# Optical Axis Perturbation Analysis for the Unit-Magnification Multipass System

Authors: Yin Guo; Liqun Sun; Zheng Yang
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

The optical axis sensitivity for the unit-magnification multipass system (UMS) is presented by using a general misaligned optical element transfer model. The generalized sensitivity factors SD1, SD2, ST1, and ST2 influenced by both the axial and angular misalignments of the objective mirrors in a UMS have been calculated for the first time. The Bernstein-Herzberg White Cells are used as an example, and their alignment tolerance and stability properties are found when their configurations change. The analysis in this paper is helpful for the design of other kinds of multipass gas cells (MGC) with high robustness and avoiding the violent vibration of the optical axis when the misalignment of each mirror is controlled within the tolerance range. Among the five possible perturbations sources, the misaligned factorsδix,δiy,θix have more effects on the output beam’s position and the perturbed sources from δix,θix and δiy,θiy have more impacts on the output beam’s slope referred to as x-axis and y-axis, respectively. Higher reflection times mean smaller tolerance range. The results benefit the multipass cell design and the precise alignment of the mirrors within the cell with the purpose of long-term stability in measurements.


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## Body

## 1. Introduction

Unit-magnification multipass system (UMS) [1] has been widely used in the absorption spectroscopy instrument for decades in the field of industrial and environmental science [2]. White cell [3], Herriott cell [4], circular cell [5], and Chernin-type systems, that is, the multipass matrix system (MMS), are the most general categories of the multipass system to obtain very long optical absorption path length in compact space to enhance the detection sensitivity. The white cell and MMS are widely known for their relatively large numerical aperture and large volume [1]. This kind of cell is commonly applied to broadband absorption spectroscopy to retrieve multiple species, such as ultraviolet differential optical absorption spectroscopy (UV-DOAS) and Fourier transform infrared spectroscopy (FTIR). On the other hand, Herriott-type cell and circular cell are commonly applied to the laser-based absorption spectroscopy and can provide very long path length in relatively small volume. These cells are usually applied to harsh in situ gas sensing environment, and highly stable conjugated alignment conditions of optical elements are required for a high throughput signal. No matter what kind of cell is used, optical system’s stability, namely, optical axis stability properties, is an important issue when used in the long term. Vibration-induced misalignment of the mirror in position will make the output signal intensity reduce. In that case, realignment of the mirror or calibration of the system is required. Tolerance data of a given UMS is calculated in this paper for the optomechanical engineers to ensure the long-term stability under harsh environment. Meanwhile, the tolerance of the reflective optics in the UMS is also investigated considering the different configuration parameters.The UMS is noncoaxial, noncoplanar, and partly asymmetric which can be regarded as the multipass 3D optical system [1]. It consists of two blocks of spherical mirrors with identical curvature radii R, objective mirrors (Mi) on one block, and the field mirrors (Fi) on the other side of block face-to-face with an interval, approximately R. The entrance aperture is placed on one side of the field mirror, and output window lies on the other side of the field mirror. After travelling multiple passes in the UMS, the light source formed multiple images with almost the same size on the field mirror. Previous researchers focused on the astigmatism calculation and its correction strategy by using reversed astigmatism compensating methods [1, 6]. It aims at obtaining less degraded beam when the beam traveled out from the output window, especially for the white cell series, that is, Bernstein-Herzberg White Cell (BHWC) [7, 8] and Pickett-Bradley White Cell (PBWC) [9]. Other various kinds of white cell are based on these two basic configurations to get enhanced spot pattern on the field mirror, that is, Chernin-type cell [10, 11] (three-objective matrix multipass system and four-objective matrix multipass system). However, the analysis of the optical axis perturbation properties for a certain UMS has not been reported before. For a given UMS, its