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Spherometer
Brief description
A spherometer is an ultra-high precision mechanical instrument designed to measure the radius of curvature of spherical surfaces, such as optical lenses and curved astronomical mirrors, as well as the thickness of microscopic plates. Invented in 1810 by French optician Robert-Agathe Cauchoix, the spherometer became the cornerstone of scientific lens making, transforming optical workshop production from guesswork into an exact mathematical science.
Use / Function
- Radius of Curvature Measurement: Determining the exact spherical curvature of convex and concave optical lenses and mirrors ($R = \frac{a^2}{6h} + \frac{h}{2}$).
- Optical Lens Grinding Control: Verifying progress during manual grinding and polishing of telescope objectives and microscope lenses against calculated specifications.
- Micro-Thickness Gauging: Measuring the thickness of thin glass slips, mica sheets, and metallic foils down to fractions of a micron.
- Surface Flatness Inspection: Detecting minute warping, dishing, or irregularities on precision optical flats and surface plates.
Operating principle
- Three-Point Kinematic Base: The instrument rests on a tripod formed by three rigid legs of identical length tipped with hardened steel points, defining an equilateral triangle of side length $a$.
- Central Micrometer Screw: A finely threaded screw passes through the central vertical axis of the tripod frame equidistant from all three outer legs.
- Point of Contact Detection: Turning the central screw lowers its bottom point toward the curved or flat surface until it makes simultaneous contact, indicated by tactile friction or optical light reflection.
- Sagitta Determination: The vertical displacement $h$ (sagitta) of the center point relative to the plane of the three outer feet is measured using a vertical linear millimeter scale and a large circular graduated micrometer disc.
- Curvature Calculation: Geometric trigonometry yields the surface’s sphere radius via the formula $R = \frac{a^2}{6h} + \frac{h}{2}$, where $a$ is the measured distance between any two outer tripod tips.
How to create it
- Fabricate the Tripod Frame: Cast and machine an equilateral triangular or circular brass frame with three precisely drilled threaded or pressed radial sockets spaced 120 degrees apart.
- Turn Hardened Outer Legs: Machine three identical steel legs with sharp or finely spherical hardened tips and secure them into the frame equidistant from the geometric center.
- Cut the Central Micrometer Thread: Tap a precision thread (typically 0.5 mm or 1 mm pitch) through the central boss of the frame on a screw-cutting lathe.
- Machine the Micrometer Screw: Turn a matching central hardened steel screw with a tapered or spherical contact tip and a knurled turning knob at the top.
- Mount the Graduated Disc: Fit a large circular brass disc (10 to 15 cm in diameter) perpendicular to the screw head, divided into 100 or 500 equal angular graduations around its rim.
- Install Vertical Scale and Zero: Fix a vertical millimeter ruler parallel to the screw to track full rotations; zero the instrument on an optically polished flat glass plate before measuring curved lenses.
Materials needed
- Essential: Brass for the main tripod body and graduated circular disc; hardened Steel for the micrometer screw, contact points, and outer legs.
- Tools: Screw-cutting lathe, dividing engine or circular dividing head, precision tap and die sets, hardened flat reference plate.
- Substitutes: Optical ring spherometers replace the three pointed legs with a precisely machined hollow circular ring to prevent scratching delicate glass surfaces.
Variants and improvements
- Three-Legged Spherometer: The standard classic Cauchoix tripod instrument with fixed steel points.
- Ring Spherometer: Uses a continuous circular ring knife-edge instead of three discrete legs, providing greater stability on small lenses and eliminating point indentations.
- Optical or Electric Contact Spherometer: Uses a weak electric circuit or optical lever to detect the exact instant of central contact without applying physical deformation pressure.
- Inverted Spherometer: Designed for measuring concave telescope mirrors without placing heavy metal frames inside delicate optical recesses.
Limits and risks
- Scratching Optical Surfaces: Hardened steel contact points can scratch soft optical crown or flint glass if pressed down with excessive force.
- Point Wear & Flattening: Blunting or deformation of the contact tips introduces systematic offset errors into the calculated sagitta $h$.
- Thermal Expansion Drift: Significant temperature variations alter the tripod leg separation and the screw pitch, requiring constant recalibration on flat glass.