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Optic equations and notes

  1. Disclaimer
  2. Optic equations and notes
    1. Diopter or Dioptre
    2. Focal Length
    3. Depth of Field
    4. Contrast Ratio
    5. Lens power or Magnification ratio
    6. Spherometers - lens Radius from arc depth
    7. Rayleigh Water Test - for testing optical flats
    8. Wavelength of LED or Laser based on band-gap voltage
    9. Colors of wave lengths
    10. Refractive Indexes 632nm
    11. Coherence time and Length
    12. Fast Lenses

Diopter or Dioptre

The diopter unit is simply the inverse of the focal length in meters.

 dio = \frac{1}{f}

Thus a +2 diopter lens has a focal length of 500mm

You can combine lenses of different diopters arranged in contact - thus a -2 diopter lens plus a +4 diopter lens would give us the same 500mm focal length.

Focal Length

If a lens has surface radii of R1 and R2 and made of glass with an n refractive index the focal length, f, is defined below

R1 and R2 = surface radii (negative for convex lens surface)

ng = refractive index of the lens glass

f = focal length of lens

 \frac{1}{f} = \left(n - 1\right) \left(\frac{1}{R_1}
        + \frac{1}{R_2} \right)

Focal length to angle of view

α = angle of view  d = size of film or sensor f = focal length


Depth of Field

Depth of Field is the minimum to maximum distance in focus.

Contrast Ratio

Contrast ratio is the difference between the brightest and darkest point in an image.

Lens power or Magnification ratio


m = Magnification ratio or Power

hi = image height

ho = object height

Di = distance from lens to image

Do = distance from lens to object

 m = \frac{h_i}{h_o} = \frac{D_i}{D_o}

Spherometers - lens Radius from arc depth

C of the Triangle is the same as Radius.

The bottom has a calculation of the Spherometers leg from the radius.

Variables in light optics Greek

V = frequency in hz

ω = radian frequency

δ = path difference (corrected for n)

φ = phase difference

P = order of interference = K

e = thickness 

τ = coherence time

l = coherence length

n = index of refraction

a = distance from center

Phase reversal on reflection

Light phase flips (E-field only) when reflecting off a material with higher refractive index than what it is traveling through.

Brewster angle reflections have no phase change.

If light reflects off an internal wall of a prism or flat - phase does not reverse.

Phase reversal is important when analyzing coatings.

Rayleigh Water Test - for testing optical flats


The interference creating bands happens between the water surface and the top of the glass.

Back to the water test - I'm still not seeing what I want to - I've started experimenting - trying to see a pattern with a single microscope slide - the reflection from the front and rear surface seems close to the water test. (The double slide glass interference is easy).

To see the single glass, the light and eye want to be very close together - a normal view point to the surface - It

Getting things level is the hardest bit - point a LASER pointer straight down - align the dots - and hope you get close. The LASER should be directly above and aimed so the dots are close to the LASER - 10mm or less.

The drill is to look without the collimating lens in place - you want to find rings - adjust so they are wide.. If the rings are close - a magnifying glass might help.  Focus on the image of the lamp - that is where you will see the first lines. (This didn't seem to work well - very careful LASER alignment was more important).

The leveling screws I used were differential - made from a M8-1.25 capscrew that was drilled out  to mount a M6-1.0 - the thread difference gives 250um per turn.

Fringes formed from a non uniform film (wedge) are called Fizeau fringes.

Newton rings are a type of Fizeau fringe - the wedge being concentric from being pressed together at a point.

Haidinger fringes are fringes of equal inclination - pin hole source best viewed with a large aperture lens. (longer path length means coherence length is critical).

Wavelength of LED or Laser based on band-gap voltage

The wave-length in lens specifications is 632.8nm - helium-neon LASER - (Comes from a persistent  neon line --  with low concentrations of a particular element relative to other substances in the source, the number of observable lines decrease with decreasing concentration until only the most persistent remain.  Should have been called a Neon-Helium LASER) but other monochrome light sources are available - LEDs and LED LASERs

Below is an estimation based on the voltage drop which is close to the band gap of a forward biased diode.

l = wave length in nm

Eg = band-gap voltage


 l = \frac{1238}{E_g}

Thus those red LEDs with a 1.9V drop run about 652 nm. (GaAs0.6P0.4)

This appears derived from ,  E_g = \frac{hc}{l}

h = Plank's Constant = 4.13 x 10-15 eV·s
c = speed of light = 2.998 x 108 m/s

Colors of wave lengths

Infrared 710nm and goes down to roughly 2000nm.

The visible range is considered to be roughly 400nm to 710nm.

390nm to 420 is Blue (or Violet),

530nm is green

 Mercury e-line  = 546.07 nm 

589nm - sodium D lines - a doublet - two lines 589.0 and 589.56nm - coherence length 0.59mm (must be one line) (588.9950 and 589.5924)

600nm to 710nm is considered red.

632.8nm - helium-neon LASER - neon - bandwidth about 0.002 nm  compared to LED bandwidth  of 24-27nm - 10,000 wider.

Refractive Indexes 632nm

Fused silica 1.457

Fused Quartz 1.46

Zerodur 1.542

Borofloat 1.472

Clearceram-Z  1.54

ULE 1.48

N-BK7 1.51

Water ~ 1.33

Coherence time and Length

Example of Coherence Length
Sodium vapor lamp yellow "D" line
• λ = 589 nm and line-width 5.1x1011 Hz
• Thus coherence time and length is

Coherence Length

The mercury line at 546.1 is .3mm ??  varies greatly with tube pressure

Coherence length test

A pen LASER has a coherence length of about 10mm - because the slide is thinner there is an interference pattern on the screen.  If in place of the slide there was a 15mm optical parallel - the pattern fails.



Bath Interferometer

The Bath interferometer is not effected by coherence length.  A bit of a puzzle why?  The trick is to think in time - rather than length - thus coherence time - and to focus on correlation. Light interference requires the light to be correlated in (average) phase (and to have some amount of spacial coherence as well).

So for a Newton rings - a type of Fizeau fringe - the time it takes to travel the two paths - is different - varies in time.  Thus if the source has a short coherence time  - and the difference is greater, the interference pattern falls apart. When the light arrives - it is no longer correlated even if at an even cycle time of the given light frequency.

But for the Bath - the time difference of the path is never greater than the defects of the mirror under test - those defects would have to be greater than the coherence length for it to fall apart. (Imagine a mirror surface with a 10mm defect - the absurdity makes it memorable <grin>)

The light doesn't care about time varying phase - as long as it is correlated between the two sources.   So if you imagine that the light source was oscillating in phase like an FM radio station - limiting the coherence time to part of one of these cycles - it doesn't matter - the light creating the pattern has the same phase when it arrives.  Coherence time means that phase changes in time - not because it traveled some length - both beams started off the same - have the same path length - so will arrive the same as well.  Both arrive highly correlated.   So coherence length matters if the time of flight differs - the key detail.

The following I think came from Dale Eason - couldn't find the original link.

Best Practices
The software requires a good “igram” for input – a photograph of the interference fringes due to interference of the test and reference beams. These guidelines are meant to supplement Dale’s instructions with a “Best Practices” for obtaining high quality, clean igrams. Much of this material is dispersed throughout years of posts in the messages section.

Fast Lenses

Translated from French http://www.dg77.net/photo/tech/fast.htm

to add post it here!
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