optical layout is determined by a given number of passes, spot pattern, input beam’s numerical aperture, and diameter. The consideration of the system’s alignment properties and optical axis stability should be carefully concerned when constructing the system and keep them in a robust condition for long-term measurements. In this case, an unstable or lack of tolerance analysis for the fabrication and alignment control to these multiple mirrors in the UMS would lead to some unexpected results, for example, temperature and pressure induced misalignment or the excess of alignment tolerance in the mechanical construction. In other words, the optimized choice of UMS configuration parameters may benefit the long-term measurements and avoid absorption path length variations induced by misalignment of the optical elements or signal obstruction. Moreover, the optical axis sensitivity analysis will also improve the alignment process.Ray matrix method has been widely used to analyze optical axis perturbations in various 3D optical systems [12–15]. Sheng [12] used a matrix approach to determine the optical axis perturbation sensitivity of an out-of-plane stable ring resonator. Based on the augmented 5 × 5 ray matrix method, Chen et al. [13] presented general calculation method of nonplanar ring resonator’s (Zero-Lock Laser Gyroscopes) beam position and proposed precise beam controlling way. Yuan and Long [14] obtained optical axis stable region in terms of axis decentration and tilt with augmented 5 × 5 matrix formulations. After that, they reanalyzed the sensitivity factors with the proposed merit indexes to characterize the nonplanar ring resonator with the same method within the defined coordinate system [15]. Therefore, by employing the augmented 5 × 5 ray matrix method, beam behavior in our multipass systems (also called UMS) can be modeled in the paraxial approximation domain (incident angle less than 5 degrees). In this paper, we develop a numerical method by incorporating 5 × 5 augmented ray matrix to analyze the optical axis sensitivity in the UMS. Any optics elements in the UMS have their possible perturbations, thus resulting in the error on the output beam in position and direction. The spherical reflection mirror’s transfer matrix is derived based on a general model of the paraxial misaligned optical element. Considering the imaging rotation due to the changing of reflection plane and coordinate system in each reflection, the rotation matrix is used. The extent of the influence of the output beam is different depending on the choice of UMS’s configuration parameters. By utilizing the defined sensitivity factors influenced by both the axial and angular misalignments of the objective mirror in the UMS, the rule of optical axis perturbation is investigated in detail. According to the different initial optical layout configurations of the UMS, the numerical simulation presents the rule of optical axis sensitivity, and the primary perturbation sources are summarized. BHWC is used as an example to analyze its optical axis sensitivity. The method proposed can also be applied to other various multipass cavity cells and their preliminary design.


## 2. General Analysis of a Misaligned Optical System

The general ray matrix of an optical system with axial and angular misalignment can be modeled by using the following form [16]:(1)roxrox′royroy′1=AxBx00ExCxDx00Fx00AyByEy00CyDyFy00001rixrix′riyriy′1,where rix,rix′,riy,riy′,1T and rox,rox′,roy,roy′,1T are the input and output ray matrix, respectively, referred to as the reference plane. rix, riy, rox, and roy are the input ray and output ray height in the reference xoz and yoz plane, respectively, and are called optical axis decentration. rix′, riy′, rox′, and roy′ are the angles (in radians) of the input ray and output ray slope in the reference xoz and yoz plane, respectively, and are referred to as optical axis tilt. Ax, Bx, Cx, and Dx are the standard ray matrix elements in the tangential plane (xoz plane). Ay, By, Cy, and Dy are the standard ray matrix elements in the sagittal plane (yoz plane).To solve the misaligned value for the transmission matrix analytically in our system (see (1)), we firstly construct a general model of a misaligned optical element [17], illustrated in Figure 1.Figure 1
