Your Answer : (Not Answered)

Correct Answer : C

The best resolving power belongs to the Fabry-Perot etalon (spelt Fabry-Perrot in the paper).

The resolving power of a Fabry-Perot etalon is the order of interference times the effective number of interfering beams. With a plate separation of a few millimetres and high-reflectance coatings this reaches a million or more, enough to resolve hyperfine structure.

A laboratory grating spectrometer reaches tens of thousands (the order times the number of lines), and a prism spectrometer or a constant deviation spectrometer, which is also a prism instrument, only a few thousand, limited by the base of the prism.

Your Answer : (Not Answered)

Correct Answer : B

The wavelength is 5890 Å.

Moving the mirror by d changes the path by 2d, and each fringe corresponds to one wavelength of path change, so \(2d = N\lambda\):

\[ \begin{aligned} \lambda &= \frac{2d}{N} = \frac{2 \times 0.0589\ \text{mm}}{200} \\ &= 5.89 \times 10^{-4}\ \text{mm} \end{aligned} \]

This is \(5.89 \times 10^{-7}\) m, or 5890 Å, the sodium D line. The other values (5885, 5895, 6000) do not follow from the given shift and count.

Your Answer : (Not Answered)

Correct Answer : D

The highest resolving power belongs to the Fabry-Perot etalon.

Resolving power \( \lambda/\Delta\lambda \) grows with the number of interfering beams and the path difference between them. A Fabry-Perot etalon works in a very high order (the plate gap is many thousands of wavelengths) with effectively dozens of beams set by the reflectivity, giving resolving powers of order \( 10^{6} \) to \( 10^{7} \) in the visible.

A good grating reaches \( mN \sim 10^{5} \), and a prism, triangular or constant deviation, only about \( 10^{4} \), set by its base length and dispersion. That is why the etalon is used to study hyperfine structure.

Your Answer : (Not Answered)

Correct Answer : D

The resolving power is approximately \(10^{4}\).

The resolving power of a grating is \(R = nN\), the order times the total number of rulings illuminated. A grating of area 2 \(cm^{2}\) has a ruled width of the order of 1.4 to 2 cm, so with 600 lines per mm:

\[ \begin{aligned} N &\approx 14\ \text{mm} \times 600 \approx 8.5 \times 10^{3} \\ N &\approx 20\ \text{mm} \times 600 = 1.2 \times 10^{4} \end{aligned} \]

Either way, in the first order (n = 1) \(R = N \approx 10^{4}\). The options 10, \(10^{2}\) and \(10^{3}\) are one or more orders of magnitude too small; \(10^{3}\) would correspond to only about 2 mm of ruled width.

Your Answer : (Not Answered)

Correct Answer : B

The coherence length is 250 μm.

The coherence length is the distance over which the wave train stays in phase, \( l_{c} = \lambda^{2}/\Delta\lambda \):

\[ \begin{aligned} l_{c} &= \frac{(500\ \text{nm})^{2}}{1\ \text{nm}} = 2.5 \times 10^{5}\ \text{nm} \\ &= 250\ \mu\text{m} \end{aligned} \]

2·5 μm is off by a factor of 100, and 1·0 μm and 100 μm do not follow from the formula.

Your Answer : (Not Answered)

Correct Answer : A

The resultant intensity is 16I.

For coherent sources, amplitudes add, and intensity goes as amplitude squared. The amplitudes are proportional to \(\sqrt{I}\) and \(\sqrt{9I} = 3\sqrt{I}\). With zero phase difference they add directly:

\[ \begin{aligned} I_{R} &= \big(\sqrt{I} + \sqrt{9I}\big)^{2} \\ &= (1 + 3)^{2} I = 16I \end{aligned} \]

Adding intensities instead would give 10I, which is the incoherent result and not an option. The options 9I and I are the individual intensities, and zero is not possible here: even with a phase difference of \(\pi\) the minimum is \((3 - 1)^{2}I = 4I\).

Your Answer : (Not Answered)

Correct Answer : C

The maximum intensity is \((A_{1} + A_{2})^{2}\).

Intensity is proportional to the square of the resultant amplitude. With phase difference \(\delta\), the resultant amplitude A satisfies \(A^{2} = A_{1}^{2} + A_{2}^{2} + 2A_{1}A_{2}\cos\delta\). At a bright fringe \(\delta = 0\), so

\[ \begin{aligned} I_{\max} &\propto A_{1}^{2} + A_{2}^{2} + 2A_{1}A_{2} \\ &= (A_{1} + A_{2})^{2} \end{aligned} \]

\(A_{1} + A_{2}\) is the maximum amplitude, not the intensity. \(A_{1}^{2} + A_{2}^{2}\) is the average intensity, what incoherent sources would give, and its square root is the corresponding amplitude.