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Abstract
We present a particular solution of the Herriott design by replacing each spherical mirror with two coaxial nested spherical mirrors with different curvatures. The analytical solution of the nested MPC parameters is derived based on the principle of the Herriott cell and the symmetry of the pattern, which enables the rapid design of nested MPCs with different optical path lengths (OPLs). The nested MPC performance is explored by parameter analysis, the ray tracing method and the transfer matrix method. We further designed and manufactured a miniaturized and long-OPL MPC, which exhibits robustness, high mirror utilization and a good q-parameter of the output beam. The miniaturized nested MPC can achieve a 10 m effective OPL at a volume of 28.4 mL, and the long-OPL MPC can obtain a 100 m effective OPL at a volume of 1.3 L.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Multipass cells (MPCs) have been applied in various tunable diode laser absorption spectroscopy (TDLAS) sensors to improve detection sensitivity [1–5]. Classical MPCs mainly include Herriott cells [6], White cells [7] and circular multipass cells [8]. Herriott cells, composed of two coaxial spherical mirrors exhibiting the same curvature, are commonly used due to their robustness, concise design and easy operation [9–14]. However, its circle or ellipse pattern is often distributed in the periphery of the mirror; hence, the Herriot cell has the disadvantages of a low effective utilization area of the mirrors, a relatively small ratio of the total optical path length to the volume (RLV) and limited detection accuracy.
To overcome the limitation of classical Herriott cells, various strategies and approaches are proposed. The astigmatic MPCs with Lissajous patterns, also invented by Herriott, can achieve longer pathlength for a given cell volume but have the drawbacks of high mirror manufacturing costs [15]. The theoretical model of this astigmatic Herriott cell was further developed by McManus to deal with problems with finite manufacturing tolerances of astigmatic mirrors [16]. Silver presented a simple method for achieving dense pattern, using low-cost cylindrical mirrors or cylindrical-spherical mirrors [17]. By combing the configurations of both White and Herriott cells, Robert divided one mirror into two to increase the total reflections by adjusting the angle between the two optical axes [18]. Stephen So developed a configuration for dense spot pattern MPC by utilizing pairs of split spherical mirrors [19]. Hudzikowski proposed a unique method to design dense astigmatic-like pattern with multiple spherical mirrors [20]. These dense spot patterns lack of regularity, the alignment is complex and OPL cannot be easily determined.
In recent years, the two-spherical-mirror-based MPCs with dense patterns, such as concentric circles, independent circles and petals, have been developed [21–27]. MPCs with concentric circle patterns have been used in various applications due to their easily identifiable number of reflections, compact structure and high RLV [21,22]. However, two-spherical-mirror-based MPCs with dense spot patterns still have several obvious shortcomings: (1) Due to the off-axis ray propagation, the distance between the mirror and the mirror size are mutually limited. The OPL is usually on the order of a few meters to dozens of meters, making it difficult to reach more than 100 m [28]. (2) The aberration makes the shape of the spot become oval, which causes the q-parameter of the output beam to be poor. (3) The lack of equation-based MPC designs for dense spot patterns makes the design process relatively complicated and time-consuming [22,28–30].
Inspired by the principle of the Herriott cell and the symmetry of the pattern, we introduce a novel nested MPC concept to alleviate the above drawback, which replaces each spherical mirror in the classical Herriott cell with two coaxial nested spherical mirrors with different curvatures. The ray reflects multiple times in the nested MPCs and forms a reentrant symmetric concentric circle pattern on the mirror. The given functional relationship between the concentric circle parameters and the nested MPC parameters ensures the rapid nested MPC design. We further analyze the effect of the concentric circle parameters on the effective utilization area of the mirrors. The robustness analysis and the q-parameters analysis of the beam are analyzed by the ray tracing and the transfer matrix methods, respectively. The nested MPCs not only have the advantages of simple design and good output beam stability, such as the Herriott cell but also overcome the disadvantages of spot deformation and the OPL limit of the two-spherical mirror-based MPC with dense spot patterns. We designed and built a miniaturized and long-OPL nested MPC. The former MPC achieves a 10 m optical path in a volume of 28.4 ml, and the latter MPC achieves a 100 m optical path in a volume of 1.3 L. The nested MPCs with reentrant symmetric concentric circle patterns will be usable in portable sensors and high-sensitivity sensors.