The general model of a paraxial misaligned optical element.A misaligned optical element in the paraxial domain can be regarded as the case when the element axis does not coincide with the optical axis. Any optical elements have six possible degrees of freedom (three in axial and three in angular error) relative to the element axis. Optical axis can be defined arbitrarily to describe ray vector. Conveniently, we take ray’s propagation in one tangential plane as an example as shown in Figure1. Ray vector referred to the element axis, and the optical axis can be written as si,si′ and ri,ri′, respectively, where si and ri denote the displacement in the reference plane; si′ and ri′ represent the angle in radian in the reference plane. For the two particular axes illustrated in Figure 1, the element axis can be expressed relative to the optical axis by using a vector ΔiΔi′, where Δi is the height of axial vector and Δi′ is the slope in a radian of the axial vector in the reference plane (i=1 denotes the objective spacing and i=2 denotes the imaging spacing). Considering the paraxial approximation when in condition of possible perturbation, we have the following relationships of the perturbed element axis between input and output plane: (2)Δ′=Δ2-Δ1L.Then, we introduce our misalignment vectors to describe our misaligned valuesP1≡Δ1Δ1′ and P2≡Δ2Δ2′, where Δ1=n1Δ′, Δ2=n2Δ′, and n1 and n2 are the refractive indices of input and output space, respectively. The connection between the two misaligned vectors can be written as(3)P2=PΔ·P1,where PΔ=1L/n10n2/n1.Moreover, ray vector referred to the optical axis and the element axis in the input and output ends can be described as(4)rin=r1r1′=s1s1′+Δ1Δ1′=sin+P1,(5)ro=r2r2′=s2s2′+Δ2Δ2′=so+P2.Because the system transmission matrix related to the element axis can be regarded as nonmisaligned system, which follows the law of traditionalABCD matrix measured on the input and output plane, we can connect sin and so by using the following formula:(6)s2s2′=ABCD·s1s1′.Up to now, we can put (6), (4), and (3) into (5); we can easily get the output ray vector written as(7)ro=r2r2′=M·sin+MΔ·P1=M·rin+MΔ-MP1=M·rin+EM,where the matrix EM2×1 is the error matrix induced by possible misalignment error; it is defined as(8)
E
M
≝
M
Δ
-
M
P
1
=
1
-
A
L
n
1
-
B
-
C
n
2
n
1
-
D
Δ
1
Δ
1
′
=
Δ
1
′
=
n
1
Δ
′
1
-
A
L
-
n
1
B
-
C
n
2
-
n
1
D
Δ
1
Δ
′
=
E
i
F
i
,where i=x and i=y represent the error matrix in tangential plane and sagittal plane, respectively.We use these results for a general misaligned paraxial optical system in one tangential plane. Thus, we arrive at 3 × 3 transmission matrix:(9)roxrox′1=AxBxExCxDxFx001rixrix′1.The matrixEM2×1 can be extended to get displacement error matrix Ex and angle error matrix Fx and thus we have solved Ex and Fx as follows:(10)Ex=1-AxΔ1+L-n1BxΔ′,Fx=-CxΔ1+n2-n1DxΔ′.According to the method described above, the specific values ofEx, Fx, Ey, and Fy in (1) can be derived in given perturbation amount for an individual paraxial optical element.Because all optical elements in the UMS are the spherical mirror, it is necessary to analyze the generalized transmission matrix of spherical mirror in case of all kinds of possible perturbation based on (10).A spherical mirrorMi (radii Ri) has been chosen as an example to demonstrate the misaligned sources in Figure 2. The incident angle is Ai (smaller than 5°); δix, δiy, and δix are its three kinds of axial displacements along the axes Tix, Tiy, and Tiz, respectively. θiz, θix, and θiy are three types of angular misalignments around the rotational axes of Rix, Riy, and Riz, respectively. The term θiz can be ignored due to the symmetry of the spherical mirror.Figure 2
Axial and radial misalignment analysis of a spherical mirrorMi. (a) Axial displacement δiz of Mi along the translational axis Tz. (b) Axial displacement δix of Mi along the translational axis Tx. (c) Definition of a mirror’s angular misalignments θix and (d) angular misalignment of the spherical mirror Mi around rotational axis Rix. Ai is the incident angle, Mi, the blue solid arc, is the mirror’s initial position, Mi′ and Mi′′ in the red solid arc in (a), (b), (c), and (d) are the position of the mirror after possible axial or angular perturbations. P1, P2, P3, P4, and P5 are the reflecting points. P0 is a point on ray Li1; O1 and O1′ are the curvature centers of Mi and Mi′. Tix, Tiy, and Tiz are three translational axes (right hand); Rix, Riy, and Riz are three rotational axes (right hand); L1i and L2i are two parallel incident rays; Li1 is an incident ray; Lo1 and Lo2 are two reflection rays; xi-yi are the coordinate axes (left hand) of the incident ray; xo-yo are the coordinate axes of the reflected ray; θ1 is the angle between line O1′P1 and line O1′P3; θ2 is the angle between line O1′P4 and line O1′P5; θ3 is the angle between line O1′P1 and line O1P1; Δx1 is the distance between P1 and P3; Δx2 is the distance between P4 and P5; Δy is the distance between P1 and P4.
