Optical Path Difference • How do we determine the quality of a lens

```Criteria for Optical Systems: Optical Path Difference
 How do we determine the quality of a lens system?
 Several criteria used in optical design Computer Aided Design
 Several CAD tools use Ray Tracing (see lesson 4)
 Then measure these criteria using the CAD tools
 Optical Path Difference (OPD) measures quality
 Measures path different from different parts of lens
 Plot OPD difference across the image relative to spherical wave
 Related to the Airy disk creation of a spot
Point Sources and OPD
 Simplest analysis: what happens to a point source
 Know that point sources should give perfect Airy disc
 Adding the OPD delay creates the distortion
 Little effect at /4
 By OPD /2 get definite distortion
  OPD point is really distorted
 Point Spread Function (PSF) is distribution of point source
 Like the response to an impulse by system in electrical circuits
 Often calculate for a system
 Again distorted by Optical path differences in the system
Wave Front Error
 Measure peak to valley (P-V) OPD
 Measures difference in wave front closest to image
 and furthest (lagging behind) at image
 Eg. in mirror system a P-V </8 to meet Rayleigh criteria
 Because P-V is doubled by the reflection in mirrors
 Also measure RMS wave front error
 Difference from best fit of perfect spherical wave front
Depth of Focus
 Depth of focus: how much change in position is allowed
 With perfect optical system </4 wave front difference needed
 Set by the angle  of ray from edge of lens
 This sets depth of focus  for this OPD </4
 

2 n sin  
2
 2  f # 
2
 Thus f# controls depth of focus
 f#:4 has 16 micron depth
 f#:2 only 2 micron
 Depth of Focus used with microscopes
 Depth of Field is term used in photography
 Depth that objects appear in focus at fixed plan
Depth of Field in Photography
 Depth of Field is the range over which item stays in focus
 When focusing close get a near and far distance
 When focusing at distance want to use the Hyperfocal Distance
 Point where everything is in focus from infinity to a near distance
 Simple cameras with fixed lens always set to Hyperfocal Distance
Depth of Field Formulas
 Every camera has the “circle of confusion” c
 Eg for 35 mm it is 0.033 mm, point & shoot 0.01 mm
 Then Hyperfocal Distance H (in mm)
f2
H
f
F# c
f is lens focal length in mm
 When focused at closer point distance s in mm
 Then nearest distance for sharp image is Dn
Dn 
sH  f 
H s2f
 Furthers distance for sharp image Df
Df 
sH  f 
H s
As get closer Depth of focus becomes very small
Get good DOF tools at http://www.dofmaster.com/
Modulation Transfer Function
 Modulation Transfer Function or MTF
 Basic measurement of Optical systems
 Look at a periodic target
 Measure Brightest (Imax) and darkest Imin
MTF 
I max  I min
I max  I min
 Contrast is simply
constrast 
 MTF more accurate than contrast
I max
I min
Square Wave vs Sin wave
 Once MTF know for square wave can get sine wave response
 Use fourier components
 If S(v) at frequency v is for square waves
 Then can give response of sine wave
M   
S   

S 3  S 5  S 7 









S


4
3
5
7

M 3  M 5  M 7 
4



M







 
3
5
7
Diffraction Limited MTF
 For a perfect optical system
MTF 
2

  cos  sin 
Where
  arccos
 

 2 NA 
Maximum or cutoff frequency v0
1

 f #
In an afocal system or image at infinity then for lens dia D
D
0 
0 
2 NA


Defocus in MTF
 Defocus MTF
2 J x 
defocus MTF  1
x
Where x is
x  2 NA
 Max cutoff is 0.017 at v=v0/2
  0   
0
MTF and Aberrations
 Eg. 3rd order spherical aberrations
 Effect goes as wavelength defect
MTF and Filling Lens
 MTF decreases as lens is not filled
 i.e. object blocking part of the lens
 Best result when image fills lens
MTF Specifications
 MTF in lenses are specified in lines per millimetre
 Typically 10 and 30 lines
 Specified separately for Saggittal and tangential
 Saggittal – vertical aberrations on focus plane
 Tangential or Meridional: horizontal on focus plane
 Camera lenses often publish MTF charts
 Below example for Nikon 18-55 mm zoom
 Plots show MTF at 10 lines/mm and 30/mm
 Shown with radius in mm from centre of image
 For a 24x15 mm image area
 Usually specified for single aperature (f/5.6 here)
 10/mm measures lens contrast
 30/mm lens resolution
Wide angle
Spatial Frequencies
10 lines/mm
30 lines/mm
Telephoto
S: Sagittal
M: Meridional
Poor MTF Charts
 Some companies give charts but little info
 Entry level Cannon 18-55 mm lens
 Chart give MTF but does not say lines/mm
 Cannot compare without that
Aerial Image Modulation Curves
 Resolution set in Aerial Image Modulation (AIM)
