The Hong Kong Polytechnic University Optics 2----by Dr.H.Huang, Department of Applied Physics1 Diffraction Introduction: Diffraction is often distinguished from interference on that: in diffraction phenomena, the interfering beams originate from a continuous distribution of sources; in interference phenomena the interfering beams originate from a discrete number of sources. If both the source of light and observation screen are effectively far enough from the diffraction aperture so that the wavefronts arriving at the aperture and observation screen may be considered plane, it is called Fraunhofer, or far-field, diffraction. When the curvature of the wavefront must be taken into account, it is called Fresnel, or near-field, diffraction.

The Hong Kong Polytechnic University Optics 2----by Dr.H.Huang, Department of Applied Physics2 Diffraction Fraunhofer Diffraction at Single Apertures: Each interval contributes spherical wavelets at P of the form,

The Hong Kong Polytechnic University Optics 2----by Dr.H.Huang, Department of Applied Physics3 Diffraction Fringe Pattern: Dark fringe: The second, third and fourth maxima of the diffraction pattern occur at  =1.43 , 2.46  and 3.47 , respectively. The central maximum represents essentially the image of the slit on a distant screen. The angular width of the central maximum Is The linear width of the central maximum is

The Hong Kong Polytechnic University Optics 2----by Dr.H.Huang, Department of Applied Physics4 Diffraction Rectangular Slits: b a a

The Hong Kong Polytechnic University Optics 2----by Dr.H.Huang, Department of Applied Physics5 Diffraction Circular Slits: The far-field angular radius of Airy disc is, Airy Disc

The Hong Kong Polytechnic University Optics 2----by Dr.H.Huang, Department of Applied Physics6 Diffraction Rayleigh’s Criterion: For a microscope, The ratio D/f is the numerical aperture.

The Hong Kong Polytechnic University Optics 2----by Dr.H.Huang, Department of Applied Physics7 Diffraction Example: Two stars have an angular separation of  radian. Find the minimum diameter of the telescope objective which can just resolve the stars in light of 550 nm wavelength. Example: Calculate the minimum angular subtense of two points which can be just resolved by an eye with a 6 mm diameter pupil in light of 555 nm wavelength.

The Hong Kong Polytechnic University Optics 2----by Dr.H.Huang, Department of Applied Physics8 Diffraction Diffraction by Small Particles: Babinet ’ s Principle  1 and  2 are complementary apertures. Suppose that monochromatic plane wavefronts are incident normally on  1 and the diffracted light is imaged on a screen. In a direction  to the normal let the magnitude of the electric vector be E 1. Replace  1 with  2 and let the magnitude of the electric vector be E 2. Apparently, E 1 +E 2 =0 and E 1 =  E 2 Therefore, It means that the diffraction patterns for  1 and  2 are identical.

The Hong Kong Polytechnic University Optics 2----by Dr.H.Huang, Department of Applied Physics9 Diffraction Fraunhofer Diffraction at Two Slits: a=2b

The Hong Kong Polytechnic University Optics 2----by Dr.H.Huang, Department of Applied Physics10 Diffraction Fraunhofer Diffraction at Two Slits: a=6b

The Hong Kong Polytechnic University Optics 2----by Dr.H.Huang, Department of Applied Physics11 Diffraction Diffraction at Many Slits: N=8 and a=3b principal maxima at

The Hong Kong Polytechnic University Optics 2----by Dr.H.Huang, Department of Applied Physics12 Diffraction Diffraction Grating:

The Hong Kong Polytechnic University Optics 2----by Dr.H.Huang, Department of Applied Physics13 Diffraction Free Spectral Range: —the non-overlapping wavelength range in a particular order. The non-overlapping spectral region is smaller for higher orders. The wavelength are better separated as their order increases. This property is described by angular dispersion, or dispersive power of a grating, Linear dispersion is, Resolving power of a grating is defined as, Using the Rayleigh’s criterion, suppose the number of grating grooves is N, we have

The Hong Kong Polytechnic University Optics 2----by Dr.H.Huang, Department of Applied Physics14 Diffraction Example: A grating has 4000 grooves or lines per centimeter. Calculate the dispersive power in the second order spectrum in the visible range. We take the mean wavelength to be 550 nm. Example: Find the number of lines (grooves) required on a grating to just resolve the two sodium lines, 1 = nm and 2 = nm, in the second order spectrum of a grating.

The Hong Kong Polytechnic University Optics 2----by Dr.H.Huang, Department of Applied Physics15 Diffraction Fresnel Diffraction:

The Hong Kong Polytechnic University Optics 2----by Dr.H.Huang, Department of Applied Physics16 Diffraction Two conclusions (1) If N is small, there is large changes in the resultant phasor A N as the contribution from each new zone is added. The resultant amplitude seems to oscillate between magnitudes that are larger and smaller than the limiting value of a 1 /2. As the aperture gradually increases, one can see oscillations between bright and dark in a fixed position of the screen. (2) If N is large, as in the case of unlimited aperture, the resultant amplitude is half that of the first contribution zone, a 1 /2.

The Hong Kong Polytechnic University Optics 2----by Dr.H.Huang, Department of Applied Physics17 Diffraction Fresnel zone plate: very other Fresnel zone is blocked The zone plate radii are approximately given by,

The Hong Kong Polytechnic University Optics 2----by Dr.H.Huang, Department of Applied Physics18 Diffraction Diffraction by Straight Edges : Use cylindrical waves.

The Hong Kong Polytechnic University Optics 2----by Dr.H.Huang, Department of Applied Physics19 Diffraction Example: Plane wave of =550 nm are incident normally on a circular aperture of radius mm. Does a bright or a dark spot appear at the point P on the axis 4 m from the hole? If the intensity of the incident light is I 0, calculate the intensity at P. Example: A 4 mm diameter circular hole in an opaque screen is illuminated by plane waves of wavelength 500 nm. If the angle of incidence is zero, find the positions of the first two intensity maxima and the first intensity minimum along the central axis. The first two maxima will occur when N=1 and 3, respectively. The first minimum occurs when N=2.

The Hong Kong Polytechnic University Optics 2----by Dr.H.Huang, Department of Applied Physics20 Diffraction Example: Plane waves of 550 nm wavelength are incident normally on a narrow slit of width 0.25 mm. Calculate the distance between the first minima on either side of the central maximum when the Fraunhofer diffraction pattern is imaged by a lens of focal length 60 cm. Example: Plane waves ( =550 nm) fall normally on a slit 0.25 mm wide. The separation of the fourth order minima of the Fraunhofer diffraction pattern in the focal plane of the lens is 1.25 mm. Calculate the focal length of the lens. Example: Light from a distant point source enters a converging lens of focal length 22.5 cm. How large must the lens be if the Airy disc is to be m in diameter? =450 nm

The Hong Kong Polytechnic University Optics 2----by Dr.H.Huang, Department of Applied Physics21 Diffraction Example: A telescope objective is 12 cm in diameter and has a focal length of 150 cm. Light of mean wavelength 550 nm from a star is imaged by the objective. Calculate the size of the Airy disc. Example: Assuming Rayleigh’s criterion can be applied to the eye, how far apart must two small lights be in order to be just resolved at a distance of 1000 m? Take the pupil diameter as 2.5 mm, the wavelength to be 555 nm, and the eye’s refractive index Assume a single surface model eye with the pupil at the surface.