(a) (b) (c) (d)The termsθix, θiy, δix, δiy, and δiz are used to describe five kinds of possible perturbation sources for a spherical mirror. In this paper, θix and θiy are angular misalignments; δix, δiy, and δiz are called axial displacements or axial misalignments.In [16], Yuan et al. have discussed and derived the generalized ray matrix for a spherical mirror reflection in a square ring resonator. They proposed generalized spherical mirror reflection-transmission matrix based on the geometrical method. In this paper, we derived the generalized spherical mirror reflection matrix based on a general misaligned optical element model to deduce the error matrix ExFxEyFyT in (1). It can be rewritten as the following equations: (11)Ex=1-AxΔ1x+L-n1BxΔx′,(12)Fx=-CxΔ1x+n2-n1DxΔx′,(13)Ey=1-AyΔ1y+L-n1ByΔy′,(14)Fy=-CyΔ1y+n2-n1DyΔy′.As is widely known for the reflection system,n1=-n2=1. For a spherical mirror, its ray transfer matrix (radius Ri with incident angle Ai) can be written as (15)AxBx00CxDx0000AyBy00CyDy=1000-fm-1100001000-fs-11,where fm=-Ricos⁡Ai/2 is the focal length in meridional plane (x-z plane) and fs=-Ri/2cos⁡Ai is the focal length in the sagittal plane (y-z plane). Therefore (11)~(14) can also be simplified as the following expression in the case of spherical reflection mirror:(16)Ex=Δ2x-Δ1x,(17)Fx=2Ricos⁡AiΔ1x-2×Δx′,(18)Ey=Δ2y-Δ1y,(19)Fy=2cos⁡AiRiΔ1y-2Δy′.To solve the terms Ex, Fx, Ey, and Fy, Figure 2 illustrates the diagram of optical axis movement. As shown in Figure 2(a), the axial displacement δiz is analyzed firstly, when Ai≠0 (the common case for the UMS). To discuss this effect in detail, two parallel incident rays L1i and L2i are chosen, for example. L1o is the reflection ray at the point P1 on Mi. L2o is the reflection ray of L2i via the reflection at the point P2 on Mi′. In the case of the displacement misalignment along Tiz axis with δiz for the mirror Mi, the ray L2i and L2o is the perturbed incident and output compared with the incident ray L1i and output ray L10 before the mirror’s perturbations. In that case, the reflecting point P2 (a point on the element axis) relative to its optical axis before reflection and after reflection is expressed as Δ1x=-δizsin⁡Ai and Δ2x=δizsin⁡Ai, respectively. We put them into (16) and obtain Ex=2δizsin⁡Ai. There is no change in slope between element axes in the input and output ends of the spherical mirror. We have Δx′=0 and then we can obtain Fx=-2δiztan⁡Ai/Ri according to (17). Similarly, in the sagittal plane, Δ2y=0, Δ1y=0, and Δy′=0 as shown in Figure 2(a). So Ey=Fy=0 in this case. Therefore, if the axial displacement along the axis Tiz is δiz, the ray matrix elements Ex and Fx should be modified into 2δiz·sin⁡Ai and -2δiztan⁡Ai/R, respectively.Next, the spherical mirror’s other two axial displacementsδix and δiy need to be considered too. As illustrated in Figure 2(b), δix is chosen as an example for analysis. The line O1′P4 is the element axis, the coordinates of point P4 (a point at the element axis) referred to as the optical axis in the input ends and output ends are Δ1x=0 and Δ2x=Δy×sin⁡2Ai, respectively. Because Δy≈Δx2·sin⁡θ2, for approximation, sin⁡θ2≈θ2≈Δx2/Ri and Δx2≈δix. Then the error matrix in this