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