Basic optics Geometrical optics and images Interference Diffraction Diffraction integral we use simple models that say a lot! more rigorous approach
Basic optics Geometrical optics and images Interference Diffraction Diffraction integral
Images Édouard Manet: A Bar at the Folies-Bergère (1882)
Mirage light rays are bent to produce a displaced image of distant objects or the sky http://epod.usra.edu/blog/2012/10/inferior-mirage-on-a-desert-road.html
Plane mirrors Point objects Mirror Object Image convention: - light entering from the left - positive distances: O, I on the left - real image: i > 0 - virtual image: i < 0 (here, we have virtual image) object distance image distance p > 0 i < 0
Plane mirrors Extended objects virtual image the same orientation and size (height) as object
Head Eye Mirror Foot
Spherical mirrors concave Real focus Central axis Focal length Radius r, f > 0 convex Central axis Virtual focus r, f < 0
Image formation Spherical mirrors Axis Mirror
Spherical mirrors Ray tracing 4 rays Concave mirror: p > f : real image p = f : image at infinity p < f : virtual image - real image: i > 0 - virtual image: i < 0
Spherical mirrors Ray tracing 4 rays Convex mirror: image is always - virtual - erect - minified - real image: i > 0 - virtual image: i < 0
Spherical mirrors Magnification m triangles ABV and DEV are similar - real image: i > 0 - virtual image: i < 0 erect image: m > 0 inverted image: m < 0
Spherical refracting surfaces Real image Real image r > 0 r < 0 Virtual image Virtual image r < 0 r > 0 - real image: i > 0 - virtual image: i < 0 r < 0 Virtual image Virtual image r > 0
Axis
Thin lens Axis Air p Glass i for both refracting surfaces (thick lens) 0 (thin lens)
Thin lens converging lens f > 0 < 0 > 0 diverging lens f < 0 Extensions < 0 > 0
Ray tracing: converging lens f > 0 3 rays
Ray tracing: diverging lens f < 0 3 rays
Thin lens (bottom line) converging lens diverging lens
Thin lens: magnification m
Simple magnifier To distant virtual image angular magnification:
Compound microscope Objective Eyepiece To distant virtual image Parallel rays The lateral magnification produced by the objective lens The overall magnification
Refracting telescope Eyepiece Objective Parallel rays from distant object To distant virtual image Parallel rays (angular magnification of the telescope)
Aberrations (image errors) examples - aberrations can be balanced - image fidelity is limited only by diffraction
Basic optics Geometrical optics and images Interference Diffraction Diffraction integral
Interference What will happen if we add waves?
Double-slit experiment (Young s experiment, 1801) Incident wave An interference pattern Superposition of waves u a suitable component of E- or H- vector
Different phases due to different paths Incident wave assume Path length difference (maxima) (minima)
Intensity in double-slit experiment Intensity (two coherent sources) (two incoherent sources) (one source) (m for maxima) (m for minima) (missing sign) (maxima) (minima)
Double-slit experiment with... particles waves http://www.feynmanlectures.caltech.edu/iii_01.html
Interference from thin films For simplicity we assume 1. Normal incidence 2. Double beam interference Ray reflected at B Phase difference Ray reflected at A Incident wave phase shift arising from reflection at B phase shift arising from reflection at A
Example: air - n 2 - air Ray reflected at B Ray reflected at A (maxima) Incident wave (minima)
Example: air - n 2 - air (maxima) (minima)
Example: glass - air - glass Incident light Air Glass Glass (Newton rings)
Temporal coherence monochromatic wave - - perfectly coherent pulse (wave-packet) - - less coherent white light - - incoherent Define: coherence length coherence time
Interference and temporal coherence coherence length coherence time
Interference and temporal coherence
Spatial and temporal coherence
Michelson interferometer Movable mirror
Interference from thin films (again) reflected wave...... transmitted wave incident wave Now we consider 1. Arbitrary incident angle 2. Multiple beam interference
Interference from thin films reflected wave...... transmitted wave incident wave geometrical series The amplitude of the resultant transmitted wave
Interference from thin films reflected wave...... transmitted wave incident wave geometrical series The amplitude of the resultant reflected wave
Interference from thin films reflected wave transmitted wave incident wave For simplicity assume symmetric structure and pure real numbers (relative transmitted intensity)
Spectral response (thin film, FP etalon) integer (relative transmitted intensity)
Spectral response (thin film, FP etalon) The free spectral range, FSR (relative transmitted intensity)
with loss Spectral response (thin film, FP etalon)
Spectral analyzer const. 0
Basic optics Geometrical optics and images Interference Diffraction Diffraction integral
Huygens-Fresnel principle Every point of a wavefront at a given instant in time, serves as a source of spherical secondary waves. The amplitude of the optical field at any point beyond is the superposition of all these wavelets. A wavefront at t = 0 The new wavefront at t = t
Huygens-Fresnel principle Every point of a wavefront at a given instant in time, serves as a source of spherical secondary waves. The amplitude of the optical field at any point beyond is the superposition of all these wavelets.
