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Final Answers
© 2000-2017   Gérard P. Michon, Ph.D.

Geometrical Optics
Rays  &  Sharp Shadows

The least part of Light, I call a Ray of Light.
Isaac Newton (1643-1727)   Opticks, 1704.

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Geometrical Optics :   Rays and Shadows

(2010-11-25)   Radius of curvature of a concave mirror...
Curvature of a mirror magnifying  k=3  times an object  d=22 cm  away?

Mirrors were the first optical systems to be analyzed mathematically...

That's a good opportunity to practice elementary geometry.
For this exercise, we shall only need two optical principles:

  • Rays from an object point emerge as if they came from its image.
  • On a mirror, the angle of incidence is equal to the angle of reflection  (Hero of Alexandria,  first century AD).

Let's choose a coordinate system where the origin O is the center of curvature of the mirror.  The mirror intersects the x-axis at a point  M,  whose abscissa  R  is the radius of curvature which we're seeking.

 Reflection off a concave mirror

Consider an object  A  of  small  height  above the point  P  on the  axis; both points are at abscissa  x.  Let  A'  be the  (virtual)  image of  A,  above a point on the axis which we'll call  P' .  If we're told that our object is magnified by a factor  k = 3 ,  we know that  P'A'  is  k  times  PA.

Any ray going through the center  O  of the sphere is reflected back onto itself,  so OA and OA' are collinear  (O, A and A' are aligned).  Therefore,  the triangles  OPA  and  OP'A'  are similar.  So, by the theorem of Thales, the abscissa  O A'  of  P'  is  k x  (it's to OA what A'P' is to AP).

On the other hand, consider the ray from  A  which is reflected at the point  M  of the mirror on the x-axis,  at abscissa  R.  Because the angle of incidence  (the inclination of MA)  is equal to the angle of reflection  (the inclination of MA' )  we have, again, two similar triangles  (MPA and MP'A' )  in a ratio  k.  So,  MP'  is equal to  k  times the distance  d  which we are given  (as the distance of  22 cm  from the mirror to the object).  This shows that the abscissa of  A'  (or  P' )  is equal to  R+kd.  Therefore:

R + kd   =   k x   =   k (R-d)

Solving this for  R,  we obtain   R   =   2 d / (1-1/k)   =   66 cm.

Focal Length of a Concave Mirror :

Importantly, the above can be forcefully recast in a standard form:

 1      =      1    +   1
 Vinculum  Vinculum  Vinculum
fp p'

Using the following equivalences:

  • p    =   x        (distance from the optical center to the real object).
  • p'   =   k x     (distance from  O  to the image).
  •  f    =   R/2    (a  positive  quantity for a  concave  mirror).

This last equation can be construed as the definition of the focal length of a  concave  mirror,  which is thus shown to obey an  optical equation  similar to what's established for a thin-lens in the next section.

(2015-06-29)   Thin-Lens Equation.  Definition of the  focal length.
Relation between the positions  p  and  p'  of an object and its image.

thin-lens  is an  ideal  system which can be approximated by an axial-symmetric thin piece of glass bounded by two polished spherical surfaces.

I use an hyphen to denote the  "thin-lens"  optical concept,  as opposed to an actual lens that happens to be a good approximation of such a thing because it's not too thick...  Actually, the best thin-lenses have a substantial thickness to them  (crystal balls are a striking example).  What really qualifies an optical system as a thin-lens is the existence of an  optical center,  as defined next.  Unfortunately, this hyphenated clarification is not universally adopted  (in fact, at this writing, I seem to be the only one advocating it).

Two physical properties of a thin-lens are sufficient to establish its ability to form real images of real objects near the optical axis, namely:

  1. The center  O  of the lens is an  optical center  (i.e., rays through it are not deflected).  That's very nearly true for rays which have low inclination with respect to the optical axis if the thickness of the glass at the center is small  (hence the qualifier  thin ).
  2. Any  incident ray  parallel to the optical axis emerges as a ray emanating from the  image focal point  F.  The distance OF is a characteristic  ( f )  of the lens called its  focal length.  (We'll see later that the value of  f  can be obtained from the  lens-maker formula.)

