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Optics Overview

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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

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**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 relates the dual particle & wave nature of light; h=Planck’s constant =6.6262⊥10-34 J sec ν is the temporal oscillation frequency of the light waves

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**Wave properties of light**

λ: wavelength (spatial period) k=2π/λ wavenumber ν: temporal frequency ω=2πν angular frequency E: electric field

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**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) “Dispersion relation” (holds in vacuum only)

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**Light in matter Speed c/n Speed c=3×108 m/sec n : refractive index**

light in vacuum light in matter Speed c/n n : refractive index (or index of refraction) Absorption coefficient α energy decay coefficient, after distance L : e–2αL Speed c=3×108 m/sec Absorption coefficient 0 E.g. vacuum n=1, air n ≈ 1; glass n≈1.5; glass fiber has α ≈0.25dB/km=0.0288/km

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**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

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**Overview of light sources**

non-Laser Thermal: polychromatic, spatially incoherent (e.g. light bulb) Gas discharge: monochromatic, (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, (e.g. Q-switched, mode-locked) ~nsec ~psec to few fsec pulse duration mono/poly-chromatic = single/multi color

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**Monochromatic, spatially coherent**

light • nice, regular sinusoid • 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

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**The concept of a monochromatic**

“ray” t=0 (frozen) direction of energy propagation: light ray wavefronts In homogeneous media, light propagates in rectilinear paths

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**The concept of a monochromatic**

“ray” t=Δt (advanced) direction of energy propagation: light ray wavefronts In homogeneous media, light propagates in rectilinear paths

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**light propagates in rectilinear paths**

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

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**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

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**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

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**The law of refraction reflected reflected incident**

Snell’s Law of Refraction

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**Total Internal Reflection (TIR)**

becomes imaginary when refracted beam disappears, all energy is reflected

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Prisms air glass air air air glass air air

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**Dispersion Refractive index n is function of the wavelength**

white light (all visible wavelengths) red air glass green blue Newton’s prism

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**Frustrated Total Internal Reflection**

(FTIR) other glass material Reflected rays are missing where index-matched surfaces touch shadow is formed Angle of incidence exceeds critical angle air gap

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**Fingerprint sensor finger air glass | air glass | finger TIR occurs**

TIR does not occurs (FTIR)

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**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) and sensing (e.g. pressure) systems

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**Refraction at a spherical surface**

point source

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Imaging a point source point source point source Lens

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**plane wave (or parallel ray bundle);**

Model for a thin lens point object at 1st FP 1st FP focal length f plane wave (or parallel ray bundle); image at infinity

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**plane wave (or parallel ray bundle);**

Model for a thin lens point image at 2nd FP focal length f plane wave (or parallel ray bundle); object at infinity

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**Types of lenses positive (f > 0) negative (f < 0)**

BICONVEX BICONCAVE PLANO CONVEX PLANO CONCAVE from Modern Optical Engineering by W. Smith POSITIVE MENISCUS NEGATIVE MENISCUS Figure The location of the focal points and principal points for several shapes of converging and diverging elements.

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**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 wavefronts

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**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.

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**Imaging condition: ray-tracing**

thin lens (+) image (real) 2nd FP 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 • The real image is inverted and can be magnified or demagnified

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**Imaging condition: ray-tracing**

thin lens (+) image 2nd FP 1nd FP object Lens Law Lateral Angular Energy magnification magnification conservation

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**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

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**the Scheimpflug condition**

Tilted object: the Scheimpflug condition LENS PLANE OBJECT LANE IMAGE PLANE OPTICAL AXIS The object plane and the image plane intersect at the plane of the thin lens.

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**Lens-based imaging • Human eye • Photographic camera • Magnifier**

• Microscope • Telescope

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**The human eye Anatomy Remote object (unaccommodated eye)**

CORNEA AQUEOUS IRIS LENS LIGAMENTS MUSCLE SCLERA Remote object (unaccommodated eye) RETINA BLIND SPOT MACULA-FOVEA OPTIC NERVE Near point (comfortable viewing) ~25cm from the cornea Near object (accommodated eye)

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**Eye defects and their correction**

Hypermetropia, farsighted Myopia, nearsighted from Fundamentals of Optics by F. Jenkins & H. White 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.

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**The photographic camera**

Bellows Focal plane Axis Object Stop meniscus lens or (nowadays) zoom lens Film or “digital imaging” detector array (CCD or CMOS)

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**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.

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**The compound microscope**

from Optics by M. Klein & T. Furtak L: distance to near point (10in=254mm) Objective magnification Eyepiece magnification Combined magnification

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**The telescope (afocal instrument – angular magnifier) from Optics**

Flg Astronomical telescope. from Optics by M. Klein & T. Furtak Flg Galilean telescope (fashioned after the first telescope).

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**The pinhole camera image object**

opaque screen image pin- hole 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; diffraction introduces undesirable artifacts in the image.

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Field of View (FoV) FoV=angle that the chief ray from an object can subtend towards the imaging system

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**Numerical Aperture Numerical Aperture (NA) = n sinθ**

medium of refr. index n Numerical Aperture (NA) = n sinθ Speed (f/#)=1/2(NA) pronounced f-number, e.g. f/8 means (f/#)=8. θ: half-angle subtended by the imaging system from an axial object

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**Resolution How far can two distinct point objects be**

before their images cease to be distinguishable?

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**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

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**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

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**Diffraction limited resolution**

object spacing δx light intensity (arbitrary units) lateral coordinate at image plane (arbitrary units) Point objects “just resolvable” when Rayleigh resolution criterion

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**Wave nature of light • Diffraction • Inteference**

broadening of point images • Inteference diffraction grating Fabry-Perot interferometer Interference filter (or dielectric mirror) Michelson interferometer • Polarization: polaroids, dichroics, liquid crystals, ...

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**Diffraction grating incident plane Grating spatial frequency: 1/Λ wave**

Angular separation between diffracted orders: Δθ ≈1/Λ “straight-through” order or DC term Condition for constructive interference: ( m integer) diffraction order

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**Grating dispersion Anomalous (or negative) dispersion polychromatic**

(white) light Glass prism: normal dispersion

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**Fresnel diffraction formulae**

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**Fresnel diffraction as a linear, shift-invariant system**

Thin transparency impulse response convolution transfer function multiplication Fourier transform (≡plane wave spectrum) g(x, y) t(x, y)

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The 4F system object plane Fourier plane Image plane

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The 4F system object plane Fourier plane Image plane

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**The 4F system with FP aperture**

object plane Fourier plane: aperture-limited Image plane: blurred (i.e. low-pass filtered)

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**The 4F system with FP aperture**

Transfer function: circular aperture Impulse response: Airy function

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**Coherent vs incoherent imaging**

field in intensity in field out intensity out Coherent optical system Incoherent

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**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) | (u,v)|: Modulation Transfer Function (MTF) (u, v): Optical Transfer Function (OTF)

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**Coherent vs incoherent imaging**

Coherent illumination Incoherent illumination

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**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 (~ μm in the visible)

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**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

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