Diffraction Grating Explorer

Grating equation

mλ = d(sin θ + sin φ), with groove spacing d = 1/N. Both angles are measured from the grating normal, and φ is positive on the same side as the incident beam, so the 0th order (the mirror reflection) leaves at φ = −θ. An order exists only when |mλ/d − sin θ| ≤ 1.

Figure

• Styled after the chart “Light Diffraction from Grating” in Diffraction Grating Light Angles.xlsm: white incident beam and 0th order, thick, medium and thin lines for the 1st, 2nd and 3rd orders, and dashed lines for negative orders.
• Each laser’s beams end in its own marker shape (circle, square or triangle), so the lasers can be told apart without colour. Every angle in the figure is also listed in the Diffraction Angles table.
• Angles are drawn to true scale. The Excel chart’s axes stretch it sideways, which makes small angles look larger.
• Beam colors are approximate screen colors for each wavelength, brightened where needed so every beam stands out from the black background. Ultraviolet and infrared lines are drawn gray.

White light

Each order is drawn as a continuous 380–780 nm spectrum that dims toward both ends. Where orders overlap their colors add: 1st-order 760–780 nm leaves at the same angles as 2nd-order 380–390 nm, because mλ is the same.

Exit slit

• An ideal, zero-width slit at angle φ receives λ = d(sin θ + sin φ)/m from every order at once.
• A slit of width w at distance L, perpendicular to the diffracted beam, spans Δφ = 2·tan⁻¹(w/2L). Each order then passes a band of width Δλ = d[sin(φ + Δφ/2) − sin(φ − Δφ/2)]/m ≈ d·cos φ·Δφ/m.
• Resolution R = λ/Δλ = (sin θ + sin φ)/[sin(φ + Δφ/2) − sin(φ − Δφ/2)]. Because λ and Δλ both scale as 1/m, R at a fixed slit angle is the same in every order. A narrower slit, a longer distance to the slit, or a larger diffraction angle raises R.
• The incident beam is treated as a collimated laser, so no entrance-slit image adds to the bandpass.

Monochromator mode

• The entrance and exit slits stay fixed, an included angle 2K apart, and the grating turns on its mount by ψ from the instrument axis that bisects them. Then θ = ψ + K and φ = ψ − K.
• Switching the mode on keeps the incident angle, the slit angle and the grating’s orientation you already have: 2K = θ − φ and ψ = (θ + φ)/2, and the grating stays flat. The Grating rotation slider then turns it up to 45° either way from there, adding the same angle to θ and φ, as long as both beams stay in front of the grating. Turning past the incident beam makes θ negative, with the beam arriving from the right of the normal. 2K is negative when the exit slit sits on the incident side, beyond the entrance; the layout is then the mirror image of the usual one.
• Substituting into the grating equation gives mλ = 2d·cos K·sin ψ, the sine drive that tunes a scanning monochromator. The wavelength reaching the exit slit is proportional to sin ψ.
• At ψ = 0 the 0th order, the mirror reflection, goes out the exit slit. K = 0 is the Littrow mount, where the diffracted light returns along the incident path.
• In the figure the slits stay put and the grating, its normal and its angle scale turn together. The dashed instrument axis and the ψ arc show the angle the sine drive uses. The Diffraction Angles table still lists angles from the grating normal.

Typing values

Type a value in a box and press Enter, or move to another control, to apply it. Escape puts the box back as it was. A value outside the allowed range is set to the nearest limit, and a message under the box says so.

Presentation mode

Present, Alt + Shift + P, or ?present=1 in the address enlarges text, controls and the figure’s lines and labels for a projector and raises their contrast. It hides the helper text, this section and the footer, and changes nothing else. Escape exits.