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Optics Overview MIT 2.71/2.710 Review Lecture p- 1 What is light? • Light is a form of electromagnetic energy – detected through its effects, e.g. heating of illuminated objects, conversion of light to current, mechanical pressure (“Maxwell force”) etc. • Light energy is conveyed through particles: “photons” – ballistic behavior, e.g. shadows • Light energy is conveyed through waves – wave behavior, e.g. interference, diffraction • Quantum mechanics reconciles the two points of view, through the “wave/particle duality” assertion MIT 2.71/2.710 Review Lecture p- 2 Particle properties of light Photon=elementary light particle Mass=0 Speed c=3⊥108m/sec According to According to Special Relativity, a mass-less particle travelling at light speed can still carry momentum! Energy E=hν h=Planck’s constant =6.6262⊥10-34 J sec MIT 2.71/2.710 Review Lecture p- 3 relates the dual particle & wave nature of light; ν is the temporal oscillation frequency of the light waves Wave properties of light λ: wavelength (spatial period) k=2π/λ wavenumber ν: temporal frequency ω=2πν angular frequency E: electric field MIT 2.71/2.710 Review Lecture p- 4 Wave/particle duality for light Photon=elementary light particle Mass=0 Speed c=3⊥108 m/sec Energy E=hν h=Planck’s constant =6.6262⊥10-34 J sec ν=frequency (sec-1) λ=wavelength (m) MIT 2.71/2.710 Review Lecture p- 5 “Dispersion relation” (holds in vacuum only) Light in matter light in vacuum light in matter Speed c=3×108 m/sec Speed c/n n : refractive index (or index of refraction) Absorption coefficient 0 Absorption coefficient α energy decay coefficient, after distance L : e–2αL E.g. vacuum n=1, air n ≈ 1; glass n≈1.5; glass fiber has α ≈0.25dB/km=0.0288/km MIT 2.71/2.710 Review Lecture p- 6 Materials classification • Dielectrics – typically electrical isolators (e.g. glass, plastics) – low absorption coefficient – arbitrary refractive index • Metals – conductivity large absorption coefficient • Lots of exceptions and special cases (e.g. “artificial dielectrics”) • Absorption and refractive index are related through the Kramers– Kronig relationship (imposed by causality) absorption refractive index MIT 2.71/2.710 Review Lecture p- 7 Overview of light sources non-Laser Thermal: polychromatic, spatially incoherent (e.g. light bulb) Gas discharge: monochromatic, spatially incoherent (e.g. Na lamp) Light emitting diodes (LEDs): monochromatic, spatially incoherent Laser Continuous wave (or cw): strictly monochromatic, spatially coherent (e.g. HeNe, Ar+, laser diodes) Pulsed: quasi-monochromatic, spatially coherent (e.g. Q-switched, modelocked) ~nsec pulse duration mono/poly-chromatic = single/multi color MIT 2.71/2.710 Review Lecture p- 8 ~psec to few fsec Monochromatic, spatially coherent • nice, regular sinusoid light • λ, ν well defined • stabilized HeNe laser good approximation • most other cw lasers rough approximation • pulsed lasers & nonlaser sources need more complicated description Incoherent: random, irregular waveform MIT 2.71/2.710 Review Lecture p- 9 The concept of a monochromatic t=0 “ray” (frozen) direction of energy propagation: light ray wavefronts In homogeneous media, light propagates in rectilinear paths MIT 2.71/2.710 Review Lecture p-10 The concept of a monochromatic t=Δt “ray” (advance d) direction of energy propagation: light ray wavefronts In homogeneous media, light propagates in rectilinear paths MIT 2.71/2.710 Review Lecture p-11 The concept of a polychromatic “ray” t=0 (frozen) energy from pretty much all wavelengths propagates along the ray wavefronts In homogeneous media, light propagates in rectilinear paths MIT 2.71/2.710 Review Lecture p-12 Fermat principle Γ is chosen to minimize this “path” integral, compared to alternative paths (aka minimum path principle) Consequences: law of reflection, law of refraction MIT 2.71/2.710 Review Lecture p-13 The law of reflection a) Consider virtual source P” instead of P b) Alternative path P”O”P’ is longer than P”OP’ c) Therefore, light follows the symmetric path POP’. mirror MIT 2.71/2.710 Review Lecture p-14 The law of refraction reflected reflected incident Snell’s Law of Refraction MIT 2.71/2.710 Review Lecture p-15 Total Internal Reflection (TIR) becomes imaginary when refracted beam disappears, all energy is reflected MIT 2.71/2.710 Review Lecture p-16 Prisms air glass air air air glass air MIT 2.71/2.710 Review Lecture p-17 air Dispersion Refractive index n is function of the wavelength white light (all visible wavelengths) red air glass green blue Newton’s prism MIT 2.71/2.710 Review Lecture p-18 Frustrated Total Internal Reflection (FTIR) glass other Reflected rays are missing where index-matched surfaces touch shadow is formed Angle of incidence exceeds critical angle MIT 2.71/2.710 Review Lecture p-19 material air gap Fingerprint sensor finger air glass | air TIR occurs MIT 