Physics · Ch 9 — Optics

Dispersion of light and prisms

9.8

Dispersion of light and prisms

The colour we perceive in light is set by its frequency, and a material's refractive index also depends on that frequency (increasing with frequency -- so violet light refracts more than red). Consequently, for an obliquely-incident polychromatic beam, each constituent colour refracts through a slightly different angle and the colours physically separate -- this is ANGULAR DISPERSION. If a polychromatic beam strikes a plane-parallel transparent SLAB obliquely, the emergent colours stay parallel to each other and to the original beam direction; the (usually tiny) perpendicular separation between two emergent colours' directions is called LATERAL DISPERSION, only appreciable over a large slab thickness and a large angle of incidence. Worked illustration: white light at $60^\circ$ incidence on a 5 cm-thick glass slab, with $n_R=1.51$ and $n_V=1.53$ : Snell's law gives refraction angles $r_R\approx35^\circ$ and $r_V\approx32^\circ25'$ ; working through the geometry, the lateral shift of each colour from the original undeviated path is about 2.58 cm for both, so the lateral DISPERSION between them turns out to be too small to detect easily by eye -- showing why a simple slab is a poor tool for observing dispersion.

To obtain OBSERVABLE dispersion, a PRISM is used instead -- two refracting surfaces inclined at an angle to each other, with the third (uninvolved) surface called the BASE. Any cross-section of a prism perpendicular to its base is its PRINCIPAL SECTION, and all rays are usually considered within this plane. For a principal section $ABC$ with refracting angle $A$ (between faces $AB$ and $AC$ ) and a ray $PQ$ striking $AB$ at incidence $i$ , refracting to $r_1$ , crossing to $AC$ and striking it with internal angle $r_2$ , then finally emerging at angle $e$ : from quadrilateral $AQNR$ , $A+\angle QNR=180^\circ$ ; from triangle $QNR$ , $r_1+r_2+\angle QNR=180^\circ$ ; combining these two gives $A=r_1+r_2$ . Since the total deviation $\delta$ is the exterior angle of triangle $XQR$ , $\delta=(i-r_1)+(e-r_2)=i+e-(r_1+r_2)$ , so $i+e=A+\delta$ .

As the angle of incidence $i$ is increased from its minimum possible value (below which NO ray can emerge at all, since $r_2$ would then exceed the critical angle, causing total internal reflection at the second face), the deviation $\delta$ is found first to DECREASE to a minimum value $\delta_m$ , and then to increase again -- tracing out an asymmetric curve (steeper on the side before the minimum than after it). Except exactly at the minimum, every value of $\delta$ corresponds to two different possible angles of incidence, and, by the reversibility of light, these two values are simply interchangeable as $i$ and $e$ . Exactly AT $\delta=\delta_m$ , $i=e$ , so $r_1=r_2=r=A/2$ , and the internal ray runs parallel to the base $BC$ . Using $i=\dfrac{A+\delta_m}{2}$ in Snell's law then gives the PRISM FORMULA, $n=\dfrac{\sin\left(\frac{A+\delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}$ .

Worked illustration: a glass prism ( $n=1.5$ ), refracting angle $A=60^\circ$ . Grazing emergence needs $r_2=i_C$ , and $\sin i_C=1/n=2/3\Rightarrow i_C\approx41^\circ49'$ ; since $r_1+r_2=A=60^\circ$ , $r_1\approx18^\circ11'$ , giving the MINIMUM angle of incidence for any emergent ray at all, $i_{min}\approx27^\circ55'\approx28^\circ$ (with $e_{max}=90^\circ$ there; by reversibility, $i_{max}=90^\circ$ corresponds to $e_{min}=27^\circ55'$ ). For minimum deviation ( $i=e$ ), the prism formula gives $1.5=\dfrac{\sin(30^\circ+\delta_m/2)}{\sin30^\circ}$ , solving to $\delta_m\approx14^\circ20'$ , and $i=e=\dfrac{A+\delta_m}{2}\approx37^\circ10'$ .

