major axis do not share this dependence on input conditions, and can therefore be chosen as described above. However, in cases where high sensitivity is required, high precision alignment may not be as feasible as using a larger cell.
Fig. 6 (a) A plot of the number of reflections that occur before any part of the beam exits the RAC for different input momenta. For certain input momenta the number of reflections is relatively small, but a small alteration to the angle at which the beam is input may greatly increase the number of reflections. Very long OPLs with more than 1500 reflections (> 75 m) are possible, but require extremely accurate alignment. (b) Number of reflections plotted versus input position.
5. Conclusion
We discussed optical trajectories in RACs used as multipass cells and numerically investigated the effects of changes in cell deformation and input conditions on the OPLs and GFTs of possible beam trajectories. Characteristics that can lead to longer OPLs were identified and used to select a trajectory with an OPL of 18 m. The stability of the optical trajectory was analyzed and shown to be favorable when compared to astigmatic Herriott cells for a given number of reflections. Because of this, RACs should be no more difficult to align than Herriott cells.
Compact multipass cells such as the one described here have many uses in defense, atmospheric studies, and medical diagnostics, and in any other application where highly sensitive measurements of trace gases must be made in the field with either fast or portable equipment. Specific applications include detection of volatile explosives and their precursors by trace gas monitoring [1–9] and diagnosis of asthma by measurement of NO concentration in exhaled breath [14]. Further topics of interest include the fabrication and testing of the RAC designed here, as well as the investigation of other types of optical modes that may be engineered in multipass cells of this type, such as whispering gallery modes.
Funding
This material is based upon work supported by the U.S. Department of Homeland Security, Science and Technology Directorate, Office of University Programs, under Grant Award 2013-ST-061-ED0001. The views and conclusions contained in this document are those of the authors and should not be interpreted as necessarily representing the official policies, either expressed or implied, of the U.S. Department of Homeland Security.
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