Glass slab refractive index theory
Understand glass slab refractive index: the equations, assumptions, supporting concepts and measurement method.
01Refraction and the raised image
Rays travelling from glass into air bend away from the normal. An observer traces the emerging rays backwards and locates a virtual image above the actual mark. The mark itself has not moved. A microscope can focus on this image even though a screen placed at its apparent location would not collect a real image there.
Does the glass physically lift the mark?
No. Refraction changes the apparent origin of the emerging rays; the object stays at the base.
02Deriving the depth ratio
Let t be real depth below a flat surface and a be apparent depth measured from that same surface. For a narrow ray bundle near the normal, ray geometry and Snell's law give a approximately equal to t divided by the glass-to-air index. This approximation becomes less reliable for strongly oblique viewing. Both depths must share the same upper reference plane.
Why must the microscope view nearly normally?
The simple depth ratio uses paraxial rays. At larger angles a single constant apparent depth is no longer an adequate description.
03Three focus positions, two depth differences
Use an upward-increasing height scale. Define R1 as the focus height of the mark with no slab, R2 as the upper surface focus height and R3 as the apparent mark focus height through glass. Then real depth is R2 minus R1 and apparent depth is R2 minus R3. The apparent rise R3 minus R1 is a different quantity. Labels vary between apparatus instructions, so identify the physical focus positions before subtracting.
If R1 = 0.00 mm, R2 = 9.00 mm and R3 = 3.00 mm, what is apparent depth?
It is 9.00 minus 3.00 = 6.00 mm. The 3.00 mm rise is not the apparent depth.
04From depth readings to a fitted index
Plot apparent depth a vertically against real depth t horizontally. A single material gives a straight line with slope s = 1/n; the fitted estimate is n = 1/s. Fit an intercept rather than forcing zero: a nonzero value can reveal a persistent focus or datum error. A ratio from one observation is a useful calculation check, but the graph combines the range of thicknesses.
What does the slope mean when both axes use millimetres?
It is a dimensionless depth ratio, equal to 1/n. The slope is unchanged if both axes are converted to metres.
05Resolution and shared errors
The simulator rounds apparent depth to 0.01 mm. That is display resolution, not a complete physical uncertainty. In a real measurement, focus judgement, backlash and scale calibration matter. Since R2 is used in both depth differences, their errors can be correlated. A basic least-squares fit treats real depth as exact; report this limitation rather than interpreting tiny residuals as proof of high precision.
Are errors in real and apparent depth necessarily independent?
No. Both contain the same upper-surface reading, so its error influences both derived quantities.
06Material identity and model limits
Refractive index depends on material and wavelength. Different thicknesses should therefore be specimens of the same glass under the same illumination. Combining unlike glasses does not test one common straight line. The reference 1.500 is a model choice, not a universal value for all glass. Air is treated as index one and the simulation ignores surface roughness, dispersion and optical aberrations.
Can changing only thickness change the model index?
No. Thickness changes both depths proportionally; material and wavelength remain fixed.
The reference glass-to-air index is n = 1.500. Real thickness t ranges from 3 to 15 mm. The generator uses a = t/n, rounded to 0.01 mm. Under the virtual height datum R1 = 0 and R2 = t, the apparent focus height would be R3 = t − a. These are illustrative heights, not three independently measured scale readings. The fitted x and y are real and apparent depth in metres; n is the reciprocal of the fitted gradient.