Download Criteria for Optical Systems: Optical Path Difference • Optical Path

Criteria for Optical Systems: Optical Path Difference
• Optical Path Difference from different part of lens sets quality
• Called OPD
• Related to the Airy disk creation
Point Sources and OPD
• Simplest analysis: what happens to a point source
• As add OPD delay get distortion
• Little effect at λ/4
• By OPD λ/2 get definite distortion
• λ OPD point is really distorted
Point Spread Function
• Point Spread Function (PSF) is distribution of point
• Often calculate for a system
• 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 <0.125 to meet Rayleigh criteria
• Because P-V is doubled by the reflection
• Reason this is doubled
• Also measure RMS wavefront error
• Difference from best fit of perfect spherical wavefront
Depth of Focus
• Depth of focus: how much change in position is allowed
• With perfect optical system <λ/4 wavefront difference needed
• Set by the angle θ of ray from edge of lens
• This sets depth of focus δ for this OPD <λ/4
δ =±
λ
(2 n sin θ )
2
= ±2λ ( 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 =
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 (ν ) =
S (ν ) =
π⎡
S (3ν ) S (5ν ) S (7ν )
⎤
(
)
ν
+
−
+
−
L
S
⎢
⎥
4⎣
3
5
7
⎦
M (3ν ) M (5ν ) M (7ν )
4⎡
⎤
(
)
M
ν
−
+
−
+
L
⎥⎦
π ⎢⎣
3
5
7
Diffraction Limited MTF
• For a perfect optical system
MTF =
2
π
(φ − cos(φ ) sin(φ ))
Where
φ = arccos⎜⎛
λν ⎞
⎟
⎝ 2 NA ⎠
Maximum or cutoff frequency v0
1
λ
λ( f #)
In an afocal system or image at infinity then for lens dia D
D
ν0 =
ν0 =
2 NA
=
λ
Defocus in MTF
• Adding defocus decreases MTF
• Defocus MTF
2 J (x )
defocus MTF = 1
x
Where x is
x = 2πδ NA
• Max cutoff is 0.017 at v=v0/2
ν (ν 0 − ν )
ν0
MTF and Aberrations
• Aberrations degrade MTF
• Eg. 3rd order spherical aberrations
• Effect goes as wavelength defect
MTF and Filling Lens
• MTF decreases as lens is not filled
• 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
Reading MTF in Camera Lenses
• 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
• Adding each item degrades system
• Also need to look at f/# for the lens
• Adding digitization degrades image
• This is 4000 dpi digitizing of negative
MTF and Coherent Light
• MTF is sharpest with coherent light
• Decreases as coherence decreases
Laser Beam Interactions with Solids
• In absorbing materials photons deposit energy
E = hv =
hc
λ
where h = Plank's constant = 6.63 x 10-34 J s
c = speed of light
• Also photons also transfer momentum p
p=
h
λ
• Note: when light reflects from a mirror
momentum transfer is doubled
• eg momentum transferred from Nd:YAG laser photon
hitting a mirror (λ= 1.06 microns)
2(6.6 x10 − 34 )
− 27
=
1
.
25
x
10
kg m / s
p= =
−6
λ
1.06 x10
h
• Not very much but Sunlight 1 KW/m2 for 1 sec
has 5x1021 photons: force of 6.25x10-6 N/m2
• Proposed for Solar Light Sails in space (get that force/sq m of sail)
small acceleration but very large velocity over time.