case becomes Ex=δix2/Ri×sin⁡2Ai. This term can be ignored in this paper due to δix≪Ri. The changing slope of the element axis between the input ends and the output is Δx′ relative to the optical axis. In our case, element axis from input ends referred to as the optical axis is the angle between the line O1P1 and the line along the incident ray Li1, while the element axis from the output ends referred to as the optical axis is the angle between the line O1′P4 and the line along the incident ray Li1. Therefore Δx′ should be the angle change between line O1P1 and O1′P4, namely, two element axes angles (initial element axis and perturbed element axis), θ2. As shown in Figure 2(b), Δx′=-θ2=-δix/Ri. The error matrix Fx=2/Ricos⁡AiΔ1x-2×Δx′=-2×-θ2=2δix/Ri. Because δix makes no contributions to Δ2y, Δ1y, and Δy′, Ey and Fy will be zero in this case. Similarly, influenced by δiy error matrix of Ey=δiy2/Ri×sin⁡2Ai, this term can also be ignored due to δiy≪Ri and Fy=2cos⁡Ai/Ri×Δ1x-2×Δx′=-2×-θ2=2δiy/Ri based on the same law. In summary, when the spherical mirror is misaligned by axial displacements δix and δiy, its ray matrix elements Fx and Fy should be modified into 2δix/Ri and 2δiy/Ri, respectively.The angular misalignmentθix is chosen as an example to show the influence of angular perturbations (the analysis of angular misalignment θiy is similar; θiz can be ignored due to the symmetry shape of the spherical mirror). Figure 2(c) shows the definition of θix. We look at the mirror Mi behind the rotation axis Rix; when the mirror Mi rotates clockwise on its rotation axis Rix, the sign of misalignment angle of θix is negative. When it rotates counterclockwise on its rotation axis Rix, the sign of misalignment angle of θix is positive. In Figure 2(d), because the reflecting point P1 on the element axis experiences no change in displacement referred to as the optical axis, we get the term Δ1x=Δ2x=Δ1y=Δ2y=0.And there is no change in slope angle for element axis inyz plane; we have Δy′=0. Element axis has slope variation between the input ends and the output ends referred to as the optical axis, from ∠P0P1O1 to ∠P0P1O1′, so the term Δx′=-θ3=-θix. According to (16)~(19), the standard ray matrix element Fx in (1) should be modified into 2θix and Fy should be modified into 2θiy with the consideration of the mirror’s angular misalignment θiy.In summary, a generalized ray matrixMRi,Ai for a spherical mirror Mi with all possible kinds of perturbation sources including θix, θiy, δix, δiy, and δiz can be written as(20)MRi,Ai=10002δiz·sin⁡Ai-2Ri·cos⁡Ai100-2δiz·tan⁡AiRi+2θix+δixRi0010000-2×cos⁡AiRi12θiy+δiyRi00001.Equation (20) is the same as Yuan et al. proposed [16], but it is explained and derived in a new way based on the modeling of a general misaligned optical element’s ray matrix (shown in Figure 1). The misaligned matrix can be understood with the clear meaning of each symbol in definitions. The derivation process of Yuan et al. was based on the geometrical compensating method. These two methods can be mutually proven in different ways. Furthermore, the general misaligned optical transfer matrix model can be easily applied to other various optics elements in different kinds of the UMS.