2. Paraxial ray propagation in multipass cells
2.1 Herriott cell
The Herriott cell is composed of two coaxial spherical mirrors placed in opposite directions (Fig. 1). Considering the case when two spherical mirrors have the same curvatures, all the reflection points projected in the mirror plane (x-y plane) form a circle or ellipse. The projection of the $i$th reflection $(x_i, y_i)$ is described by the projection of the incident point $(x_0, y_0)$ and the 1st reflection $(x_1, y_1)$, the distance between mirrors , and the radius of curvature $R$:
Equation (1) has another form of expression:
The parameters $(A, B, \beta, \gamma )$ in Eq. (3) can be represented by the initial conditions $(x_0, y_0, x_1, y_1, d, R)$, and Eq. (3) is widely used for its simple form. When the product of the angle $\theta$ and the total reflection number $i_{total}$ is an integer multiple of $2\pi$, namely,
then the beam is reflected $i_{total}$ times and returns to the incident point position.In the case of two spherical mirrors with different curvatures, the coordinate $(x_j, y_j)$ of the $j$th point is determined by the reflection process at the $(j-1)$th point:
The radius $R$ is the radius of curvature of the mirror where the $(j-1)$th reflection point is located. $\cos {\theta }$ and $\sin {\theta }$ can be represented by $d$ and $R$ according to Eq. (2), and Eq. (5) is rewritten in the form:
2.2 Nested MPC
Based on the principle of the Herriott cell, ray propagation is closely related to mirror curvature, so we use nested mirrors with different inner and outer curvatures to form a new MPC called a nested MPC, as shown in Fig. 2. The nested MPC forms the reentrant symmetric concentric circle patterns, which exhibit the following characteristics: (1) The patterns formed on both mirrors are the same. (2) The output and input point positions coincide. (3) Spots are evenly distributed on each circle. (4) The pattern exhibits axial symmetry and rotational symmetry.
Fig. 2. Nested MPC based on two equal and coaxial nested mirrors. The curvature and size of the outer and inner spherical mirrors are $R_1$, $R_2$, $r_1$, $r_2$, respectively.
Figure 3 shows the schematic diagram of forming a symmetrical concentric circle spot pattern, which consists of several rotating semiellipses. Considering a special Herriott cell, see Fig. 3(a), each of the four reflection points are located on the same circle centered on the origin, forming n concentric circles altogether. The ellipse spot pattern is symmetric with respect to the long and short axes, and the total number of reflections of the ellipse spot pattern is $4n$. If rotation of the axial symmetric semiellipse is achieved, uniformly distributed symmetric concentric circle pattern will be obtained. Therefore, the inner mirror size $r_2$ should be set in the middle of the $n$th circle and the $(n-1)$th circle. And the special curvature of the inner mirror can realize semiellipse pattern rotation each time the beam reaches the inner spherical mirror. The concentric circle pattern is determined by the ratio of the semiminor axis to the semimajor axis $B/A$ and the total number $K$ of the unit semiellipse pattern.
Fig. 3. Process diagram of the concentric circular spot pattern formation. (a) The beam propagating between two spherical mirrors with radius $R_1$ forms the axial symmetric ellipse spot pattern, and the yellow area is the unit semiellipse spot pattern. The black and red dots represent reflection points on both mirrors. (b) The semiellipse pattern symmetry and rotation are realized by the inner spherical reflection. Line $h$ is the axis of symmetry, and the angle $\alpha$ is the angle of rotation. (c) The concentric circular spot pattern of $K$ = 5 and $\alpha = 4\pi /5$.
There are no other reflection points between two consecutive reflection points in the axial symmetric ellipse spot pattern, which is the case when $k = 1$ in Eq. (4). The initial point $P_0$ is set at the point on the outermost circle in the first quadrant, then the $1$st to $n$th reflection points are exactly on the $1$st to $n$th concentric circles (from outside to inside). $P_i$ represents the $i$th reflection point, and its projection coordinates $(x_i, y_i)$ are:
$A, B$ are the lengths of the semimajor axis and semiminor axis of the ellipse, respectively. The outer mirror size $r_1$ should be slightly larger than the length of the semimajor axis $A$, $r_1>A$. According to Eq. (2), the radius $R_1$ satisfies the equation:
$P_n$ is the intersection of the beam with the inner spherical mirror for the first time, and it is used as a new entrance point to produce reflection points $P_i^{'}$ on the new semiellipse. When $P_n$ and $P_{n+1}^{'}$ are at an equal distance from the optical axis, two semiellipse spot patterns are axisymmetric according to the principle of reversibility of light. Because the semiellipses themselves are also axisymmetric patterns, the two semiellipses have a rotational symmetry relationship (Fig. 3(b)), and this symmetry ensures the subsequent semiellipse rotations. According to Eq. (6), the coordinates of $P_{n+1}^{'}$ can be obtained:
Due to the rotational symmetry relationship between the two semiellipses, semiellipse 1 rotates angle $\alpha$ around the z-axis to coincide with semiellipse 2, and the coordinates of $P_{n+1}^{'}$ can also be expressed as:
From Eqs. (7), (10) and (11), we find that
Figure 3(c) shows the reentrant concentric circle pattern when $K$ = 5 and $\alpha = 4\pi /5$.
