Optics 010 · Ray worlds, boundaries, and natural optics

N-BK7 Prism Spectrometer Observatory

How do you set up a glass prism so that where each color lands tells you its wavelength? Turn the prism to minimum deviation for a known reference line, then leave it fixed. At minimum deviation, light of that wavelength passes through the prism symmetrically. Every other wavelength now lands at its own spot on the detector, set by how the glass's refractive index changes with wavelength, so position reads as wavelength.

On a goniometer, an N-BK7 prism whose shape changes with its apex angle becomes a calibrated spectrometer. White light crosses both fixed faces at once, a deviation scan finds the symmetric stationary point, and known atomic lines calibrate the detector before an unknown wavelength is measured.

Primary spectral solution

\mathcal S_1

$0.50$

Angular or detector result

\mathcal S_2

$50\%$

Geometry or inversion check

\varepsilon_{\mathrm{spec}}

$0.00\pi$

Model regime $\text{valid model regime}$

Physics tutorial

Audit the physics of N-BK7 Prism Spectrometer Observatory

BackgroundA prism spectrometer starts from material dispersion: the refractive index changes with wavelength. For SCHOTT N-BK7 glass, the index comes from the published Sellmeier relation $n^2(\lambda)=1+\sum_{j=1}^{3}\frac{B_j\lambda^2}{\lambda^2-C_j}$ . The Lab then applies Snell's law at the entrance and exit faces of one fixed triangular prism and finds where each emerging ray hits one fixed detector plane.

Why it mattersHow do you set up a glass prism so that where each color lands tells you its wavelength?

Start with the essentials

Focus question: How do you set up a glass prism so that where each color lands tells you its wavelength?
One-sentence intuition: Minimum deviation is a symmetric, stationary arrangement that holds for one wavelength at a time: $r_1=r_2=\frac{A}{2},\qquad i=e$ . Setting a reference line there fixes the prism orientation. Every other wavelength then passes through the prism at that same orientation; the prism is not quietly rotated for each color.

Core mathematical model

Three-term Sellmeier dispersion for N-BK7

n^2(\lambda)=1+\sum_{j=1}^{3}\frac{B_j\lambda^2}{\lambda^2-C_j}

Wavelength enters in micrometres, with the coefficients published for this glass. The angular spread of colors comes from the derivative of this same relation; no separate curve is fitted to space the colors in the picture.

Symmetric path at minimum deviation

n(\lambda)=\frac{\sin\!\left[(A+\delta_{\min})/2\right]}{\sin(A/2)},\qquad \delta_{\min}=2\arcsin\!\left[n(\lambda)\sin\!\left(\frac{A}{2}\right)\right]-A

At the stationary point, the ray inside the prism divides the apex angle equally. The drawn surface normals, the entrance, internal and exit segments, and the deviation scan all come from this one solution.

Where a ray meets the detector plane

t=\frac{(\mathbf c-\mathbf x_2)\cdot\hat{\mathbf n}_d}{\hat{\mathbf k}_{\mathrm{out}}\cdot\hat{\mathbf n}_d},\qquad x_d=\bigl(\mathbf x_2+t\hat{\mathbf k}_{\mathrm{out}}-\mathbf c\bigr)\cdot\hat{\mathbf s}_d

Once the reference line is set, the detector center, normal and tangent stay fixed. The measured position then follows from the exit point and the direction of the emerging ray. The map from wavelength to position is monotonic, so it can be inverted to recover the wavelength.

Common difficulties

Putting every color at its own minimum deviation

Typical misconceptionTo draw a white-light spectrum, compute the minimum deviation separately for each wavelength and send every colored ray out from the center of the prism.

Better mental modelThat amounts to rotating the prism once for every color, and it throws away both refractions. Here one reference line sets the orientation. All colors enter at the same point, travel to their own computed exit points, refract again and land on the same detector.

Run the experiment

01
Scene 1: White-light dispersion
In the white-light scene, sweep the apex angle from its smallest to its largest value. Check that the prism itself changes shape, then follow the violet and red rays through both fixed faces to the detector plane.
What to observe: A larger apex angle widens the prism base and increases both the deviation and the spacing of colors on the detector. The selected marker moves across the same sensor; the sensor does not move to meet it.
02
Scene 2: Minimum-deviation lock
In the minimum-deviation scene, change the wavelength. Compare the dashed face normals, the equal internal angles, the readout of equal incidence and emergence angles, and the marked bottom of the full deviation scan.
What to observe: At minimum deviation the two internal refraction angles agree to numerical precision, and the curve of deviation against incidence angle has a stationary bottom. Away from that orientation the angles are no longer equal, though Snell's law still holds at both faces.
03
Scene 3: Line-spectrum calibration
In the calibration scene, move the unknown line between the reference anchors and then past them. Read its detector coordinate and recovered wavelength together, and judge accuracy from the inversion residual, not from color alone.
What to observe: Shorter wavelengths see a larger index in N-BK7 and bend more. The two close yellow mercury lines stay close on the detector, while the red and blue hydrogen lines give a wider span for calibration.