2.71/2.710 Review Lecture p-20 glass | finger TIR does not occurs (FTIR) Optical waveguide • Planar version: integrated optics • Cylindrically symmetric version: fiber optics • Permit the creation of “light chips” and “light cables,” respectively, where light is guided around with few restrictions • Materials research has yielded glasses with very low losses (<0.25dB/km) • Basis for optical telecommunications and some imaging (e.g. endoscopes) MIT 2.71/2.710 and Review Lecture p-21sensing (e.g. pressure) systems Refraction at a spherical surface point source MIT 2.71/2.710 Review Lecture p-22 Imaging a point source point source point source Lens MIT 2.71/2.710 Review Lecture p-23 Model for a thin lens point object at 1st FP 1st FP focal length f plane wave (or parallel ray bundle); image at infinity MIT 2.71/2.710 Review Lecture p-24 Model for a thin lens point image at 2nd FP focal length f plane wave (or parallel ray bundle); object at infinity MIT 2.71/2.710 Review Lecture p-25 Types of lenses positive (f > 0) BICONVEX PLANO CONVEX POSITIVE MENISCUS BICONCAVE PLANO CONCAVE NEGATIVE MENISCUS Figure 2.12 The location of the focal points and principal points for several shapes of converging and diverging elements. MIT 2.71/2.710 Review Lecture p-26 negative (f < 0) from Modern Optical Engineering by W. Smith Huygens principle Each point on the wavefront acts as a secondary light source emitting a spherical wave The wavefront after a short propagation distance is the result of superimposing all these spherical wavelets optical wavefront s MIT 2.71/2.710 Review Lecture p-27 Why imaging systems are needed • Each point in an object scatters the incident illumination into a spherical wave, according to the Huygens principle. • A few microns away from the object surface, the rays emanating from all object points become entangled, delocalizing object details. • To relocalize object details, a method must be found to reassign (“focus”) all the rays that emanated from a single point object into another point in space (the “image.”) • The latter function is the topic of the discipline of Optical Imaging. MIT 2.71/2.710 Review Lecture p-28 Imaging condition: ray-tracing thin lens (+) 2nd FP image (real) 1nd FP object • Image point is located at the common intersection of all rays which emanate from the corresponding object point • The two rays passing through the two focal points and the chief ray can be ray-traced directly MIT 2.71/2.710 • The Review Lecture p-29 real image is inverted and can be magnified or demagnified Imaging condition: ray-tracing thin lens (+) image 2nd FP 1nd FP object Lens Law MIT 2.71/2.710 Review Lecture p-30 Lateral magnification Angular magnification Energy conservation Imaging condition: ray-tracing thin lens (+) Image (virtual) 2nd FP 1st FP object • The ray bundle emanating from the system is divergent; the virtual image is located at the intersection of the backwards-extended rays • The virtual image is erect and is magnified • When using a negative lens, the image is always virtual, erect, and demagnified MIT 2.71/2.710 Review Lecture p-31 Tilted object: the Scheimpflug condition OBJECT LANE LENS PLANE IMAGE PLANE OPTICAL AXIS The object plane and the image plane intersect at the plane of the thin lens. MIT 2.71/2.710 Review Lecture p-32 Lens-based imaging • Human eye • Photographic camera • Magnifier • Microscope • Telescope MIT 2.71/2.710 Review Lecture p-33 The human eye Anatomy CORNEA AQUEOUS LENS IRIS LIGAMENTS MUSCLE SCLERA Remote object (unaccommodated eye) RETINA BLIND SPOT MACULA-FOVEA OPTIC NERVE Near point (comfortable viewing) ~25cm from the cornea MIT 2.71/2.710 Review Lecture p-34 Near object (accommodated eye) Eye defects and their correction Hypermetropia, farsighted Myopia, nearsighted FIGURE 10K Typical eye defects, largely present in the adult population. Farighted eye corrected Nearsighted eye corrected FIGURE 10L Typical eye defects can be corrected by spectacle lenses. MIT 2.71/2.710 Review Lecture p-35 from Fundamentals of Optics by F. Jenkins & H. White The photographic camera Bellows Focal plane Axis Object Stop meniscus lens or (nowadays) zoom lens “digital imaging” CMOS) MIT 2.71/2.710 Review Lecture p-36 Film or detector array (CCD or The magnifier from Optics by M. Klein & T. Furtak FIGURE 10I The angle subtended by (a) an object at the near point to the naked eye, (b) the virtual image of an object inside the focal point, © the virtual image of an object at the focal point. MIT 2.71/2.710 Review Lecture p-37 The compound microscope from Optics by M. Klein & T. Furtak L: distance to near point (10in=254mm) Objective magnification Combined magnification Eyepiece magnification MIT 2.71/2.710 Review Lecture p-38 The telescope (afocal instrument – angular magnifier) Flg. 3.40 Astronomical telescope. from Optics by M. Klein & T. Furtak Flg. 3.41 Galilean telescope (fashioned after the first