 Combines the lens and the detector (eg film or digital sensor)
 Measures the smallest resolution detected by sensor
 Sensor can significantly change resolutions
Film or Sensor MTF
 Film or sensor has MTF measured
 Done with grating directly on sensor
 Eg Fuji fine grain Provia 100 slide film
 50% MTF frequency (f50) is 42 lp/mm
MTF/AIM and System
 Also need to look at f/# for the lens
 This is 4000 dpi digitizing of negative
MTF and Coherent Light
 MTF is sharpest with coherent light
 Decreases as coherence decreases
Low Power Laser Applications: Alignment & Measurement
Circularizing Laser Diodes
 Laser diodes are important for low power applications
 But laser diodes have high divergence & asymmetric beams
 Get 5-30o beam divergence
 Then compensate for asymmetry
 Use cylindrical lens beam expander
 Cylindrical lenses: curved in one axis only unlike circular lenses
 Expands/focuses light in one direction only (along curved axis)
 Results in circular collimating beam
 Often put detector on object being aligned to laser
 Use 4 quadrant detector Silicon photodiode detector
 Expand beam so some light in each quadrant
 Amount of photocurrent in each quadrant proportional to light
 Detect current difference of right/left & top bottom
 Higher current side has more beam
 Perfect alignment null current for both sides
Laser Leveling
 Lasers used to project lines of light
 Accuracy is set by the level of the beam source
 Used in construction projects: lines and cross lines
 Get vertical and horizonal
 Laser diodes give low cost levels now
 More complex: reflect light back from object
 Make certain light is reflected along the same path
 Called Autocolation
Laser Size Gauging
 Gauging is measuring the size of objects in the beam
 Simplest expand beam the refocus
 Object (eg sphere) in beam reduces power
 Estimate size based on power reduction
 More accurate: scanning systems
 Scan beam with moving mirror (focused to point)
 Then measure time beam is blocked by object
 Knowing scan range then measure size of object
Laser & Linear Detector Array
 Use laser diode to illuminate a linear or 2D detector array
 Laser diode because creates collimated beam
 Expand beam to fill area
 Image is magnified or shrunk by lens
 Use pixel positions to determine object profile
 Low cost pixel arrays makes this less costly to gage scanners
Laser Scanner to Detect Surface Defects
 Laser beam scanned across surface of reflective (eg metal) sheets
 Detect reflected light
 Flaws result in reduce or increase light
 Timing (when scanning) determines defect size
 Instead of spot use cylindrical expander to beam line of light
 Moving sheet (eg metal, glass, paper) crosses beam
 Use line or 2D images to detect changes
 Use both reflection and transmission depending on material
 Transmission can detect changes in thickness or quality
Bar Code Scanners
 Diode laser now widely used in Bar code scanners
 Typically use two axis scanner
 Laser beam reflected from mirror on detector lens
 Bar code reflected light comes back along same path
 Detect rising and falling edge of the pattern
 Note: have the laser beam & return light on same path
 Use small mirror or beam splitter to put beam in path
Laser Triangulation
 Lasers aimed at precise angles depth/profiles using triangulation
 Single spot for depth measurement
 Laser spot focused by lens onto detector array
 Change in laser spot depth position z
 Gives change in position z’ at detector
 Change set by magnification caused by lens
  laser to lens angle
  angle between detector an lens axis
 Resulting equations
 sin  
z
 sin  
 Get real time measurement of distance changes
z   m
Laser Profileometry
 Use cylindrical lens to create line of laser light
 Use 2D detector array (imager) & lens to observe line
 If object is moving get continuous scan of profile
 Problems: Background light eg sunlight
 Changes in surface reflectance makes signal noisy
 Eg log profileometry for precise cutting of logs
 Problem is log surface changes eg dark knots, holes
Lidar
 LIDAR: LIght Detection And Ranging
 Can use pulses & measure time of flight (like radar)
 But only hard to measure <10-10 sec or 3 cm
 Better phase method
 Modulate the laser diode current with frequency fm
 Then detector compares phase of laser to detector signal
 Phase shift for distance R is

2
m
2 R 
c  m f m
and
 Then the distance is
R
c
4 f m

 If > modulation wavelength m need to get number of cycles
 In extreme phase changes in the laser light
 That requires a very stable (coherent) laser: HeNe not diode
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