WO.2 · Optical instruments, path differences and refraction

A gas cell in one arm of a Michelson interferometer 干涉仪 is crossed twice. Counting only one passage halves the inferred refractive-index change.

Prerequisites: 3, 33.

• Calculate telescope magnification and grating resolving power 分辨本领
• Relate Michelson fringe counts to double-pass optical path changes
• Trace surface normals before applying Snell and reflection laws

For an ideal astronomical telescope adjusted for relaxed viewing at infinity, the objective forms its focal-plane image and the eyepiece collimates the emerging light. Lens separation is f_objective+f_eyepiece; angular magnification magnitude is f_objective/f_eyepiece. The usual two-converging-lens telescope produces an inverted image, represented by a negative signed angular magnification under a consistent convention. A finite final-image distance changes lens separation; do not use the infinity adjustment without checking the condition.

Resolving power R=λ/Δλ describes the smallest distinguishable wavelength separation near λ. For an ideal diffraction grating in order m with N illuminated slits, R=mN. This is a resolution criterion, not simply the angular position formula d sinθ=mλ. Increasing illuminated slit count sharpens the principal peaks; increasing slit spacing mainly changes their angular locations. A measured λ=600 nm and Δλ=3 nm gives R=200, independent of converting both lengths to metres because their ratio uses matching units.

Michelson interference depends on the optical path difference between the arms. If one arm contains a gas cell of geometric length L and index changes from n to 1 upon evacuation, a double passage changes path by 2L(n−1). If N fringes pass, the magnitude of this change is Nλ, so n−1=Nλ/(2L). For a moving mirror instead, displacement Δx gives path change 2Δx. Fringes count phase cycles; one fringe is one wavelength of optical path, not one wavelength of mirror motion.

Snell’s law n1 sinθ1=n2 sinθ2 uses angles from the local surface normal. Reflection has equal incoming and outgoing angles from that same normal. In a rectangular block, normals of neighbouring faces are perpendicular: an incidence angle measured from one normal is complementary to the angle of that same ray from the other. Draw the ray and normals before substitution. For incidence from index n into air, total internal reflection occurs only when sinθ>1/n; equality is the critical grazing case. A stated partly refracted ray must satisfy the transmission condition.

A Michelson interferometer sends light twice through a gas cell of length L. Evacuation changes path by $2L(n-1)$. N fringes correspond to path change $N\lambda$.

Thus n = 1.00030. Moving a mirror is also double-pass; a fringe is not one wavelength of mirror motion.

Use optical path, not just geometric distance. Refraction angles are measured from the normal of the actual surface, and telescope focal-length addition assumes a final image at infinity.

resolving power: Wavelength divided by the smallest resolvable wavelength separation.

optical path length 光程: Geometric path weighted by refractive index along the ray.