2.5 Future improvements
3.2.2 Misalignment analysis
Misalignment of multipass cells, especially during field measurements due to vibrations, ac-celerations (shocks) and ambient temperature and pressure variations, is known to degrade the performance of TDL absorption spectrometers considerably [Dyroff et al., 2004; Roller et al., 2006]. Subtle alignment changes can dramatically affect the retrieved trace gas con-centrations, mainly due to changing spectral background structures, but also due to changes in optical power transmitted through the system. In the following, the effect of alignment in-stabilities is estimated to quantify the tolerances during cell alignment. Misalignment errors that are caused by a) changes in mirror separation ∆d , b) transverse mirror displacements ∆x and ∆y, and c) tilt of the mirrors δx and δy [Fig. 3.5 (a)] are considered. To quantify these misalignment errors an ABCD matrix description of the cell optics is used.
Figure 3.3: (a) Ray tracing simulation of the multipass cell for the first 22 passes. The outer electrode has been removed for clarification. (b) Spot pattern on the front mirror as observed from inside the cell as a result of a complete ray tracing simulation. (c) Photograph of the cell. (d) Photograph of the front mirror with the beam of a red trace laser aligned through the cell. The outer electrode has been removed.
In an ideal centered collinear optical system, one can use a 2 × 2 ray-transfer matrix to represent the transformation of a ray between two reference planes (RP) according to the matrix equation
x1 tan(α1)
=A B C D
x0 tan(α0)
, (3.9)
where x0and tan(α0) are the components of the incident ray vector, denoting the position of the ray and its slope with respect to the optical axis, respectively. This is illustrated in Fig. 3.4 for an arbitrary transformation matrix with elements A, B, C, and D, where both reference planes are located at the same position on the optical axis (RP).
Applying the transformation matrix to the incident ray vector yields the resulting position x1and slope tan(α1) after the optical element described by the transformation matrix. In an actual system the position of the optical axis in the second reference plane may be displaced slightly from its assumed position by a small distance ∆x . Also, the direction in which the optical axis is pointing may deviate slightly by a small amount δx.
Figure 3.4: Propagation of a ray vector between two coincident reference planes.
One can represent this misalignment by adding misalignment terms to Eq. 3.9, and Gerrard and Burch[1994] suggest to add a third dummy equation,
x1= Ax0+ B tan(α0) + ∆x tan(α1) = Cx0+ D tan(α0) + δx.
1 = 0 + 0 + 1
(3.10)
These equations can again be written in matrix form,
x1 tan(α1)
1
=
A B ∆x C D δx
0 0 1
x0 tan(α0)
1
. (3.11)
In the description of the multipass cell optics the 3 × 3 matrix in Eq. 3.11 denotes the reflection matrix Rx
Rx=
1 0 ∆x
−2/r 1 δx
0 0 1
, (3.12)
where r is the radius of curvature of the mirror, ∆x is a small transverse displacement of the mirror axis, and δxis a small tilt of the mirror axis with respect to the optical axis. Rxdescribes the reflection in the zx-plane and a similar quantity Ryis valid in the yz-plane.
The actual optical system of the multipass cell contains two different matrices, the reflec-tion matrix Rxof each of the two mirrors and a matrix Dxof a plane-parallel plate of air, which separates the reference planes of both mirrors,
Dx=
1 d + ∆d 0
0 1 0
0 0 1
. (3.13)
The term ∆d accounts for changes in the separation of the two multipass cell mirrors. The matrix for a single pass through the cell, e.g. from the coupling hole to the rear mirror, is
calculated by Mx= RxDx. The ray starts at the first reference plane of the front mirror. It then passes through the plane-parallel plate of air (Dx) and is reflected by the rear mirror (Rx). The second reference plane refers to the rear mirror. To calculate the position of the beam on the mirrors after n passes, Mxis raised to the power of n and multiplied by the incident ray vector at the first reference plane Z0= {x0; tan(α0); 1}T:
Zn|x= MxnZ0. (3.14)
An equivalent formulation is valid for the y-direction. Together, Zn|x and Zn|y form the ray vector Zn, that contains the coordinates xn, ynand the slopes tan(αn), tan(βn) of the ray at the nth reference plane. Here tan(βn) denotes the slope of the ray in the yz-plane after n passes.
Again, spots with even numbers n lie on the front mirror, spots with odd numbers n lie on the rear mirror.
Based on the formalism described above the misalignment of Zn has been calculated and compared to that of a confocal Herriott cell with approximately the same mirror separation and a circular spot pattern of equivalent diameter while introducing the three sources of misalign-ment listed above. It must be noted that the matrix description discussed above only describes the propagation of the center ray and does not fully describe the propagation of a practical laser beam. Furthermore, this description assumes a paraxial approximation and neglects higher or-der effects. The results are based on coupling into the cell in a way to obtain a circular spot pattern which is a prerequisite for the present design. Changes in the angles α and β would lead to slightly different results (compare with Eq. 3.5).
Changes of mirror distance ∆d are usually related to thermal drifts of the cell or pressure changes inside or outside of the cell (e.g. for an airborne instrument during ascent/descent of the aircraft). The coefficient of thermal expansion of stainless steel is 16 × 10−6K−1 at 20◦C [Kuchling, 1984], and the base length of the present cell is d = 466.4 mm. An assumed temperature increase of the cell by 2 K leads to an increase in base length of ∆d = 15 µm.
The calculations demonstrate that the ray at the coupling hole after N = 86 passes is displaced by only 13.5 µm along the x-direction. This corresponds to a ratio of ∆x/∆d = 0.9. The beam pointing angle αnchanges by ∆αn/∆d = 4.5 × 10−3deg/mm. The beam displacement in y-direction (∆y) is negligible (∆y/∆d < 0.02), whereas the beam pointing angle βnchanges at a ratio of ∆βn/∆d = 3.1 × 10−2deg/mm. With a misalignment of ∆d = 15 µm we find for the confocal cell that the beam is displaced in x-direction by 81 µm (∆x/∆d = 5.4) with
∆αn/∆d = 0.61 deg/mm. In y-direction one calculates a displacement of 7.2 µm (∆y/∆d = 0.48), whereas the beam pointing angle changes by ∆βn= 0.66 deg/mm. The calculated ratios are linear approximations that are valid for small changes ∆d. For a better comparison of the cell characteristics the data are summarized in Tab. 3.2.
Maximum transverse mirror displacements and mirror tilt put constraints on construction tolerances and accuracy of alignment. The calculations show that both, position and pointing angles, do not change with transverse mirror displacement and/or mirror tilt. This holds true for the confocal arrangement as well. The spot pattern becomes distorted, i.e. elliptical. The level of ellipticity is very similar for the confocal and non-confocal configuration when dis-placing one or both mirrors transversally. However, introducing mirror tilt reveals much less ellipticity of the pattern for the confocal arrangement. Therefore, the initial alignment of the Stark multipass cell is more demanding than that of a classical confocal Herriott cell.
Table 3.2: Comparison of beam displacement ∆x,∆y and beam pointing angle ∆αn,
∆βn due to changes in mirror separation d for the present Stark modulation cell and a confocal Herriott cell with a circular spot pattern of the same diameter. The ratios are linear approximations which are valid for small changes ∆d.
Stark modulation Cell Herriott Cell
Volume 0.8 l ∼2 l
Path length 40 m 40 m
∆x/∆d 0.9 5.4
∆y/∆d ∼ 0 0.48
∆αn/∆d 4.5 × 10−3deg/mm 0.61 deg/mm
∆βn/∆d 3.1 × 10−2deg/mm 0.66 deg/mm