Optical diagrams are intended to portrait the situation near the optical axis but exaggerated radial distances are used for clarity.  The usual convention is to make the optical axis horizontal, with light shining from left to right.

A converging thin-lens is represented by a vertical line with two outward-pointing arrows  (they would be inward-pointing for a diverging lens).  Objects and images  (usually, only one of each)  are vertical arrows originating on the horizontal optical axis.

Here, we consider an object  A  above a point  P  on the axis,  at distance  p  from the optical center  O.  Its image  A'  is located below a point  P'  on the axis,  at a distance  p'  from  O.  The point  W  is where a ray from  A  parallel to the optical axis meets the central plane of our lens.

 Proof of the thin-lens equation

The heights of the similar triangles  APO  (or OWA)  and  A'P'O  are proportional to  p  and  p'.  With this in mind,  we apply the theorem of  Thales  again to the triangles  FOW  and  FP'A'  and obtain this relation:

f / p   =   ( p'- f  ) / p'

It boils down to the following celebrated relation between  p  and  p' :

The  Thin-Lens  Equation :
 1      =      1    +   1
 Vinculum  Vinculum  Vinculum
fp p'

We only derived that formula in the case of a converging lens  (positive focal length)  real object  (positive p)  and  real image  (positive p' ).  However, it remains valid in all other cases,  with the following sign conventions:

  • For a  divergent lens,   the focal length   f   is negative.
  • For a  virtual object,   p  is negative.
  • For a  virtual image,   p'  is negative.

Thin-lens equation  by  Rod Nave.

(2015-07-01)   Hyperfocal Distance
Position  of the nearest in-focus objects when the lens is set to infinity.

It's convenient to define the  position  of an object as the parameter  p  which appears in the  thin-lens equation  (or its counterpart for more general optical systems, analyzed later).  This is only indirectly related to the  distance  used by  photographers  (the actual distance between the film/sensor and the object, which may or may not be in sharp focus).

The distinction is made between objects in  sharp focus  (whose images are precisely located on the sensor)  and other  in-focus  objects which project a pencil of light thats intersects the plane of the sensor on a  spot  whose diameter does not exceed the diameter of the accepted  circle of confusion.

In traditional  35 mm  photography,  the diameter of the circle of confusion is commonly taken to be  0.03 mm.  For crop-sensor cameras  (with a crop factor around 1.5)  that would be equivalent to  0.02 mm,  which corresponds to the width of  about  5  pixels in the Nikon D5500  DSLR.  It's just a single pixel in an image resized to 1200 by 800 pixels.

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 still working on this one...

Hyperfocal Distance :
H    =      f 2
A e

Depth of field

(2010-11-26)   Paraxial Optics.  Ray Transfer Matrix  (RTM).
Each optical component acts on the distance and inclination of a ray.

Elementary geometry is great in simple cases but fails to give the rules by which complicated optical systems can be constructed...  Let's give some method to our optical madness:

We're only considering optical systems endowed with  cylindrical symmetry  around a line called the  optical axis  (i.e., the optical system is unchanged in any rotation around the  optical axis).  Because of that symmetry, light travels in a straight line along the optical axis.

Almost all commercial lenses are designed this way.  One exception are the panoramic lenses used in the movie industry, which squeeze laterally a panoramic image into the standard frame of  35mm film.

meridional ray  (or  tangential ray  is a ray contained in a plane which includes the  optical axis.  Other rays are called  skew rays  (this includes  sagittal rays  whose direction is perpendicular to the optical axis but do not intersect it).

Meridional rays that are close to the optical axis are called  paraxial rays

At the location of a given plane orthogonal to the optical axis, a  paraxial rays  is described by two parameters:  Its distance from the optical axis and its inclination with respect to the optical axis.  There is a linear relation between the description of a ray at one location and the description of the same ray at another location.  That relation is made  unimodular  (i.e., the determinant of its matrix is unity)  if we describe a ray by a nnormalized vectorial quantity whose second coordinate is the angular inclination while the first coordinate is the distance to the optical axis  multiplied into the index of refraction  at the specified location along the optical axis.

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 still working on this one...

