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Ideas

Sampling

From CHEP 2015 Proceedings

First experience of vectorizing electromagnetic physics models for detector simulation

Alias sampling is suggested as better for parallelisation than acceptance-rejection method.

This may well be the case, but the elephant in the room (for optical physics anyhow)
is geometrical intersection, not rng sampling

Geometry

Prism TIR

Implement Prism (triangular half cube)

Cartesian Oval Surface of Revolution

l0 n1 + li n2 = constant

Practicalities

  • enable test geometry to somehow fallback to triangulated when no anlytic

Optical Tests

Spherical Interface Refraction

p133 Hecht, refraction at spherical interface:

.
                             /
                            A
                           /
      SA = l0             /           AP = li
                         /
                         |
S -----------------------V             C        P
.                        .                      .
|         so             |          si          |



S : point source
C : center of sphere
P : point where ray crosses axis

phi        : angle SCA
180. - phi : angle PCA


law of cosines for triangles SAC and ACP
and angle relation: SCA + PCA = pi

optical path length,  OPL = n1 l0 + n2 li

Fermats principle, dOPL/dx = 0   (derivative with position variable)

n1 R (s0 + R) sin(phi)      n2 R (si - R ) sin(phi)
----------------------  -   ------------------------  = 0
    2 l0                             2 li

yielding relation between parameters of ray going from S to P via refraction at spherical interface

n1     n2       1  /  n2 si     n1 so   \
--  +  --   =   -- |  ----- -  -------  |
l0     li       R  \    li       l0     /

small angle assumption A close to V,  cos(phi) ~ 1  sin(phi) ~ phi
(this assumption corresponds to paraxial rays and is known as Gaussian Optics)

l0 ~ s0
li ~ si

n1     n2      n2 - n1
--  +  --  =   -------
s0     si         R

Thin Gaussian Lens

p138 Hecht, spherical lens assuming small angles from optical axis (paraxial rays)

/|

/ |

/ |

C2 V1 | V2 C1
| /
| /
|/
| d | |
R2 |
R1 |

C1 - R1 + d = C2 + R2

d = R2 + R1 - (C2 - C1)

nm nm / 1 1 nl d — + — = (nl - nm) | - - - | + ———— so1 si2 R1 R2 / (si1 - d)sil

Thin lens assumption removes the d term, and simplify with air/vacuum nm=1 get relation between object and image distances:

1      1         1                /  1      1   \
--  +  ---   =   --  =   (nl - 1) |  --  -  --  |
so     si        f                \  R1     R2  /


                     =   2 (nl - 1 )        for R1 = -R2 = R
                         -----------
                              R

With parallel rays, 1/so = 0:

si = f =   R / 2(nl - 1)

For example Vacuum/Pyrex:

ggv --mat Pyrex   # index 1.458

si = f = R * 1.0917

In [2]: 1./(2*(1.458-1.))
Out[2]: 1.091703056768559

In [3]: 700./1.091703056768559
Out[3]: 641.1999999999999

Pick radius to make focus at edge of box:

local test_config=(
             mode=BoxInBox
             analytic=1

             shape=B,L

             boundary=Rock//perfectAbsorbSurface/Vacuum
             parameters=-1,1,0,700

             boundary=Vacuum///Pyrex
             parameters=641.2,641.2,-600,600

           )

Visually at least, get the expected focus point.

TODO:

  • numerical check of focus coordinates, using the record data, incorporating lens thickness

Dispersing Prisms

Hecht p163, deviation angle as function of prism apex angle, refractive index and incident normal angle. Minimum deviation occurs where ray traverses symmetrically.

How to define a symmetric prism

  • apex angle A, height h, depth d
.
                 A  (0,h)
                /|\
               / | \
              /  |  \
             /   h   \
            /    |    \ (x,y)
           M     |     N
          /      |      \
         C-------O-------B

               (0,0)
      (-a/2,0)         (a/2, 0)


  angles B = C = (180 - A)/2



               a/2
  tan(A/2) = --------
                h

  a/2 = h tan(A/2)

  need plane eqns of faces

  AB direction : ( 0, h) - (a/2, 0)  = (-a/2, h)    ON direction (h, a/2)
  AC direction : ( 0, h) - (-a/2, 0) = ( a/2, h)    OM direction (h, -a/2)

  (-a/2, h ).( h, a/2 ) = 0



  ON. A = (h, a/2)  . (0, h ) =  ah/2
  OM. A = (h, -a/2) . (0, h ) = -ah/2


  hmm can I calc the planes whilst calulating the bounds...

  plane N
      (h, a/2, 0 )     ah/2

  plane M
      (h, -a/2, 0)     -ah/2

  plane O
      (0, -1,  0)       0

  plane F
      (0,  0,  1)       d/2

  plane B
      (0,  0, -1)      -d/2





  plane containing

  A    (0,  h,0)
  B    (a/2,0,0)
  B'   (a/2,0,d)

Dispersion

  • dispersion angle calculation yields refractive index, so predict the refractive index as a function of wavelength from the angle and compare
    • or fabricate a material with a linear refractive index