For a THIN prism ( $A\lesssim10^\circ$ ) with small angles of incidence too, $\sin\theta\approx\theta$ throughout, giving $i\approx nr_1$ and $e\approx nr_2$ , so $\delta=i+e-A\approx n(r_1+r_2)-A=nA-A=A(n-1)$ -- a deviation that is INDEPENDENT of the angle of incidence, i.e. constant for a given thin prism. …

Figure 9.14Fig 9.14: Angular dispersion at a single refracting surface

What this figure shows. A single flat refracting surface (interface between two media) with a single incident ray of WHITE (polychromatic) light striking it obliquely from the rarer medium side. On refraction, the beam is shown SPLITTING into several separate coloured rays fanning out at slightly different angles of refraction below/beyond the surface, with the two extreme colours explicitly labelled -- violet (V), refracted at the largest angle from the normal (bent most), and red (R), refracted at the smallest angle (bent least) -- with the other spectral colours implied to lie between them; the small angular gap between the violet and red refracted rays is marked as the …

Figure 9.15Fig 9.15: Lateral dispersion due to a plane parallel slab

What this figure shows. A plane-parallel transparent glass slab of breadth (thickness) shown edge-on, with a single incident ray of white light striking its upper (longer) surface obliquely at angle of incidence i. Inside the slab the beam has split into red and violet component rays (refracting at slightly different angles r_R and r_V), which travel through the slab and, on reaching the lower parallel surface, refract back out into air, both emerging PARALLEL to each other and parallel to the original incident ray's direction (since the two surfaces are parallel), but now laterally displaced/offset from one another. Points labelled V, R, T and M mark where the red and violet emergent rays cross a line perpendicular to the original incident direction; the perpendicular distances VM (=L_V) and RT (=L_R) are marked as the lateral deviations of the violet and red rays respectively from the original undeviated direction, and the small perpendicular gap between the violet and …

Misc Ex.9.8Angular deviation and lateral dispersion of red and violet light through a glass slab

Worked out. A fine beam of white light strikes the longer face of a plane-parallel glass slab of breadth 5 cm at an angle of incidence of 60 degrees. With refractive indices for red and violet of 1.51 and 1.53 respectively, Snell's law gives the angles of refraction inside the slab as r_R about 34 degrees 28 minutes and r_V about 32 degrees 35 minutes (via sin(r)=sin(60)/n for each colour). The angular deviations within the slab (i-r for each colour) and, using the slab's breadth and these angles, the lateral shifts L_R and L_V of each colour from the original ray's straight-through path both work out to about 2.58 cm, so the lateral DISPERSION between red and violet (the small difference L_V-L_R) is shown to be too small to detect with the naked eye -- explaining why plane slabs are a poor way …

Figure 9.16Fig 9.16: A prism -- three plane refracting surfaces forming a triangle

What this figure shows. A triangular prism shown in cross-section (its principal section) as a triangle ABC. Two of the three faces, AB and AC, meeting at vertex A, are the REFRACTING surfaces (through which light actually enters and exits), inclined to each other at angle A -- labelled as the 'angle of prism' or 'refracting angle'. The third face, BC (opposite vertex A), is labelled as the BASE of the prism and is NOT involved in the refraction of the ray being traced. The figure establishes the basic vocabulary (angle of prism = A, base = BC) used throughout the rest of the …

Figure 9.17Figs 9.17(a) and (b): Refraction through a prism for monochromatic and white light

What this figure shows. Part (a), monochromatic light: a single-colour ray PQ strikes refracting face AB of a triangular prism ABC obliquely, refracts at Q (bending toward the normal MQN drawn at Q), travels through the glass as QR, strikes the second refracting face AC at R, refracts again at R (bending away from the normal MRN drawn at R) and emerges as ray RS into the air, with the net effect that the emergent ray RS is deviated from the original incident direction PQ (extended as a dashed line QT) by an angle delta, marked at their point of intersection X. Part (b), white (polychromatic) light: the same basic geometry (incident ray, two refractions at the two faces) but the beam entering at Q is shown splitting inside and after the prism into a small FAN of separately-coloured emergent rays (violet deviated most, red deviated least), with the angular spread between the extreme emergent colours marked as the angular dispersion produced by the whole prism, distingu …

