Double slit and diffraction grating
Light through two narrow slits spreads out and overlaps with itself. Where the waves arrive in step you see a bright fringe; where they cancel, a dark one.
Intensity on the screen
读数
发生了什么
Each slit acts as a new source of waves, and the two sets of waves are coherent because they come from the same light. At a point on the screen the waves have travelled different distances. If that path difference is a whole number of wavelengths, d sin θ = nλ, crests meet crests and the light is bright; halfway between, crests meet troughs and it is dark.
For small angles the bright fringes are equally spaced, w = λD/d, so longer wavelengths and a more distant screen spread them out, while wider-spaced slits squeeze them together. Add more slits and the bright lines stay in the same places but become much narrower and brighter: that is a diffraction grating, used to measure wavelengths precisely. Each slit also has a width a, and its own single-slit pattern acts as an envelope over the fringes; when d/a is a whole number some orders fall exactly on a dark point of the envelope and go missing.
- Diffraction gratingd sin θ = nλ
- Fringe spacingw = λD / d
- Highest ordernₘₐₓ = ⌊d / λ⌋
- Intensity patternI = I₀ sinc²(πa sin θ / λ) [sin(Nφ/2) / N sin(φ/2)]²
考试大纲
- A-Level Physics AQA, Edexcel, OCRYoung's double-slit experiment, w = λD/s or λD/d, path difference and the diffraction grating d sin θ = nλ
- First-year university opticsN-slit interference and the single-slit envelope
挑战
Predict first: green light (λ = 550 nm) passes through two slits 0.25 mm apart onto a screen 2.0 m away. How far apart are the bright fringes? Then switch to a grating with 600 lines per mm: at what angle is the first-order maximum?
w = λD/d = 550 × 10⁻⁹ × 2.0 / 0.25 × 10⁻³ = 4.4 mm. For the grating d = 1/600 mm = 1.667 μm, so sin θ = 550 × 10⁻⁹ / 1.667 × 10⁻⁶ = 0.33 and θ = 19.3°.
系统地学
GCSE Combined and Separate Science 中讲这个主题的课：先看分步讲解，再做考试风格的题目，每次数字都不同。 课程为英文。
常见问题
How do I find the wavelength from the fringe spacing?
Rearrange w = λD/d to λ = wd/D. Fringes 4.4 mm apart from slits 0.25 mm apart on a screen 2.0 m away give λ = 4.4 × 10⁻³ × 0.25 × 10⁻³ / 2.0 = 5.5 × 10⁻⁷ m, which is 550 nm (green). Pick Wavelength λ under Solve for to see the working.
What is the formula for fringe spacing?
For small angles the bright fringes are equally spaced by w = λD/d, where λ is the wavelength, D the slit to screen distance and d the slit separation. With λ = 550 nm, D = 2 m and d = 0.25 mm, w = 4.4 mm.
Why are diffraction grating maxima so sharp?
With N slits, light from all N must arrive in step to give a maximum, and a tiny change in angle already puts the slits out of step with each other. The width of each maximum shrinks roughly as 1/N while the peak intensity grows as N², so a grating with thousands of lines gives very sharp, bright lines.
How many orders can a grating produce?
sin θ cannot exceed 1, so the highest order is nₘₐₓ = d/λ rounded down. A 600 lines per mm grating has d = 1.667 μm; with 500 nm light that gives 3.33, so orders up to n = 3 appear on each side.
Why do some orders go missing?
Each slit has a width a, and its single-slit pattern is zero where a sin θ = λ, 2λ and so on. If d/a is a whole number, say 3, then the 3rd, 6th and later orders land exactly on those zeros and disappear.