Optique matricielle (Université du Maine)
  Wikipedia :   Ray   |   Optical axis   |   Paraxial approximation
Ray transfer matrix analysis   |   Cardinal points

Meghan  (via Yahoo!  2011-01-05)   Crystal Balls   (spherical lenses)
A solid sphere of glass  (radius R, index n)  has focal length  f = R/(2n-2)

There are several ways to obtain this result.  The easiest one is probably to notice that  the lens-maker's formula  (originally intended for  thin  lenses only)  applies directly to this particular case of a  thick  lens, because of the existence of an  optical center  (a point through which light rays are not deflected at all).

 Ray tracing in a transparent 
 sphere of radius R and index n.

 We may also do it the  hard way,  without even using the small angle approximation:

For an incident ray at a distance  u < R  from the center  O  of the sphere, we consider the plane  xOy  where the x-axis is parallel to the ray  (whose direction is that of increasing values of  x  at a constant value of  y = u > 0).  See above figure.

The ray enters the sphere at point  I  = ( x0 , y0 )  at an angle of incidence denoted  i  (that's the angle with respect to the normal to the surface).

x0  =  - ( R 2 - u 2 )½'             y0  =  u  =  R sin i

The refracted ray emerges from  I  at an angle  r  (with respect to the normal)  whose sine is equal to  u/nR  (according to Snell's law).  At this point, the ray's inclination with respect to the x-axis is  a  (which is a negative angle).

sin r   =   (1/n) sin i  =  u / nR           a  =  r - i  =  Arcsin (u/nR) - Arcsin (u/R)

Using a dummy parameter  z,  the equations of the ray inside the sphere are:

x  =  x0 + z cos a       &       y  =  y0 + z sin a

The exit point J is at the nonzero value of z for which  x2 + y2 = R:

R 2   =   ( x0 + z cos a )2  +  ( y0 + z sin a )2
0   =   z 2  +  2 z  [ x0 cos a  +  y0 sin a ]

Therefore, we must plug  z  =  -2 [ x0 cos a  +  y0 sin a ]   into the previous expressions to obtain the coordinates  (x1 , y1 )  of the exit point  J :

x1  =  x0 - 2 [ x0 cos a  +  y0 sin a ]  cos a   =   - x0 cos 2a  +  y0 sin 2a
y1  =  y0 - 2 [ x0 cos a  +  y0 sin a ]  sin a   =   - x0 sin 2a  +  y0 cos 2a

We could have obtained the same result geometrically...

 Come back later, we're
 still working on this one...

Ransom (2010-11-26)   Lens-Maker's  Equation   (with index  n = 1.44)
Focal length of a lens with two concave faces of radii 0.300 & 0.970 m.

The following formula gives the focal length  ( f )  for a  thin lens  made from stuff of index  n  (relative to the surrounding medium)  bounded by two surfaces whose radii of curvature are respectively  R1  and  R2

Lens-Maker's  Formula
 1      =   (n-1)  [    1    +   1   ]
 Vinculum  Vinculum  Vinculum
f R1 R2

The curvatures are counted positively when the surface bends toward the denser medium and negatively otherwise.  Similarly, the resulting focal length is positive for a converging lens and negative for a diverging one.

In the above case of a  plastic  biconcave lens  (n = 1.44)  the radii of curvature are both negative  (-0.300 and -0.970).  So is the focal length given by the above formula:  f = -0.521 m

(2014-12-14)   Gullstrand's formula
Combined power of two coaxial lenses separated by a distance.

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 still working on this one...

Gullstrand's Equation   |   Allvar Gullstrand (1862-1930; Nobel 1911)

(2016-12-24)   Light Falloff   &   Center Filters
Natural dimming away from the center of a photographic image.

This darkening of the sides of a photographic image is small for long lenses or for  retrofocal  wide-angle lenses  (as used in small-format SLR cameras).

However,  the effect is very noticeable with wide-angle lenses in large-format cameras.  It can be corrected with professional  center filters  costing hundreds of dollars.  Such filters  (dark in the center and clear near the rim)  must be manufactured with precision to match a given lens.

