Basic optics. Geometrical optics and images Interference Diffraction Diffraction integral. we use simple models that say a lot! more rigorous approach

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1 Basic optics Geometrical optics and images Interference Diffraction Diffraction integral we use simple models that say a lot! more rigorous approach

2 Basic optics Geometrical optics and images Interference Diffraction Diffraction integral

3 Images Édouard Manet: A Bar at the Folies-Bergère (1882)

4 Mirage light rays are bent to produce a displaced image of distant objects or the sky

5 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

6 Plane mirrors Extended objects virtual image the same orientation and size (height) as object

7 Head Eye Mirror Foot

8 Spherical mirrors concave Real focus Central axis Focal length Radius r, f > 0 convex Central axis Virtual focus r, f < 0

9 Image formation Spherical mirrors Axis Mirror

10 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

11 Spherical mirrors Ray tracing 4 rays Convex mirror: image is always - virtual - erect - minified - real image: i > 0 - virtual image: i < 0

12 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

13 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

14 Axis

15 Thin lens Axis Air p Glass i for both refracting surfaces (thick lens) 0 (thin lens)

16 Thin lens converging lens f > 0 < 0 > 0 diverging lens f < 0 Extensions < 0 > 0

17 Ray tracing: converging lens f > 0 3 rays

18 Ray tracing: diverging lens f < 0 3 rays

19 Thin lens (bottom line) converging lens diverging lens

20 Thin lens: magnification m

21 Simple magnifier To distant virtual image angular magnification:

22 Compound microscope Objective Eyepiece To distant virtual image Parallel rays The lateral magnification produced by the objective lens The overall magnification

23 Refracting telescope Eyepiece Objective Parallel rays from distant object To distant virtual image Parallel rays (angular magnification of the telescope)

24 Aberrations (image errors) examples - aberrations can be balanced - image fidelity is limited only by diffraction

25 Basic optics Geometrical optics and images Interference Diffraction Diffraction integral

26 Interference What will happen if we add waves?

27 Double-slit experiment (Young s experiment, 1801) Incident wave An interference pattern Superposition of waves u a suitable component of E- or H- vector

28 Different phases due to different paths Incident wave assume Path length difference (maxima) (minima)

29 Intensity in double-slit experiment Intensity (two coherent sources) (two incoherent sources) (one source) (m for maxima) (m for minima) (missing sign) (maxima) (minima)

30 Double-slit experiment with... particles waves

31 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

32 Example: air - n 2 - air Ray reflected at B Ray reflected at A (maxima) Incident wave (minima)

33 Example: air - n 2 - air (maxima) (minima)

34 Example: glass - air - glass Incident light Air Glass Glass (Newton rings)

35 Temporal coherence monochromatic wave - - perfectly coherent pulse (wave-packet) - - less coherent white light - - incoherent Define: coherence length coherence time

36 Interference and temporal coherence coherence length coherence time

37 Interference and temporal coherence

38 Spatial and temporal coherence

39 Michelson interferometer Movable mirror

40

41 Interference from thin films (again) reflected wave transmitted wave incident wave Now we consider 1. Arbitrary incident angle 2. Multiple beam interference

42

43

44 Interference from thin films reflected wave transmitted wave incident wave geometrical series The amplitude of the resultant transmitted wave

45 Interference from thin films reflected wave transmitted wave incident wave geometrical series The amplitude of the resultant reflected wave

46 Interference from thin films reflected wave transmitted wave incident wave For simplicity assume symmetric structure and pure real numbers (relative transmitted intensity)

47 Spectral response (thin film, FP etalon) integer (relative transmitted intensity)

48 Spectral response (thin film, FP etalon) The free spectral range, FSR (relative transmitted intensity)

49 with loss Spectral response (thin film, FP etalon)

50 Spectral analyzer const. 0

51 Basic optics Geometrical optics and images Interference Diffraction Diffraction integral

52 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

53 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.

54 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.

55 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

56 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

57 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)

58 ... back to diffraction from a single slit x Incident wave? z Screen

59 Diffraction from a single slit (results) x Incident wave? z Screen (minima)

60 Diffraction from a single slit (results) (minima)

61 Diffraction from a circular aperture

62 Diffraction from a circular aperture first minimum diameter Airy rings

63 Resolution of imagining systems Rayleigh s criterion for the minimum resolvable angular separation

64 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

65 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

66 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

67 Diffraction gratings (multiple slits) Path length difference (grating orders) (maxima)

68 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

69 Diffraction gratings (multiple slits) Diffraction factor due to the diffraction by a single slit Interference factor due to the interference from N slits

70 X-ray diffraction Incident x rays (Bragg s law)

71 Electron diffraction Incident electron beam Davisson, C. J., "Are Electrons Waves?," Franklin Institute Journal 205, 597 (1928) (Bragg s law)

72 Basic optics Geometrical optics and images Interference Diffraction Diffraction integral

73 Angular spectrum representation in homogeneous medium = +z arbitrary wave = superposition of plane waves

74 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)

75 Propagation of waves known? +z paraxial approximation

76 Propagation of waves paraxial approximation

77 (calculation of the integral)

78 Diffraction integral known? +z Fresnel-Kirchhoff diffraction formula Fraunhofer approximation: only for

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