Q1.1 Find the wavelength of light used in this 2- slits interference. Note: Not drawn to scale. 3.16mm 0.666mm 0.000633mm 0.200mm Ans: λ=633nm (typical of red He-Ne laser)

Phasor review Will need phasors to understand diffraction gratings.

Diffraction grating and the sun (first gateway to QM) Joseph von Fraunhofer (1787-1826) QM means quantum mechanics Unexpected dark lines due to the absorption of elements in the sun’s atmosphere. In the 20th century it was realized that atomic energies are quantized. The spectrum is not continuous !

Crystallography by X-ray diffraction Protein x-ray crystallography setup at the SLAC LCLS (Linac Coherent Light Source) in Stanford, CA. First x-ray diffraction image of DNA by Rosalind Franklin (Fig 36.24 of the text) Other similar facilities in Hamburg Germany and Harima Science City, Japan

Goals for Chapter 36 (Diffraction) To see how a sharp edge or an aperture affect light To analyze single-slit diffraction and calculate the intensity of the light To investigate the effect on light of many closely spaced slits To learn how scientists use diffraction gratings (e.g. in astrophysics) To see what x-ray diffraction tells us about crystals, proteins… To learn how diffraction places limits on the resolution of a telescope Holograms !

Diffraction, Chapter 36 What is Diffraction: the process by which a beam of light or other system of waves is spread out as a result of passing through a narrow aperture or across an edge, typically accompanied by interference between the wave forms produced. A picture helps Diffraction of water waves

Diffraction from a single slit (Physical Optics) In the Figure below, the prediction of geometric optics in (a) does not occur. Instead, a diffraction pattern is produced, as in (b). The narrower the slit, the broader the diffraction pattern. Light is a wave (EM waves) and hence diffracts. Diffraction phenomena also occur for EM waves, sound, electrons (QM) Geometric optics Physical Optics

Diffraction What physical parameters control whether single slit diffraction occurs ? Ans: λ (wavelength), a (slit spacing). But this is not the full story. Ans: 2nd part, distance to the screen (near field, Fresnel diffraction) and (far field, Fraunhofer diffraction)

Diffraction According to geometric optics, a light source shining on an object in front of a screen should cast a sharp shadow. Surprisingly, this does not occur because of diffraction.

Diffraction and Huygen’s Principle Huygens’s principle can be used to analyze diffraction. Fresnel diffraction: Source, screen, and obstacle are close together. Fraunhofer diffraction: Source, screen, and obstacle are far apart. The figure below shows the diffraction pattern of a razor blade.

Diffraction and Huygen’s Principle What is Huygens’s principle ? The Huygens-Fresnel principle states that every point on a wavefront is a source of wavelets. These wavelets spread out in the forward direction, at the same speed as the source wave. The new wavefront is a line tangent to all of the wavelets. So what is interfering in single slit diffraction ?

Fresnel and Fraunhofer diffraction by a single slit The figure below shows Fresnel (near-field) and Frauenhofer (far-field) diffraction for a single slit.

Locating the dark fringes Review the single-slit diffraction discussion in the text. The figure below shows the geometry for single slit Fraunhofer diffraction. (Imagine the aperture is made of many tiny slits). What is the path difference between the two strips ? Ans: path difference =a/2 sin(θ) Dark fringe If path difference to P is λ/2, what do we see at Point P ?

Locating the dark fringes in single slit diffraction In general, at what angles do we find the dark fringes ? Ans: If path difference to P is mλ/2, we will find dark fringes Dark fringes

Locating the dark fringes in the single slit diffraction pattern

Examples of single-slit diffraction The figure on the left is a photograph of a Fraunhofer pattern of a single horizontal slit. What features are especially notable ? Example 36.1: You pass 633-nm light through a narrow slit and observe the diffraction pattern on a screen 6.0 m away. The distance at the screen between the center and the first minima on either side is 32 mm long. How wide is the slit?

Intensity in the single-slit pattern Follow the text’s discussion of the intensity in the single-slit pattern using the phasor diagrams below.

Quantitative Intensity in the single-slit pattern The angle b is the phase angle of the ray from the top of the slit, while the phase angle from the bottom of the slit is 0. The vectors lie along a circle whose center is at C, so Ep is a chord of the circle. The arc length E0 is subtended by this same angle b, so the radius of the circle is E0/b. From the diagram, Since We have (sinc function)

Quantitative Intensity in the single-slit diffraction pattern Intensity of a single slit diffraction pattern

Intensity maxima in a single-slit pattern The figure on the right shows the intensity versus angle in a single-slit diffraction pattern. The minima occur when β is a multiple of 2π, i.e. at The location of the maxima are found by taking the derivative of and setting it to zero. Surprisingly, these are not precisely where In fact, there are no maxima for m = 0 in this expression. The central maximum is wider than the others, and occurs at q = 0. Using these approximate values of β in the intensity, we find

Width of the single-slit pattern The width of the single-slit diffraction pattern depends on the ratio of the slit width a to the wavelength λ.