Interference of Light Intensity of double-slit pattern Three slits Questions: what is the intensity of the double-slit pattern at an arbitrary position? What about three slits?

Young’s Double Slit Experiment Screen at infinity θ d Δx Path difference Δx = d sin θ Maximum intensity at d sinθ = m λ, m = 0, 1, 2, … Minimum intensity at d sinθ = (m +½ ) λ, m = 0, 1, 2, …

Find the intensity of the double-slit interference pattern as a function of position on the screen. Two waves: E1 = E0 sin(w t) E2 = E0 sin(w t + f ) where:

E1 + E2 = E0 [sin(w t) + sin(w t + f )] Trigonometry: sin a + sin b = 2 cos [(a-b)/2] sin [(a+b)/2] E1 + E2 = E0 [sin(w t) + sin(w t + f )] = 2 E0 cos (f / 2) sin(w t + f / 2)] Resultant amplitude ER Resultant intensity, (a function of angle q to the screen)

I position fringes are wide, with fuzzy edges equally spaced (in sin θ or small θ) equal intensity

Quiz With only one slit open, the intensity in the centre of the screen is 100 W/m 2. With both (identical) slits open together, the intensity at locations of constructive interference will be zero 100 W/m2 200 W/m2 400 W/m2

Example Two narrow parallel slits are separated by 0.85mm and are illuminated by light of a 600nm wavelength. What is the phase difference between the two interfering waves on a screen 2.8m away at a point 2.5mm from the central bright fringe? What is the ratio of the intensity at this point to the intensity at the center of a bright fringe?

Three Equally-Spaced Slits θ is the phase difference between consecutive slits d Δx = dsinθ d Total,

Three Slits Pattern The total displacement is where  = 2 (d sin )/ When d sin = m  and , ER = 3E0 (maximum value); so the bright fringes are in the same locations as with two slits. There are minima (zero intensity) at  = 120° and  = 240° (where d sin = /3, 2/3 ); to see this, add the first and third term above). At  = 180° (where d sin = /2) there is a small maximum: ER = E0

I 9I0 3 slits 2 slits 4I0 I0 f = In wavelengths Δx =λ /3 1/3 1/2 2/3 1 2π/3 rad Δx =λ /3 In wavelengths

Differences between 2-slit and 3-slit patterns: 3 Slits: Main peaks become narrower & brighter One “extra” faint secondary peak in between Main peaks ( = m2π) are in the same places As the number of slits becomes large, the main interference fringes become even narrower and brighter; the additional numerous faint peaks become less and less noticeable in comparison.

Many Slits (“diffraction grating”) -1 1 m=(d sin )/

Quiz Suppose the multiple slits from the previous slide were illuminated with white light, instead of red. What would the pattern on the screen look like? The three bright lines would be white, with a little colour at the edges, but otherwise the same Each of the three lines would spread into a wide coloured spectrum Two of the three lines would spread into a wide coloured spectrum, with the centre one narrow and white