• Russian Cosmos 1 solar sail
Failed to reach 500 km orbit June 2005
Absorbing Solids
• Beam absorbed as it enters the material
• For uniform material follows Beer Lambert law
I ( z ) = I 0 exp( −α z )
where α = β = absorption coefficient (cm-1)
z = depth into material
• Absorption coefficient dependent on
wavelength, material & intensity
• High powers can get multiphoton effects
• Rayleigh scattering, Brillouin scattering, Raman scattering
• Tissues, liquids, have low absorption but high scattering
• Add a scattering coefficient αs & to absorption αa & modify law
I ( z ) = I 0 exp( −[α a + α s ] z )
• I then is amount of light transmitted unscattered
• However scattered light still emerges but contains no information
• Turbid (scattering) media where αs >> αa (100x greater)
• Most laser processing done on non-scattering materials αs ~ 0
Single Crystal Silicon
• Absorption Coefficient very wavelength dependent
• Argon light 514 nm α = 11200/cm
• Nd:Yag light 1060 nm α = 280/cm
• Hence Green light absorbed within a micron
1.06 micron penetrates many microns
• Very temperature dependent
• Note: polycrystalline silicon much higher absorption
: at 1.06 microns α = 20,000/cm
Absorption Index
• Absorbing materials have a complex index of refraction
c
nc = n − ik
v=
nc
where n = real index of refraction
k = absorption index or extinction coefficient
• The Electric field then becomes
r
ω nc z ⎞⎤
⎡ ⎛
E (t , z ) = iˆE 0 exp ⎢ j ⎜ − ω t +
⎟⎥
c
⎠⎦
⎣ ⎝
ω nz ⎤ ⎞ ⎛ ω kz ⎞
⎛ ⎡
E( t , z ) = E0 exp⎜ i ⎢ω t −
⎟ exp⎜ −
⎟
⎥
c ⎦⎠
⎝ λ ⎠
⎝ ⎣
• The k can be related to the absorption coefficient by
α=
4π k
λ
where wavelength is the vacuum value
Absorption Index & Electrical Parameters
• k and n are related to the dielectric constant ε
and the conductivity σ of the material
n2 − k 2 = ε
nk =
σ
ν
where ν= the frequency
• High conductivity Metals have high k relative to n: hence high R
• Note n can be less than 1 for absorbing materials but nc>1
• Insulators: k=0 when transparent
Opaque Materials
• Materials like metals have large numbers of free electrons
• High conductivity, reflectivity and absorption
• Reflectivity given by (for normal incidence of light)
2
(
n − 1) + k 2
R=
(n + 1)2 + k 2
• For opaque materials light absorbed A is
A = 1− R
• R and k are very wavelength dependent
• Also these are very dependent on
the absence of other materials from the surface.
Temperature Dependence
• Absorption and reflectivity are very temperature dependent
• Often undergo significant changes when material melts
• eg Silicon, steel becomes highly reflective on melting
Laser Processing of Materials
• Use the laser as a local heat source for most applications
Basic Processes
• Laser cutting (removing material)
• Laser Welding (joining materials)
• Laser heat treatment (modify materials)
• Laser disassociation (cause material to break down)
With increasing beam power the material
• Heats
• Melts
• Boils
• Forms a plasma (ionized material)
Laser Beam Impact on Surface
• Laser absorbed into the surface
• Creates local hot spot
• If hot enough melts to some depth
(function of thermal and optical properties
• Material removed - forms keyhole in material
• Keyhole allows beam to penetrate further
Laser Cutting: Simple Thermal
• Six basic ways
Vaporization
• beam heats local area above melting point
• material boils and ejects
Melting and Blowing
• Beam melts surface, jet of inert gas removes material
Burning in Reactive Gas
• Beam melts surface, jet of reactive gas removes material
• gas (usually oxygen) reacts with material, adds energy
Laser Cutting: Other Effects
Thermal stress Cracking
• Heat/cool cycle creates thermal stress, material breaks
• Requires least energy
Scribing
• Cut creates small stress point, breaks with force
Cold Cutting
• Laser photons in UV cause material to disassociate
Lasers in Microelectronics
• Laser cutting, heat processing
widely used in microelectronics now
• Oldest application, resistor trimming
• Polysilicon resistors are deposited thin film resistors
• However polySi resistance varies considerably with process
• Use laser cutting to Trim resistance to desired values
• This is how A/D & D/A are made accurate
• Growing applications in laser microsurgery, defect avoidance
Laser Defect Correction
• Using laser circuit modification to make
postfabrication changes in Integrated Circuits
• Generally cutting lines and making connections
Laser Microsurgery
• Repair defects on chip sized structures
Laser process not part of original design
• Chip repair
Laser process as part of original design
• Used at production repair process (eg DRAM's)
• Laser chip customization
Wafer Scale Restructurable Silicon system
• Using laser corrections to build more complex systems
• System design integrated closely with the laser repair technique
• Done at Lincoln Lab in 1985
Laser Line Cutting
• Used in Laser microsurgery:Laser acts as a local heat source
• Laser power used to segment signal/power lines
• Cutting of Metal & Polysilicon
• Call each laser pulse on a point a "Zap"
• Effects of intermetal insulators and scratch protect important
• Usually cause openings in Scratch Protect
Factors Affecting Laser Line Cutting Parameters
Power delivered to cut point
• Reflectivity/Absorption of light
• Thermal flow in line
Laser Light Wavelength
• Poly Silicon must be visible (usually Green)
• Metals have low wavelength sensitivity
Materials
• Metals highly reflective & high thermal conductivity
• Mixed alloys can significantly
Layer thicknesses
• Thicker metal layers, more heat flow
• Thicker intermetal layer below, less heat to substrate
• Thicker intermetal/insulator above: more power to move layer
• (PolySi) thin layers mean less energy absorbed
Laser Line Cutting
Laser Beam
Laser focused spot
Scratch Protect &
Intermetal Insulators
Silicon Substrate
Metal Line