telescope). MIT 2.71/2.710 Review Lecture p-39 The pinhole camera opaque screen image pinhole object • The pinhole camera blocks all but one ray per object point from reaching the image space an image is formed (i.e., each point in image space corresponds to a single point from the object space). • Unfortunately, most of the light is wasted in this instrument. • Besides, light diffracts if it has to go through small pinholes as we will see later; MIT 2.71/2.710 diffraction Review Lecture p-40 introduces undesirable artifacts in the image. Field of View (FoV) FoV=angle that the chief ray from an object can subtend towards the imaging system MIT 2.71/2.710 Review Lecture p-41 Numerical Aperture medium of refr. index n θ: half-angle subtended by the imaging system from an axial object MIT 2.71/2.710 Review Lecture p-42 Numerical Aperture (NA) = n sinθ Speed (f/#)=1/2(NA) pronounced f-number, e.g. f/8 means (f/#)=8. Resolution How far can two distinct point objects be before their images cease to be distinguishable? MIT 2.71/2.710 Review Lecture p-43 Factors limiting resolution in an imaging system • Diffraction • Aberrations • Noise Intricately related; assessment of image quality depends on the degree that the “inverse problem” is solvable (i.e. its condition) 2.717 sp02 for details – electronic noise (thermal, Poisson) in cameras – multiplicative noise in photographic film – stray light – speckle noise (coherent imaging systems only) • Sampling at the image plane – camera pixel size – photographic film grain size MIT 2.71/2.710 Review Lecture p-44 Point-Spread Function Light distribution near the Gaussian (geometric) focus (rotationally symmetric wrt optical axis) Point source (ideal) The finite extent of the PSF causes blur in the image MIT 2.71/2.710 Review Lecture p-45 light intensity (arbitrary units) Diffraction limited resolution object spacin g δx lateral coordinate at image plane (arbitrary units) Point objects “just resolvable” when MIT 2.71/2.710 Review Lecture p-46 Rayleigh resolution criterion Wave nature of light • Diffraction broadening of point images • Inteference diffraction grating Fabry-Perot interferometer Michelson interferometer Interference filter (or dielectric mirror) • Polarization: polaroids, dichroics, liquid crystals, ... MIT 2.71/2.710 Review Lecture p-47 Diffraction grating incident plane wave Grating spatial frequency: 1/Λ Angular separation between diffracted orders: Δθ ≈1/Λ “straight-through” order or DC term Condition for constructive interference: (m integer) MIT 2.71/2.710 Review Lecture p-48 diffraction order Grating dispersion Anomalous (or negative) dispersion polychromati c (white) light MIT 2.71/2.710 Review Lecture p-49 Glass prism: normal dispersion Fresnel diffraction formulae MIT 2.71/2.710 Review Lecture p-50 Fresnel diffraction as a linear, shift-invariant system Thin transparency t(x, y) g(x, y) impulse response convolutio n Fourier transform (≡plane wave spectrum) MIT 2.71/2.710 Review Lecture p-51 transfer function multiplication Fourier transform The 4F system object plane MIT 2.71/2.710 Review Lecture p-52 Fourier plane Image plane The 4F system object plane MIT 2.71/2.710 Review Lecture p-53 Fourier plane Image plane The 4F system with FP aperture object plane MIT 2.71/2.710 Review Lecture p-54 Fourier plane: aperture-limited Image plane: blurred (i.e. low-pass filtered) The 4F system with FP aperture Transfer function: circular aperture MIT 2.71/2.710 Review Lecture p-55 Impulse response: Airy function Coherent vs incoherent imaging MIT 2.71/2.710 Review Lecture p-56 field in Coheren t optical system field out intensity in Incoherent optical system intensity out Coherent vs incoherent imaging Coherent impulse response (field in ⇒field out) Coherent transfer function (FT of field in ⇒ FT of field out) Incoherent impulse response (intensity in ⇒intensity out) Incoherent transfer function (FT of intensity in ⇒ FT of intensity out) ~ MIT 2.71/2.710 Review Lecture p-57 | H (u,v)|: Modulation Transfer Function (MTF) ~ H (u, v): Optical Transfer Function (OTF) Coherent vs incoherent imaging Coherent illumination MIT 2.71/2.710 Review Lecture p-58 Incoherent illumination Aberrations: geometrical Paraxial (Gaussian) image point Non-paraxial rays “overfocus” Spherical aberration • Origin of aberrations: nonlinearity of Snell’s law (n sinθ=const., whereas linear relationship would have been nθ=const.) • Aberrations cause practical systems to perform worse than diffraction-limited • Aberrations are best dealt with using optical design software (Code V, Oslo, Zemax); optimized systems usually resolve ~3-5λ (~1.5-2.5μm in the visible) MIT 2.71/2.710 Review Lecture p-59 Aberrations: wave Aberration-free impulse response Aberrations introduce additional phase delay to the impulse response Effect of aberrations on the MTF unaberrated (diffraction limited) aberrated MIT 2.71/2.710 Review Lecture p-60