Diffraction Every point of a wavefront at a given instant in time, serves as a source of spherical secondary waves. The amplitude of the optical field at any point beyond is the superposition of all these wavelets.
Diffraction Every point of a wavefront at a given instant in time, serves as a source of spherical secondary waves. The amplitude of the optical field at any point beyond is the superposition of all these wavelets. Incident wave Diffracted wave Screen
x Diffraction from a single slit Incident wave? in the Fraunhofer region (far-field region) x z z a source at x Path length difference Screen radiates a wavelet The superposition of all these wavelets: some constant Amplitude of diffracted wave
x Fourier transform and diffraction Incident wave? z aperture function Screen The amplitude of diffracted wave is proportional to the Fourier transform of the field distribution across the aperture ( = the aperture function). (we will prove it later)
... back to diffraction from a single slit x Incident wave? z Screen
Diffraction from a single slit (results) x Incident wave? z Screen (minima)
Diffraction from a single slit (results) (minima)
Diffraction from a circular aperture
Diffraction from a circular aperture first minimum diameter Airy rings
Resolution of imagining systems Rayleigh s criterion for the minimum resolvable angular separation
x Diffraction from a double slit substitution z Diffraction factor due to the diffraction by a single slit Interference factor due to the interference between two slits
Diffraction from a double slit diffraction by a single slit interference between two slits diffraction from a double slit Diffraction factor due to the diffraction by a single slit Interference factor due to the interference between two slits
Diffraction from a double slit interference fringes for a double slit system diffraction by a single slit Diffraction factor due to the diffraction by a single slit Interference factor due to the interference between two slits
Diffraction gratings (multiple slits) Path length difference (grating orders) (maxima)
Diffraction gratings (multiple slits) Path length difference Diffraction factor due to the diffraction by a single slit Interference factor due to the interference from N slits
Diffraction gratings (multiple slits) Diffraction factor due to the diffraction by a single slit Interference factor due to the interference from N slits
X-ray diffraction Incident x rays (Bragg s law)
Electron diffraction Incident electron beam Davisson, C. J., "Are Electrons Waves?," Franklin Institute Journal 205, 597 (1928) (Bragg s law)
Basic optics Geometrical optics and images Interference Diffraction Diffraction integral
Angular spectrum representation in homogeneous medium = +z arbitrary wave = superposition of plane waves
Angular spectrum representation (more details) Wave function: real complex for EM waves scalar approximation Plane wave Superposition of plane waves: For we choose + sign, i.e., we assume propagation in +z (possible reflections are neglected) (IFT) (FT)
Propagation of waves known? +z paraxial approximation
Propagation of waves paraxial approximation
(calculation of the integral)
Diffraction integral known? +z Fresnel-Kirchhoff diffraction formula Fraunhofer approximation: only for