Optical instruments, path differences and refraction
| English | 中文 | Pinyin |
|---|---|---|
| resolving power/rɪˈzɒlvɪŋ ˈpaʊə/ | 分辨本领 | fēn biàn běn lǐng |
| optical path length | 光程 | guāng chéng |
A decision before an answer
- A gas cell in one arm of a Michelson interferometer is crossed twice. Counting only one passage halves the inferred refractive-index change.
- Your goal: Calculate telescope magnification and grating resolving power.
Read the relationship
- 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.
- Relate Michelson fringe counts to double-pass optical path changes.
A relaxed-view telescope has separation 72 cm and eyepiece focal length 12 cm. Angular magnification magnitude is:
Objective focal length is 72−12=60 cm; 60/12=5.
Use the defining rule
- 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.
- Trace surface normals before applying Snell and reflection laws.
A grating just resolves 400 nm and 402 nm. Approximate resolving power is:
R≈400/2=200; wavelength and separation use the same units.
Check the conditions
- 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.
- Trace surface normals before applying Snell and reflection laws.
A relaxed-view telescope has lens separation 80 cm and eyepiece focal length 16 cm. Its objective focal length is 64 cm and magnification magnitude is 4. Evacuating a 5 cm gas cell causes 80 fringes at wavelength 500 nm: n−1=80·500×10⁻⁹/(2·0.05)=0.0004. The initial gas index is 1.0004, not 1.0008.
Moving one Michelson mirror by 150 nm changes round-trip optical path by ____ nm.
The light crosses the changed distance outbound and returning.
Apply the task format
- 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.
- Trace surface normals before applying Snell and reflection laws.
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.
Which answer fits this case?
Calculate telescope magnification and grating resolving power
Snell’s-law angles should be measured from the surface itself.
They are measured from its normal; the surface angle is complementary.
Keep the distinctions
- resolving power 分辨本领 — Wavelength divided by the smallest resolvable wavelength separation.
- optical path length 光程 — Geometric path weighted by refractive index along the ray.
- Calculate telescope magnification and grating resolving power.
- Relate Michelson fringe counts to double-pass optical path changes.
- Trace surface normals before applying Snell and reflection laws.
Match each term with its precise meaning in this lesson.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.