Figure 9.18Fig 9.18: Deviation through a prism -- full angle geometry

What this figure shows. A detailed geometric diagram of monochromatic refraction through prism ABC (refracting angle A at vertex A, base BC). Ray PQ strikes face AB at Q with angle of incidence i, measured from the normal MQN drawn at Q; it refracts to angle r1 and travels as QR to the second face AC, striking it at R with angle of incidence r2 (measured from the normal MRN at R), refracting there to angle of emergence e and leaving as ray RS. The original incident ray PQ is extended forward as a dashed line QT; the emergent ray RS is extended backward as a dashed line; these two extended/backward lines meet at a point X, and the angle TXS between them, marked at X, is the total angle of deviation delta. The quadrilateral AQNR (vertex A, the two normal-feet Q and N-related points, and R) and the triangle formed at X are the …

Figure 9.19Fig 9.19: Deviation curve for a prism

What this figure shows. A graph with angle of incidence i on the horizontal axis and angle of deviation delta on the vertical axis, plotted as a smooth curve starting at the minimum angle of incidence i_min (where delta is relatively large, at the point the ray just begins to emerge), DECREASING as i increases, reaching a single minimum point (labelled delta_m, the angle of minimum deviation, occurring at some intermediate angle of incidence), and then INCREASING again as i increases further toward 90 degrees. The curve is explicitly NOT a symmetric parabola -- the portion of the curve after the minimum (for larger i) has a visibly SHALLOWER slope than the portion before the minimum, an asymmetry the text calls out explicitly. A horizontal dashed line at the height delta (some deviation value other than the minimum) is drawn crossing the curve at two distinct points, illustrating th …

Misc Ex.9.9Range of possible incidence angles and angle for minimum deviation, for a 60-degree glass prism

Worked out. For a glass prism (n=1.5) with refracting angle A=60 degrees, the example finds: the critical angle ic (sin ic = 1/n = 2/3) is about 41 degrees 49 minutes; for grazing emergence (e=90 degrees, i.e. r2=ic exactly) with A=r1+r2=60 gives r1 about 18 degrees 11 minutes, and via Snell's law at the first face this corresponds to the MINIMUM possible angle of incidence for any emergent ray at all, i_min about 27 degrees 55 minutes (with e_max=90 degrees there); by reversibility, i_max=90 degrees corresponds to e_min=27 degrees 55 minutes. For the special case i=e (minimum deviation), using the prism formula n=sin((A+delta_m)/2)/sin(A/2) with A=60, n=1.5 gives delta_m about 14 degrees 20 minutes, and correspondingly i=e=(A+delta_m)/2 ab …

Figure 9.20Fig 9.20: Angular dispersion through a prism for a polychromatic beam

What this figure shows. A triangular prism ABC with a single incident ray of WHITE light striking face AB. Inside and after the prism, the beam is shown splitting into a small fan of differently-coloured emergent rays leaving face AC at slightly different angles of emergence -- the two extreme rays, violet (deviated the MOST, i.e. bent farthest from the original incident direction) and red (deviated the LEAST), are explicitly labelled, with the angular gap between these two extreme emergent rays marked as the prism's angular dispersion for the full visible spectrum, and a middle ray (implicitly yellow, the conventional 'mean' colour) drawn roughly midwa …

Misc Ex.9.10Angular deviation and dispersive power of a dense flint glass prism

Worked out. A dense flint glass prism of refracting angle A=10 degrees has refractive indices nR=1.712 for red and nV=1.792 for violet. Using the thin-prism deviation formula delta=A(n-1) for each colour, delta_R=10x0.712=7.12 degrees and delta_V=10x0.792=7.92 degrees, so the angular dispersion delta_VR=delta_V-delta_R=0.8 degrees. The dispersive power omega=(delta_V-delta_R)/delta_Y (with delta_Y taken as the mean, (delta_V+delta_R)/2=7.52 degrees) works out to 2x0.8/15.04 which is about 0.1064 -- notably higher than the roughly 0.03 dispersive power of ordinary popular crown glass, consistent with dense flint glass being chosen when strong dispersion …

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