Position (mm)
4.6 Use of Non-Planar Ring Resonators to Obtain Unidirectional, Single-Frequency Lasing
4.6.3 Calculation of the Amount of Rotation
4.6.3.1 Beam-based Method
If we assume that beam 1 in Figure 4.20 is linearly polarised in the horizontal plane {i.e. in the plane of the paper in the "plan" view) the effect of the non-planarity of the ring is such that in moving from point P to point Q on the path M1-M2-M3-M4, the electric field vector is rotated so that at Q it makes some angle with the horizontal
plane. This “characteristic rotation” of a given arrangement {i.e. for given values of L, D, h, Q) was denoted 6 c . The calculation of 0^ is a geometrical problem.
In the first method of calculation, a beam-based approach was taken, /. e. using coordinate systems set in the beams as shown in Figure 4.21 for the first reflection. An example program for the method is given in Appendix 1. In Figure 4.21, n i is the mirror normal and Ni is the normal to the plane of incidence. The frames xiyizi and X] ’y 1 ’zi ’ are the beam-based coordinate systems before and after the first reflection repectively. To implement this method, the four angles of incidence have to be determined and also the angles between consecutive plans of incidence. In order to find these two sets of angles, the whole system is referred to the coordinate system ijk (see Figure 4.20) in which the j direction is collinear with the straight line joining the two plane mirrors and the k direction is in the vertical direction. The angle a can be calculated given L, D and h. In addition the unit vectors representing the incident and reflected ray directions for each of the four reflections can be found. The scalar product of these two vectors leads to the angle of incidence while the vector products allows the normal to the plane of incidence to be found. The scalar products of such normals give the angles between the various planes of incidence.
The normal to the plane of incidence and the incident and reflected ray vectors can also be used to calculate the mirror normals. Although these are not required for the beam-based method they are required for the vector method described in section 4.6.3.2 below.
With these geometrical preliminaries completed, the rotation calculation can be started. This begins on line 99 of the program given in Appendix 1. Starting with the part of beam 1 immediately before mirror M l, the field is taken here as horizontally polarised. A system of coordinates XYZ is set up with Z as the direction of propagation andX and Y parallel and perpendicular to the horizontal plane, as shown in Figure 4.22. In this system, the initial unit vector of the E field is (1,0,0). This is easily converted to the ijk system using the angle a. The scalar product of the E vector in the ijk system and the unit vector normal to the first plane of incidence (Nj) gives the angle between the horizontal plane and the first plane of incidence, and this is used to construct a matrix which transforms the E field components in the XYZ frame to the frame xiyizj in the first plane of incidence. Given the E field components in this system, those in the xfyi'zi' frame (the coordinate system immediately after the first reflection) are the same except that the x component is reversed in sign. These components are then transformed into the frame which is the frame used for the polarisation of the incident ray of the second reflection. The direction x% is parallel to the second plane of incidence, while the direction y 2 is perpendicular to this plane. Z] coincides with the beam direction between mirrors Ml and M2. To carry out this transformation the
M2
M3
Nd:YAG
M1
plate
X
ELEVATION
M2
M1
k out
of paperi
M4
MS
L
^ef
PLAN
Figure 4.20. Elevation and plan views o f the non-planar ring resonator.
out of
paper paperout of
incident ray reflected ray
M1
out of paper
out of paper Beam 1 angle a
M1
Beam 1M4
out of paperFigure 4.22. Relation between XYZ and ijk systems.
incidence. This process of reflection plus coordinate tranformation between consecutive planes of incidence is carried out for all four reflections. Finally the components have to be transformed from the frame to xiyizi so that the initial and final polarisation states can be compared. The scalar product of the initial and final unit E field vectors gives the amount of polarisation rotation produced by the ring 6 c .
Given the complexity of this method for this particular ring, a simple check on the various transformations was built into the program. If C(a,b) is the transformation between planes of incidence a and b {i.e. to transform between frames x j y j z j and xyyyzy ) and R(a) is the transformation for reflection at mirror a (i.e. to transform b e tw e e n fram es XgyaZa a n d x j y j z j ) th e n the m a trix
C(4,1)R(4)C(3,4)R(3)C(2,3)R(2)C(1,2)R(1) should be equal to the identity matrix, and this was found to be the case (see lines 190 - 192 in the program in Appendix 1).
The form of a transformation matrix C(a,b) is
^ cos{ ± sin { 0^ + sin§ cos{ 0
.
0
0
1,
i.e. a rotation about the Za axis where ^ is the angle between planes of incidence a and 6, while that of a reflection R(a) is
^ 1 0 0
0 cos(;r~2zJ ± sin (;r-2 /^) 0 + sin (;r~ 2 z j c o s(;r-2 z J
i.e. a rotation about the Xa axis where ia is the angle of incidence for the ath reflection. The choice of ± depends on the particular sense of coordinate-frame rotation.
As the non-planar angle Ü is reduced, two critical points are reached where the form of some of the matrices in the program need to be changed. The first is when beams 1 and 2 (and beams 3 and 4) lie in a vertical plane; this occurs for £2 -13°. The second, and more interesting, point is when the angle bewteen the first two (and last two) planes of incidence is 180°. This happens when £2= a , i.e. when beams 1 and 3 are parallel. At this point the net rotation produced by the ring is zero; further reduction
of the non-planar angle O causes polarisation rotation in the opposite sense. Thus although the ring is never exactly planar (due to the displacement of beam 1 by the NdrYAG slab) there is nevertheless a point of zero net rotation. To cover the three regimes, two others programs were written, both very similar to that given in Appendix 1, but with the forms of some of the matrices slightly modified. The results of the calculation for the two rings under consideration are shown in Figure 4.23.
c
o
4-*(0 4-*o
oc 10 H—o
4-» c 3 OE
<
o o o(S o CO oNon-Planar A ngle i
Figure 4.23. Polarisation rotation versus non-planar angle. 4.6.3.2.Vector Method
Although the beam-based method is easily applied to simple non-planar geometries, it is clearly unwieldy for the four-mirror system here. A much simpler vector method for the calculation of polarisation rotation was therefore developed. In this method, the polarisation vector incident on a given mirror is resolved into two orthogonal components in the plane containing the mirror normal n and the incident polarisation (electric field) vector p, as shown in Figure 4.24. One component is parallel to the mirror normal n, and the other is in the direction (pxn)xn. This latter component is reversed in sign upon reflection while the former remains unchanged. The input and output polarisation vectors ip and p' respectively) for a reflection are thus
,/v ... f,. ( p x n ) x n | ( p x n ) x n , .... [ . ( p x n ) x n ] ( p x n ) x n
This procedure is applied to the four mirrors in turn, the output polarisation vector from one reflection forming the input polarisation vector for the next. The entire calculation is
method the amount of polarisation rotation is obtained from the scalar product of the initial and final polarisation unit vectors. The program for this method is given in Appendix 2, and produced exactly the same results as the beam-based method, although it is clearly much shorter.
Figure 4.24. Vector method for calculating polarisation rotation.