1 Chapter 10 Diffraction March 9, 11 Fraunhofer diffraction: The single slit 10.1 Preliminary considerations Diffraction: The deviation of light from propagation in a straight line. There is no essential physical distinction between interference and diffraction. Huygens-Fresnel Principle: Every unobstructed point of a wave front serves as a source of spherical wavelets. The amplitude of the optical field at any point beyond is the superposition of all these wavelets, taking into account their amplitudes and phases. Fraunhofer (far field) diffraction: Both the incoming and outgoing waves approach being planar. a 2 / R<<1  where R is the smaller of the two distances from the source to the aperture and from the aperture to the observation point. a is the size of the aperture. The diffraction pattern does not change when moving the observation plane further away. Fresnel (near field) diffraction: Plane of observation is close to the aperture. General case of diffraction. The diffraction pattern changes when the observation plane moves. S P a R1R1 R2R2

2 Coherent line source: y x P (x,y) dy' r -D/2 D/2  L is the source strength per unit length. This equation changes a diffraction problem into an integration (interference) problem. Mathematical criteria for Fraunhofer diffraction: The phase for the rays meeting at the observation point is a linear function of the aperture variables. S y' P y' sin  Waves from a point source: Harmonic spherical wave: A is the source strength.

Fraunhofer diffraction The single slit y x P (x,y) y' r -D/2 D/2 R  The slit is along the z-axis and has a width of D. In the amplitude, r is approximated by R. In the phase, r is approximated by R-y' sin  if D 2 /R  <<1. Fraunhofer diffraction condition  The overall phase is the same as a point source at the center of the slit. Integrate over z gives the same function.

4 y x P (x,y) y' r -D/2 D/2 R   I/I(0)= (©WIU OptoLab)

5 Phasor model of single slit Fraunhofer diffraction: rolling paper

6 Read: Ch10: 1-2 Homework: Ch10: 3,7,8,9 Due: March 27

The double slit z x P (x,z) R-a sin  R  a b The result is a rapidly varying double-slit interference pattern (cos 2  ) modulated by a slowly varying single-slit diffraction pattern (sin 2  /  2 ). March 23 Double slit and many slits

8 (©WIU OptoLab) Single-slit diffraction Two-slit interference Envelope Fringes Question: Which interference maximum coincides with the first diffraction minimum? “Half-fringe” (split fringe) may occur there. Our author counts a half-fringe as 0.5 fringe. half-fringe

Diffraction by many slits z x P (x,z) R-a sin  R  a b R-2a sin 

10  Principle maxima: Minima (totally N-1): Subsidiary maxima (totally N-2):

11 Phasor model of three-slit interference: rotating sticks

12 Read: Ch10: 2 Homework: Ch10: 10,11,12 Due: April 3

The rectangular aperture Coherent aperture: dS=dydz P(Y,Z) r R x y z Y Z March 25 Rectangular aperture Fraunhofer diffraction condition X

14 Rectangular aperture: dS=dydz P(Y,Z) r R x y z Y Z a b

15 Y minimum: Z minimum:

16 Read: Ch10: 2 Homework: Ch10: 14,17 Due: April 3

The circular aperture Importance in optical instrumentation: The image of a distant point source is not a point, but a diffraction pattern because of the limited size of the lenses. Bessel functions:. March 27 Circular aperture  P(Y,Z) R x y z Y Z  q   a

18 J0(u)J0(u) J1(u)J1(u) u q1q1 Radius of Airy disk: D P f

19 Read: Ch10: 2 Homework: Ch10: 25,28 Due: April 3

20 March 30 Resolution of imaging systems Equivalence between the far field and the focal plane diffraction pattern Two coherent point sources:  P a sin  R a y This applies to any number of arbitrarily distributed point sources in space. Far field and focal plane produce the same diffraction pattern, but with different sizes. R is replaced by f in the focal plane pattern.  A lens pulls a far-field diffraction pattern to its focal plane, reduces the size by f/R.  P' a sin  a  f y' L

Resolution of imaging systems Overlap of two incoherent point sources: Rayleigh’s criterion for bare resolution: The center of one Airy disk falls on the first minimum of the other Airy disk. We can actually do a little better. Angular limit of resolution: D P f Image size of a circular aperture: D P2P2 S1S1 S2S2 P1P1 far awayf Image size of a far point source: D P f

22 Question: Comparing the circular with the square aperture, why does the square aperture produce a smaller diffraction pattern? ( /D vs /D) Wavelength dependence: CD  DVD Angular limit of resolution: Our eyes: About 1/3000 rad Spot distance on the retina: 20 mm/3000=6.7  m Space between human photoreceptor cells on the retina: 5-7  m. Pixel size of a CCD camera: ~7.5  m. Pupil diameter Focal length Human cone photoreceptor cells 150  m

23 Read: Ch10: 2 No homework

24 Diffraction grating: An optical device with regularly spaced array of diffracting elements. Transmission gratings and reflection gratings. Grating equation: ii mm a ii mm a Blazed grating: Enhancing the energy of a certain order of diffraction. Blaze angle:  Specular reflection: April 1, 3 Gratings m= ii rr 00 a  specular 0th

25 Grating spectroscopy: Angular width of a spectral line due to instrumental broadening. Inversely proportional to Na. Angular dispersion: dmdm d Angular width for a spectral line: N-slit interference  Between two minima, (N-1)  /N to (N+1)  /N.

26 Limit of resolution: Resolving power: Question: Why does the resolving power increase with increasing order number and with increasing number of illuminated slits? Barely resolved two close wavelengths:  separation   width

27 Free spectral range: sin  m m =1 m =3 m =2 fsr  In higher order diffraction the spectrum is more spread in angle. This results in a higher resolving power but a narrower free spectral range.

28 Read: Ch10: 1-2 Homework: Ch10: 32,33,34,39,41 Due: April 10