In any camera with a  thin-lens,  the illumination of the image falls off as the fourth power of the  cosine  of the angular distance  q  to the optical axis  (as measured from the center of the lens)  for  3  combined reasons:

  • The intensity of light is inversely as the square of the distance to the source  (loosely speaking,  the aperture of the lens acts as a source of light).  For a fairly narrow aperture,  that distance is just inversely proportional to  cos q  so the intensity is proportional to  cosq.
  • The amount of light received per unit area on the plate is proportional to the cosine of its tilt from the rays.  That's another factor of  cos q.
  • Finally,  the aperture is observed from the plate tilted at an angle of  q  and its  apparent area  is thus proportional to  cos q.

The maximum angle  q  is half the field of view  (corner to corner).  For a  normal lens,  the field of view is,  by definition,  close to the field if view of the human eye,  say  2q = 45°,  which lets the above dimming factor be:

cos4 22.5°   =   ( 3 + 2 Ö2 ) / 8   =   0.72855339...   =   2 -0.4568934

This means that the corners are about  0.457  f-stops  dimmer than the center of the image.  Less than half a stop is noticeable but not striking  (for longer lenses, the effect is even harder to detect).  However, the situation is very different for wide-angle lenses,  as shown by the following table:

Light falloff at corners,  compared to the center of the image  (for a thin-lens)
Field of View2 q 120° 107° 90° 65.5° 45° 27.4°
Falloff  (in f-stops)- 4 lg cos q 4 3 2 1 0.457 1/6
Focal Length  / mm35 mm 12.49 16.00 21.63 33.61 52.23 88.72
Nikon DX  8.14 10.43 14.10 21.90 34.04 57.81

(2016-12-24)   Retrofocus Wide-Angle Lens Design   (Angénieux, 1950)
How to keep a lens with short focal length away from the focal plane.

The yet-to-be-named idea was first applied in the 1930's by  Taylor Hobson  for their  Technicolor® cameras,  to provide space for the beam-splitter required by the Technicolor system.

This is also critical for single-lens-reflex  (SLR)  cameras, to allow room for the flip-up mirror behind the lens.

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Angénieux retrofocus (1950)   |   Pierre Angénieux (1907-1998)

(2016-10-24)   Microscopy   (Antonie van Leeuwenhoek)
Optical Compound Microscope.

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 Antonie van Leeuwenhoek

Optical Microscopy   |   Antonie van Leeuwenhoek (1632-1723)

(2015-06-01)   Numerical Aperture.
This doesn't depend on the refractive index of the propagation medium.

The concept was introduced in microscopy by the celebrated German optician  Ernst Abbe (1840-1905).

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Nikon microscopy   |   Numerical aperture (Wikipedia)

(2015-06-01)   Resolving Power   (Lagrange  &  Abbe)
Best possible resolution is inversely proportional to aperture.

This isn't part of proper geometrical optics but it's good to know what limit to the sharpness of lenses is imposed by diffraction  (due to the wavelike nature of light).

The following formula gives the smallest angular distance between two points that can be barely distinguished according to the conventional  Rayleigh's criterion.  For other conventions, a slightly different coefficient would be substituted for the  Rayleigh factor  (1.220).

q   =   1.220  l / D

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Wikipedia :   Angular resolution   |   Airy disk (Wikipedia)

(2008-10-26)   Shadow Hiding.  The  Opposition Effect.
The cause of extra brightness directed back to the source of illumination.

When illuminated, a smooth enough  dull  surface sends back in all directions an intensity of light which is proportional to its apparent area in the direction of the observer  (Lambert's Law).

However, some features of a rough surface may be large enough to cast shadows on deeper patches which reduce the percentage of the surface that's illuminated.  This can reduce significantly the albedo of the surface of a rocky planet whenever it's not observed directly at opposition.

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 Honeycomb grid
(2015-05-22)   Honeycomb Grid
Attenuating diverging ray from a light beam.

The device essentially consists of many circular tubes with reflective walls parallel to the central axis of the beam.

A ray entering such a tube isn't modified if it's parallel to the axis.  Otherwise, the ray is reflected  n  times off the walls of the tube and emerges with the same angle  (up to a change of sign when  n  is odd, which we may ignore if we assume the system to be symmetric with respect to the central axis, since two symmetric rays simply switch rôles in that case).

Because the material isn't perfectly reflective, each reflection reduces the intensity of a ray by a factor  k  < 1.  The total attenuation is  kn.

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