## 3. Analysis of the BHWC

BHWC [7, 8] is one of the traditional and widely known optical schemes in the UMS. Another commonly used optical layout is the PBWC [9]. Other types of complicated multipass gas cell are the multiple usages of BHWC or PBWC to increase the number of reflection times when incorporated with different reentrant way. In this paper, we just discuss the BHWC in detail as an example.As shown in Figure3, we can use a parameter set n,R,dc,dr to establish a UMS. N is the total reflection times of the multipass cell. As for BHWC, N=4n(n=1,2,3,…), where n is the number of spots for the bottom row on the field mirror; for example, in Figure 3, n=4.Figure 3
Schematic diagram of BHWC. Two objective mirrorsM1 and M2 lay on one side and a field mirror F locates on the other side with an interval close to R. All mirrors have identical radii of curvature R. Ci(i=1,2,F) is the curvature center of each mirror. Two-row images are formed on the field mirror with the column spacing dc=2c and row spacing dr=h, where c is the distance of curvature center of M1 and M2. p is the center shift of the input beam which represents the center point on the mirror M1.As is well known, the UMS are noncoaxial, noncoplanar, and partly asymmetric because the meridional planes of each reflection are not coplane. The optical transmission matrix can be modeled as several cascades misaligned optical elements. An individual reflection on a spherical mirror with an off-axis incident angleAi can be considered as a transmission through a thin astigmatic lens with its optical axis in the direction of a chief ray. The transfer matrix for the jth reflection Mj can be written as [18](21)Mj=RφjMR,AjTLj,where MR,Aj is the generalized spherical mirror’s reflection matrix (described by (20)). TLj is 5 × 5 ray transfer matrix for free space propagation in a homogeneous medium and Rφj is the coordinate rotation matrix when considering the change of each reflective meridional plane, where φj is the meridional plane rotation angle relative to the last meridional plane. Aj is the incident angle of the jth reflection on a spherical mirror. TLj and Rφj are denoted as(22)TLj=1Lj00001000001Lj00001000001,(23)Rφj=cos⁡φj0sin⁡φj000cos⁡φj0sin⁡φj0-sin⁡φj0cos⁡φj000-sin⁡φj0cos⁡φj000001.Therefore, the UMS transmission ray matrix can be represented by a general 5 × 5 transfer matrix as(24)M=TLn·∏i=1i=NRφi·MiR,Ai·TLi.The system transfer matrixM exhibits beam properties of the bundle of the narrow beam along the chief ray of the UMS. The input ray matrix rin=00001T; then the output ray matrix r0=M·rin=roxrox′royroy′1T.To analyze the influence of all possible perturbations, the deviations of the output beam in position and direction should be our concerns. The relative amount of change in terms of the deviations of the output beam after the perturbations in position and direction can be calculated numerically from the termro.We define the position deviationLDo=rox2+roy2 and direction deviations Lθx=rox′ and Lθy=roy′. The symbols LDo, Lθx, and Lθy are denoted as the output beam’s position and orientation error referred to as the ideal output beam’s orientation (no perturbations included).Once given the configuration parameters of BHWC, the calculation of each reflection’s parameterAi,Li,φi can be deduced from Snell’s law in vector form.The parameter setAj,Lj,φj are used to characterize the jth reflection-transmission matrix, where φj is the dihedral angle between the two consecutive incident planes of the jth and (j+1)th reflection and Aj is the incident angle and Lj is the propagating distance of the chief ray between the two reflections. Figure 4(b) describes the method to figure out φj based on the known incident ray vector ri1→ and reflection ray vector rf1→ calculated based on single ray tracing method. Thus, for example, P1→=rf1→×ri1→ and P2→=rf2→×ri2→, where ri2→=rf1→; we can obtain φj using(25)φj=±cos-1⁡P1→·P2→P1→·P2→.Figure 4
Rotation angleφ for an off-axis reflection between two consecutive off-axis reflections onto the mirror m1 and m2. (a) Reflections and corresponding coordinate system for the UMS in the jth reflection. (b) Demonstration of the rotation angle between the two successive reflection planes.
(a) (b)The absolute value ofφj from (25) is always smaller than 90°. Considering that the coordinate system will change once between the left-handed system and right-handed system for each reflection, we define the initial coordinate system as left-handed and φj should be positive for the even number of reflection times for the UMS and negative for the odd number of reflection times for the UMS.