In summary, the nested MPC parameters $R_1,R_2,P_0,P_1$ can all be represented by the variables $(A,\alpha,\theta,d)$. Variables $A$, $\alpha$, and $\theta$ are the parameters that characterize the features of the concentric circular pattern, representing the pattern size, the spot density, and the concentric circle number, respectively. Therefore, in the process of nested MPC design, appropriate cell parameters can be selected according to the specific concentric circular pattern parameters $(A,\alpha,\theta )$.
3. Analysis of the nested MPC
The effective utilization area of the mirrors, the q-parameter of the output beam and the robustness of the cell are essential factors in MPC design. The corresponding relationship between the concentric circular spot pattern parameters and the nested MPC design parameters has been given in Section 2, in which the parameter $A$ value can be set as a constant in the following discussion because it only influences the spot pattern size. According to Eq. (8), $\theta$ is a function of $n$. Therefore, in this part, we mainly explore the impact of parameters $(n, \alpha )$ on the performance of the nested MPC, introduce the analysis method in detail, and show the results through numerical simulation.
3.1 Analysis of the reentrant condition and mirror utilization
To satisfy the reentrant condition, the product of the angle $\alpha$ and the integer $K$ should be an integer multiple of $2\pi$, namely,
where $m$ is an integer less than $K$, and $K$ and $m$ have no common divider other than 1. $\alpha$ is a function of $(K,m)$. When $m$ has multiple values, the larger $m$ value corresponds to a larger angular interval $\alpha$ between two consecutive semiellipses, and the different reentry modes are obtained.The effective utilization area of the mirrors depends critically on the distribution of spots and the total number of spots. A high effective utilization area of the mirrors can improve the RLV under the avoidance of light spot overlap conditions [31]. The total number of reflections of the concentric circle spot pattern $i_{total}$ is equal to the product of the number of spots on the unit semiellipse $2n$ and the number of semiellipses $K$, namely, $i_{total} = 2nK$. With fixed values of K and n, the effective utilization area of the mirrors is determined by m. With $n$ = 4 and $K$ = 11, the value of $m$ can be 2 to 5 under the condition of satisfying Eq. (13). The larger the m value is, the greater the spacing between the concentric circles, and the higher the effective utilization area of the mirrors (Fig. 4). The detection sensitivity of the measurements can be greatly degraded by undesirable interference fringes, if any adjacent spot on the same side leaks out from the mirror hole [32]. We select the widely spaced spot patterns with high mirror utilization and set the coupling hole on the outermost circle to avoid the spot overlap.
Fig. 4. Concentric circular pattern projection in the x-y plane for various $m$ values with $n$ = 4 and $K$= 11. The semiellipses are numbered according to the order of formation. The dots of the same color are located in the same mirror, and the six-mount star represents the incident point.
3.2 Analysis of robustness
The robustness of the MPC is related to the position change of the reflection points caused by the variation in environmental factors. The line-sphere equation can be used to solve the exact position of the intersection point between the beam and the spherical mirror, and it can verify the principle of the nested MPC. The line-sphere model regards the beam as a line and the spherical mirror as a spherical crown to solve the intersection point position and the reflection direction. The relative position of the line and sphere has an influence on the coordinate equation of the intersection point [33]. For the nested MPC, the incident point is inside the sphere, due to the unidirectional light propagation, the line has only one intersection $P$ with the sphere [33]:
An important factor leading to the deformation of the spot pattern is the variation in the first reflection point $P_1$, which is caused by the offset of the incident beam. The variation range is set on a circle centered on the original $P_1$ position, and the radius length is 1% of the semimajor axis $A$. To obtain a similar number of reflections, $(n, K)$ are (2, 32), (3, 22), (4, 16) and (5, 13). The numbers of reflections are 128, 132, 128, and 130, respectively. To obtain a high effective utilization area of the mirror, the corresponding $m$ values are set to 15, 9, 7, and 6, respectively. Figure 5 shows the robustness of the concentric pattern with the above parameters when the $P_1$ position deviates by 1% of the $A$ value positively to the y-axis. We can see that the output point position always coincides with the input point position, although the $P_1$ position varies slightly. Similar results were found when the direction of the $P_1$ position deviated in other directions. The nested MPC with a small $n$ value has better stability at every point position.
