Optical Bench in 3D: Refraction, Thin Lens and Slit Diffraction

What is modelled

Three optics experiments on one bench, each with its own station. This is the 3D companion of the 2D page "Light Bench: Refraction, Thin Lens and Diffraction"; both use the same closed-form optics and the same four checks.

Equations

• Refraction (station A). Snell's law n1 sinθ1 = n2 sinθ2. The reflected power is the mean of the Fresnel s and p terms, Rs = ((n1cosθ1 − n2cosθ2)/(n1cosθ1 + n2cosθ2))², Rp = ((n2cosθ1 − n1cosθ2)/(n2cosθ1 + n1cosθ2))². Critical angle asin(n2/n1) for n₁ > n₂; Brewster angle atan(n2/n1). Beam brightness shows the power split.
• Thin lens (station B). 1/f = 1/do + 1/di, m = −di/do. Each of the three drawn rays is bent at the lens by the paraxial rule u' = u − y/f; a virtual image (d_i < 0) is shown as a blue ghost with dashed back-extensions.
• Diffraction (station C). Fraunhofer far field I = I0 sinc²(π a sinθ/λ), for two slits times cos²(π d sinθ/λ). The pattern on the screen 1.000 m behind the slit plate is painted from this formula; first minimum at sinθ = λ/a, fringe spacing λL/d.

Checks run in the page

• Snell's law by bisection on the angle against the arcsine closed form.
• Energy: R + T = 1, with T built from the amplitude transmission coefficients.
• Lens: two numerically traced rays cross where the lens equation predicts.
• Diffraction: a brute-force sum of phasors over the aperture against the closed form.
• The Blender bench geometry: slit plate to screen distance equals the 1 m used in the formulas, and the object base sits on the lens axis.

Honest simplifications

• Scalar waves, no dispersion (the index does not depend on colour), no absorption, paraxial thin lens with no aberrations; the lens model in the picture only changes its thickness with 1/f.
• The beams in station A stop at the tank wall; the real exit refraction at the glass wall is not drawn. The two liquids and the glass walls are only tinted by their index.
• The slits are micrometres wide, so the glowing slit marks on the plate are drawn 200 times too large; the pattern itself is true to scale at L = 1 m.
• Fresnel reflectance is for unpolarised light and plane interfaces. The colour of a wavelength is an approximate display mapping.

What to try

• Station A: raise the angle beyond the critical angle in "Glass to air" and watch the refracted beam vanish; at Brewster's angle p-polarised reflection is zero, so R is at a minimum, not zero for unpolarised light.
• Station B: move the object inside the focal length to see the virtual, magnified image.
• Station C: shrink the slit width and the central maximum widens; add the second slit to see the fringes.