This is a first post of an introduction to optics series and it explain how to create a 2D ray tracing model for a spherical mirror.
This section deals with the creation and charting of the mirror using a parametric equation for a circle.
Introduction to Geometrical Optics – a 2D ray tracing Excel model for spherical mirrors – Part 1
by George Lungu
– This is a tutorial explaining the creation of an exact 2D ray tracing model for both
spherical concave and spherical convex mirrors.
– The model is 2D in the sense that the ray tracing is done in the median x-y plane of
symmetry of the mirror
– This is an exact model in the sense that no geometrical approximations are used, however
the model does not take into consideration diffraction effects.
The ideal reflection laws:
There are two reflection laws:
1. The incident ray the normal to the surface and
the reflected ray are situated in the same plane
2. The angle of incidence and the angle of
reflectance are equal
Creating the chart of a spherical mirror section:
– This being a 2D ray tracing program, the spherical
mirror will be represented by a section through its x-y
median plane which is a circular arc. R R*sin()
-We use a parametric Cartesian representation of the circular arc based on the definition of the trigonometric
functions on the trigonometric circle.
the angle being proportional to a parameter index
Conventions and Excel implementation:
– In our model the light will travel from left to right before the
– By an ad-hoc convention the radius of the curvature will be positive
for a convex mirror and negative for a concave mirror
– Column A will contain labels.
– Parameters xM and yM are the position of the vertex of the mirror
and are located in range B2:B3. We name cell B2 “xM” and B3 “yM”.
– The radius is placed in cell B4 and we name that cell “Radius”
– Cell B5 contains the diameter of the mirror and we name that cell
– Cell B6 contains the length of the hachured area behind the
reflective surface and we name the cell “Back”
Creating the spherical mirror:
– Range A42:A62 will contain the index parameter which will
have the function of scanning a mirror angle (measured from the
center of curvature) starting with the top (d/2) of the mirror
and ending with the bottom (-d/2) in 21 steps.
– While increment “i” varies from -10 to 10 in increments of 1
the (x,y) coordinates described in the formulas above will trace an
arc of circle with a size (length) of “d”, the radius of “Radius”,
and the vertex placed at coordinate (xM,yM) then copy A42 down to A62
– B42: “=Radius*(1-COS((A42/10)*ASIN(d/(2*Radius)))) +xM” then copy
B42 down to cell B62
– C42: “=Radius*SIN((A42/10)*ASIN(d/(2*Radius)))+yM” then copy C42
down to cell C62
Chart the mirror:
– Select rage B42:C62 => Insert => Chart =>
Scatter Chart => Finish => delete the legend
– Right click the horizontal axis => Format
Axis => Scale => Minimum=-5, Maximum=5,
Max Unit=1, Min Unit=1
– Right click the vertical axis => Format Axis
=> Scale => Minimum=-3, Maximum=3, Max
Unit=1, Min Unit=1
– Make the gridlines visible and change their color and the background color to something
-Double click the curve and
Line” and make sure to str
Create the hachure on th
– We will use the existing co
Hachure the back of the mirror.
– A65: “=0”, A68: “=A65+
– B65: “=OFFSET(B$42,$A
– B66: “=B65+Back”, C66:
– Copy range B65:C66 into
– Copy range A67:C69 into
The mirror back – continuation:
– Extend the data range of the chart from
B42:C62 to B42:C1216 and name the series
– Above there is a snapshot of the resulting chart
and the hachure formula table.
– Conical mirrors (parabolic, elliptic and hyperbolic)
are very close in shape to spherical mirrors and are
all used in the construction of astronomical
Technician examining a mirror that will be used in the HESS (High Energy
Stereoscopic System) array in Namibia. The HESS array is used to
investigate gamma ray sources such as supernova remnants and pulsars.
The setup: a brief review of the light source parameters:
– There are many ways of simulating an optical system.
– One of the simplest, yet effective analysis options would be to have a point (artificial
star) emitting a bundle of rays towards the mirror and visualizing the bundle behavior
after the reflection from the mirror.
– We are interested in both on axis and off axis behavior
– We are also interested in both near and far effects (the star can be within a few focal lengths from M(xM,yM) the mirror but also thousands of focal lengths from it.
– We would like to keep the angle of incidence constant while we change the x coordinate of the light source.
– Because of this, we will set two input parameters for the light source: xL and incident.
– We choose to have 21 rays emitted by the star and the rays will be uniformly covering the
mirror (there is a constant angle difference between two consecutive rays). The first and the
21st ray hit the edges of the mirror. The diagram above shows only 15 rays out a total of 21.
to be continued…