## 4. Numerical Analysis Results

According to the method presented above, the generalized spherical reflection ray matrix is applicable only when the input ray shooting on the spot is superimposed with the point on the perturbation or element axis. The feature of the UMS indicates that the reflecting point on the objective mirrorsM1 and M2 is almost superimposed on the same point which is the center of the mirror’s aperture while the reflecting points on the field mirror F are distributed with two-row images pattern with defined column and row spacing. The misaligned reflection ray matrix (matrix in (20)) cannot be used for the analysis of the field mirror F. In essence, the UMS is a conjugated resonator in which all mirrors have specific relative relationship in 3D space. So the perturbation of the field mirror can be regarded as the perturbation of the objective mirrors; it is enough to present the optical axis sensitivity for the objective mirrors M1 and M2.The analysis of axis’s perturbation in this paper is based on the coplane condition between the reflection plane and the translational perturbation plane (Tix-O-Tiz plane defined in Figure 5). However, the reflection plane (meridional plane, namely, Tix-O-Tiz plane) for each reflection in the UMS varied each time, and the Tix-O-Tiz plane is fixed as a precondition in the derivation based on Figure 5. So it is necessary to convert the perturbation vector δixδiyθixθiy1T to the perturbation vector δixjδiyjθixjθiyj1Tin the reflection plane for jth reflection by using the coordinate rotation matrix R(αj) from (23); we modified the perturbation vector as(26)δjixδjiyθjixθjiy1T=Rαj·δixδiyθixθiy1T.Figure 5
The definition of the objective mirror’s translational and rotational perturbation axisTix-Tiy-Tiz and Rix-Riy-Riz (i can be 1 or 2 indicating the mirrors M1 and M2, resp.). X-Y-Z is the global coordinate system. CF, CM1, and CM2 are the centers of curvature of the mirrors F, M1, and M2, respectively.The definitions of four dimensionless sensitivity factorsSD1, SD2, ST1, and ST2 are given as [14, 19](27)SD1xyoδxδyδzj=∂rxyo∂δxyzj,SD2xyoθxθyj=1R·∂rxyo∂θxyj,ST1θxθyoδxδyδzj=R·∂rxyo′∂δxyzj,ST2θxθyoθxθyj=∂rxyo′∂θxyj,where rxyo and rxyo′ are the induced x or y component of optical axis displacement and optical axis tilt (in radians) on the output aperture as the result of jth mirror’s axial decentration along the translational axis Tix, Tiy, or Tiz with an amount of δxyzj or angular misalignment around rotating axis Rix or Riy with an amount θxyj (in radians).The sensitivity of optical axis perturbation can be obtained by solving (27) One thing to note is that the incident angle for each reflection should be smaller than 5° in a defined BHWC when choosing the initial parameters n,R,dr,dc; if not, the ray matrix method will result in large errors.In our numerical simulations, we choose the given parametersR,n,dc and vary dr to establish different UMS configurations. We observe the relationship between sensitivity factors SD1, SD2, ST1, and ST2 with the ratio of dr/dc.In BHWC, we have fixed parametersR = 600 mm, n=5, and dc = 15 mm and varied dr/dc ranging from 0 to 3; we calculate the sensitivity factors of the objective mirrors M1 and M2 with possible independent perturbation sources δjx,δjy,δjz,θix,θiy with very small value.It can be seen from the simulation results (Figures6–9) that the movement of optical axis of the objective mirrors demonstrates some characteristics:(1)
The sensitivity factor of the system is not dependent on the choice ofdr/dc but is dependent on the choice of parameter n, that is, the reflection times N=4n (BHWC) or N=4n-2 (PBWC). The larger the reflection times, the larger the value of the sensitivity factor, which means the lower tolerance of the objective mirror’s alignment is required.(2)
In terms of SD, the axial misalignments along the translational axesTix and Tiy for the objective mirrors M1 or M2 have the same effects on the position error of the output beam, which means SD1roxi=SD1royi(i=1,2), and the angular misalignment around Rix has similar effects on the output beam’s position when compared with the axial misalignment along Tix or Tiy, which means SD2roθxi=SD1roxi(i=1,2). It can be summarized as(28)SD1roxi=SD1royi=SD2roθxi.(3)