Fig. 5. Deformation of concentric circle patterns on the incident end mirror when the y-offset of $P_1$ is 1% of the $A$ value. The black and red dots represent reflection spots before and after the offset, respectively, and the output point position always coincides with the input point position.
3.3 Analysis of the q-parameter and the spot size on the mirror
Maintaining the same q-parameters of the input and output Gaussian beams is one of the conditions needed to characterize the stable ray propagation in the MPC, and it is also helpful to improve the signal-to-noise ratio [34–37]. For the concentric circle pattern formed on the mirrors of the two-spherical mirror-based MPCs, spots will become oval due to the aberrations [21,22,30]. However, for the concentric circle pattern formed on the mirrors of the nested MPCs, the paraxial condition makes the spot shapes circular, and the reentry condition allows this configuration to maintain the q-parameter of the input and output beams.
The spherical mirror has the same focal length in the x and y directions under paraxial conditions, so the q-parameter can be solved with $2 \times 2$ transfer matrices [34]. The q-parameter and the spot radius $w$ are the characteristic parameters of the Gaussian beams and are correlated with the distance $z$ from the beam waist, the beam waist radius $w_0$, and the laser wavelength $\lambda$. Taking one pass as an example, one pass consists of a reflection and a transmission, the matrix $M$ can be written as:
The discussions presented in Sections 3.1 and 3.2 show that the stability and mirror utilization area of the 3-concentric-circle pattern are superior, so we chose the pattern with the parameters $(n, K, m)$ of (3, 27, 13) to explore the effect of incident conditions on spot size. The ABCD matrix in the ray tracing algorithm is the same as the matrix for calculating the q-parameter in the paraxial approximation condition, so the q-parameter approximation of the input and output beam remains unchanged under the reentry condition. The above conclusion was verified by MATLAB simulation of the propagation of Gaussian beams in nested MPCs, the waist radius $w_0$ and the laser wavelength $\lambda$ of the incident beam are set to 5% of the $A$ value and 520 $nm$. At the same time, it is found that the focus position of the incident beam will seriously affect the size of the intermediate spot. When the near-collimated beam with a beam waist near the incident aperture enters the MPC, all radii of the reflection spots are relatively small, and other focal lengths will cause the rapid deterioration of the q-parameter of the intermediate spot. However, the beam waist radius and the collimation distance of Gaussian beams are a pair of parameters that are difficult to balance. Therefore, for the long-OPL MPC with two mirrors separated far apart, we should choose incident light with small divergence angles for the sake of an appropriate spot size. Figure 6 shows the spot pattern simulation results of the collimated incident beam and incident light with a small divergent angle, which all meet the conditions necessary to avoid spot overlap.
Fig. 6. Simulation diagrams of the spot size on the incident end mirror surface when $(n, K, m)$ is (3, 27, 13). (a) the simulation result when the incident beam is the collimated beam, and (b) is the result when the incident beam has a small divergence angle. The little black circle in the pattern represents the incident hole.
4. Practical design example
A schematic diagram of the nested mirror is shown in Fig. 7. A central hole with a radius of $r_2$ is dug on the spherical mirror M1 with the curvature radius of $R_1$ and the size of $r_1$, where a new spherical mirror M2 with the curvature of $R_2$ can be placed, and the other set of mirrors M3 and M4 is identical to it. The entrance and exit holes can be set on the same mirror or on both sides, and we experimented with both structures. The inner and outer mirrors are made of BK7 glass in 10 mm thickness. These mirrors have a surface quality 60/40 and are coated with multilayer dielectric coating. The reflectivity of the mirrors is > 99.5 % @520 nm. They are both spherical mirrors, which have the advantages of easy fabrication and low cost. The outer mirror was manufactured with through hole whose diameter is slight larger than the inner mirror within a tolerance of < 0.05 mm. The inner and outer mirrors are nested in a coaxial configuration, and then some mounting adhesives are used for gluing both inner and outer mirrors together. The mirrors can be adjusted in a way that is similar to the alignment and adjustment of the traditional Herriott cell. A laser diode at 520 nm was utilized to reproduce the visible spot patterns of symmetric concentric circle pattern. The laser beam is collimated by Thorlabs FiberPort (CFC5A-A, PAF2A-11A) for the miniaturized and long-OPL versions, and the laser spot diameter is around 1mm and 2mm respectively.
