Calculation of Round-trip Matrix

The ray-matrix formalism supposes the elements’ matrices are multiplied one-by-one in the direction opposite to the beam propagation direction.

General elements

Each element characterizes by a pair of ray matrices. One matrix is for the tangential (T) plane, and the other is for the sagittal (S) plane. Round-trip matrices are calculated independently for both planes.

SW

If the schema is the standing wave resonator, then elements’ matrices are multiplied in the next way:

  • from the reference element to the first element of the schema,

  • from the first element to the last one,

  • from the last element to an element next to the reference.

_images/round_trip_sw_1.png

Sample result (M4 is the reference element):

M0 = M4 × M3 × M2 × M1 × M2 × M3 × M4 × M5 × M6 × M7 × M6 × M5

RR

If the schema is the ring resonator, then elements’ matrices are multiplied in the next way:

  • from the reference element to the first element of the schema,

  • from the last element to an element next to the reference.

_images/round_trip_rr_1.png

Sample result (M4 is the reference element):

M0 = M4 × M3 × M2 × M1 × M7 × M6 × M5

SP

If the schema is the single-pass system, then ray passes it from the first element to the last one. Therefore, matrices are multiplied in the reverse order (from the last element to the first one).

_images/round_trip_sp_1.png

Sample result (M4 is the reference element):

M0 = M4 × M3 × M2 × M1

Elements having length

Some functions can analyze elements over their length, caustic functions, for example. The reference (being analyzed) element is divided into two sub-elements for such functions. The first sub-element is a part of the original element from its left edge till the current point (in which the beam size is calculated, for example). The second sub-element is a part of the original element from the current point till the right edge. Each sub-element characterizes by its ray matrix pair.

SW

If the schema is the standing wave resonator, then elements’ matrices are multiplied in the next way:

  • from the left reference sub-element to the first element of the schema,

  • from the first element to the last one (the reference element is accounted as whole at this step),

  • from the last element to the right reference sub-element.

_images/round_trip_sw_2.png

Sample result (M4 is the reference element):

M0 = M41 × M3 × M2 × M1 × M2 × M3 × M4 × M5 × M6 × M7 × M6 × M5 × M42

RR

If the schema is the ring resonator, then elements’ matrices are multiplied in the next way:

  • from the left reference element to the first element of the schema,

  • from the element to the right reference sub-element.

_images/round_trip_rr_2.png

Sample result (M4 is the reference element):

M0 = M41 × M3 × M2 × M1 × M7 × M6 × M5 × M42

SP

If the schema is the single-pass system, then matrices from the left reference sub-element to the first element are multiplied.

_images/round_trip_sp_2.png

Sample result (M4 is the reference element):

M0 = M41 × M3 × M2 × M1

See also

Elements Matrices