regarding ST, we have ten lines to represent the sensitivity factor in direction error; there are four couple lines showing the same relationships with the termdr/dc; they can be summarized as(29)ST1θxoxi=ST2θxoθxi,ST1θyoyi=ST2θyoθyi,ST1θyoxi=ST2θxoθyi,ST1θxoyi=ST2θyoθxi,wherei=1,2 representing the objective mirrors M1 and M2.(4)
The sensitivity factors ofM1 and M2 manifest the feature of symmetry regarding ST1θxoxi and ST1θyoxi, which can be explained by the symmetry position of the BHWC’s objective mirrors.(5)
All possible perturbationsδjx,δjy,δjz,θix,θiy influence the direction and the position of the output images. However, they have different weights on the influence of the optical axis’s variations related to the ideal output orientation. Considering the factor SD, the influence from δjx,δjy,θix plays an important role in changing the position of the output beam when compared with terms δjz,θjy. Moreover, considering ST, the angular misalignment in x direction is mainly influenced by element’s axial misalignment along Tix and angular misalignment rotating around Rix. The angular misalignment in y direction is mainly influenced by element’s axial misalignment along Tiy and angular misalignment rotating around Riy.Figure 6
Sensitivity factors SD1, SD2, ST1, and ST2 characterizing the movement of the optical axis on the mirrorM1 with fixed column spacing dc = 15 mm, R = 600 mm, and n=5. We change the ratio of dr/dc between 0 and 3 to get different configurations of BHWC.Figure 7
Sensitivity factors SD1, SD2, ST1, and ST2 characterizing the movement of the optical axis on the mirrorM2 with fixed column spacing dc = 15 mm, R = 600 mm, and n=5. We change the ratio of dr/dc between 0 and 3 to get different configurations of BHWC.Figure 8
Sensitivity factors SD1, SD2, ST1, and ST2 considering the possible perturbation of the optical axis on the mirrorM1 with parameters R = 600 mm. The parameter n varies ranging from 2 to 21 to get different configurations of BHWC.Figure 9
Sensitivity factors SD1, SD2, ST1, and ST2 considering the possible perturbation of the optical axis on the mirrorM2 with parameters dc=dr = 15 mm and R = 600 mm. The parameter n changes ranging from 2 to 21 to get different configurations of BHWC.


## 5. Conclusion

In summary, the generalized ray matrix for the spherical mirror reflection has been derived in a new view, which is based on the general model of a paraxial misaligned element. The spherical mirror transfer matrix considers all possible perturbation sources, two kinds of angular misalignments,θix and θiy, and three kinds of axial displacements, δix, δiy, and δiz. BHWCs have been chosen as examples to present the optical axis alignment properties of the multipass cell for the first time. The sensitivity factors SD1, SD2, ST1, and ST2 are calculated numerically when BHWCs are in different configurations. The main factors that result in the deviation of the output beam’s position and direction have been found for the first time. Regarding the position error of the output beam, the axial displacement along Tix and Tiy axes and the angular rotation around Rix of the objective mirror have more weights than other kinds of perturbation sources. Regarding misalignment amount of the direction of the output beam, the misaligned slope deviation about x-axis is mainly influenced by the axial displacement δix and angular rotation θix of the objective mirror. The misaligned slope deviation about y-axis is mainly influenced by the axial displacement error δiy and angular rotation error θiy of the objective mirror. The increase in the number of the reflection times will decrease the stability of the system and make the system more sensitive to the perturbations. The misalignment of Tiz axis (δiz) has the least contributions to the output beam. The choice of the column spacing dc or the row spacing dr does not have relations to the optical axis sensitivity of the objective mirrors. The method can also be applied to the analysis of various kinds of multipass cells, such as PBWC, three-objective/four-objective multipass matrix system, and Herriott cell. These results are important for the multipass cell’s design and long-term stability control for the high accuracy alignment improvement for the multipass gas cell’s construction.



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*Source: 9528713-2017-02-20.xml*