Fig. 7. Nested mirror drawings. (a) and (b) show the projection diagrams of the z-y and x-y planes, respectively. M1 and M3 are two large spherical mirrors with a center hole, and M2 and M4 are two small spherical mirrors. $C_1, C_2, C_3$ and $C_4$ are their centers of curvature, respectively.
The nested MPC design can be determined by five independent variables $(A, n, K, m, d)$, where $A$ and $d$ determine the mirror size and spacing, $K$ and $n$ determine the number of reflections, and $m$ determines the spot pattern. Once the parameters are optimized, the nested MPC can also be scaled by applying linear scaling law to achieve different OPLs. Considering the analysis of the nested MPC performances in Section 3, the design parameters are determined and shown in Table 1. The miniaturized nested MPC based on two one-inch mirrors separated 56 mm apart achieves an OPL of 10 m, which has a smaller volume and a longer OPL compared to the palm-sized MPC [3,25]. The long-OPL MPC based on two two-inch mirrors separated 641 mm apart can achieve a 100 m OPL, and the volume is 1.3 L.
Table 1. The nested MPC parameters.
The nested MPC parameter values can be calculated according to the equations in Section 2. For the miniaturized nested MPC, the value of B/A is 0.282, the inner and outer curvatures of the nested mirror are 303 mm and 418 mm. The incident beam is injected from $P_0$ (10.63, 0.8) and strikes the other mirror at $P_1$ (10.63, -0.8). For the long-OPL MPC, the values of B/A, $P_0$ and $P_1$ are 0.284, (19.32,1.47), (19.32,-1.47) respectively. The inner and outer curvatures of the nested mirror are 3337 mm and 4785 mm. The total number of reflections of the long-OPL MPC under reentry conditions is 162. With the entrance and exit holes set on both sides, the 156th reflection point with high stability and a good q-parameter is selected as the exit point. The spot pattern simulation and experimental diagrams of the miniaturized and long-OPL MPCs are shown in Fig. 8. The experimental diagrams agree well with the simulation, which demonstrates the correctness of the principle. To investigate the mechanical and thermal stability, we follow the procedure for stability analysis of the final point as described in a previous paper [29]. The distance between mirrors, the x-offset and y-offset of the first reflection point $P_1$ was varied to evaluate the effects of small perturbations of the incident angles and distance. The tolerances are calculated by the MATLAB-based ray tracing program to calculate whether any intermediate point hits the inner border ($r_1$) or outer border ($r_2$) of mirror or the final point cannot pass through the exit hole. The tolerances of the miniaturized and long-OPL MPCs are shown in Table 2.
Fig. 8. Simulation diagrams and actual diagrams of the concentric circle pattern. (a)(b)(e)(f) are the spot patterns of the miniaturized nested MPC gas chamber. (c)(d)(g)(h) are the spot patterns of the long-OPL nested MPC.
Table 2. Stability analysis of the nested MPC
5. Conclusion
We propose a novel MPC consisting of two equal and coaxial nested mirrors. Light propagation in the nested MPC can form a reentrant symmetric n-concentric circle spot pattern with different OPLs. We present the equation-based nested MPC rapid design method and analyze the influence of independent variables on reflection times and mirror utilization. The robustness and the q-parameters of the beam are further explored by ray tracing and the transfer matrix method. The suitable cell parameters and the near-collimation incident beam enabled nested MPCs with high stability and good q-parameter incident light.
We designed and built a miniaturized and long-OPL nested MPC, which achieved OPLs of 10 m and 100 m with less than 200 reflections, and the volumes were only 28.4 mL and 1.3 L, respectively. The visible spot patterns aligned with a green trace laser are in good agreement with the theory that proves the correctness of the principle. Simple, stable and compact nested MPCs with reentrant symmetric concentric circle patterns are suitable for the development of various sensors, including portable sensors and high-sensitivity sensors.
Funding
Beijing Normal University (10100-312232102).
Acknowledgments
This work was supported by the Scientific Research Foundation of the High Level Scholars of Beijing Normal University.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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