Wave Optics Set-5

Q1. In Young's double-slit experiment two slits separated by $d=0.25\ \text{mm}$ are illuminated by monochromatic light of wavelength $\lambda=600\ \text{nm}$. The screen is at distance $D=2.0\ \text{m}$. The fringe width $\beta$ on the screen (distance between adjacent bright fringes) is:

Q2. In a Young's double-slit setup the slit separation is $d=0.50\ \text{mm}$ and the screen distance is $D=1.20\ \text{m}$. Two wavelengths $\lambda_1=600\ \text{nm}$ and $\lambda_2=605\ \text{nm}$ produce overlapping fringe systems whose central maxima coincide at the centre. Find the smallest positive distance $y$ from the central maximum at which bright fringes of the two wavelengths coincide again.

Q3. Two coherent waves from slits have intensities in the ratio $I_1:I_2=1:4$. If $I_{\max}$ and $I_{\min}$ are the maximum and minimum intensities of the resulting interference pattern, the visibility $V=\dfrac{I_{\max}-I_{\min}}{I_{\max}+I_{\min}}$ equals:

Q4. In a Michelson interferometer illuminated by light of wavelength $\lambda=600\ \text{nm}$, one mirror is mounted on a metal rod of length $L=5.00\ \text{cm}$ with coefficient of linear expansion $\alpha=1.2\times10^{-5}\ \text{K}^{-1}$. The other mirror is fixed. If the rod's temperature is uniformly increased by $\Delta T=10.0\ \text{K}$, how many fringes $m$ will move past a fixed point on the detector? (Assume mirror displacement equals rod expansion along the optical axis.)

Q5. A thin coating of refractive index $\mu=1.30$ is deposited on a glass substrate of refractive index $\mu_s=1.50$. Monochromatic light of wavelength $\lambda=600\ \text{nm}$ in air is normally incident. Taking into account phase reversals upon reflection, the minimum non-zero thickness $t_{\min}$ of the coating that produces zero reflected intensity is:

Q6. Two slits separated by $d=0.20\ \text{mm}$ are illuminated by light of wavelength $\lambda=600\ \text{nm}$. A screen is at distance $D=1.5\ \text{m}$. The fringe width is given by $\beta=\dfrac{\lambda D}{d}$. What is the value of $\beta$?

Q7. In a Young's double-slit experiment with wavelength $\lambda=600\ \text{nm}$, a thin glass plate of thickness $t=4.8\ \mu\text{m}$ and refractive index $n=1.5$ is placed in front of one slit. By how many fringes does the interference pattern shift? (Use $N=\dfrac{(n-1)t}{\lambda}$.)

Q8. In a double-slit experiment the slit separation is $d=0.60\ \text{mm}$ and slit width is $a=0.20\ \text{mm}$. Monochromatic light $\lambda=600\ \text{nm}$ falls on the slits and the interference pattern is observed on a screen at distance $D=1.5\ \text{m}$. Some interference maxima are missing because they coincide with diffraction minima. What is the distance between two successive missing interference maxima on the screen? (Use $\beta=\dfrac{\lambda D}{d}$ and note missing orders occur when $m/p=d/a$.)

Q9. A Michelson interferometer is illuminated by a source with central wavelength $\lambda=600\ \text{nm}$ and spectral width $\Delta\lambda=0.20\ \text{nm}$. The coherence length may be estimated by $l_c\approx\dfrac{\lambda^2}{\Delta\lambda}$. If one mirror is displaced by $\Delta x$, the optical path difference changes by $2\Delta x$. What is the maximum mirror displacement $\Delta x_{\max}$ for which fringes can still be observed?

Q10. A plane diffraction grating is used in the third order ($m=3$) to just resolve the sodium doublet at $589.0\ \text{nm}$ and $589.6\ \text{nm}$. What is the minimum number of illuminated grating lines $N$ required? (Use $R=\dfrac{\lambda}{\Delta\lambda}=mN$, with $\Delta\lambda=0.6\ \text{nm}$.)

Q11. In Young's double-slit experiment the two slits are separated by $d=0.5\ \text{mm}$ and the screen is at a distance $D=2.0\ \text{m}$. Monochromatic light of wavelength $\lambda=600\ \text{nm}$ is used. What is the fringe spacing $\beta$ on the screen?

Q12. Light of wavelength $\lambda=600\ \text{nm}$ in air is normally incident on a thin film of refractive index $n=1.4$ deposited on a glass substrate of refractive index $n_s=1.6$. The film thickness is small compared to the coherence length. What is the smallest non‑zero thickness $t$ of the film for which the reflected light shows constructive interference?

Q13. In a Michelson interferometer using light of wavelength $\lambda=600\ \text{nm}$, a thin glass plate of thickness $t=0.12\ \text{mm}$ and refractive index $n=1.5$ is inserted normally into one arm (the beam passes through the plate twice). To restore the central fringe to its original position the mirror in that arm must be moved by a distance $\Delta x$. What is $\Delta x$?

Q14. A diffraction grating has $300\ \text{lines/mm}$ and is illuminated over a width $1.5\ \text{mm}$. It is used in the $m=2$ order to attempt to resolve two spectral lines near $\lambda=600\ \text{nm}$ separated by $\Delta\lambda=0.1\ \text{nm}$. Can the grating resolve the two lines? If not, what is the minimum illuminated width required to resolve them in the same order?

Q15. In Young's double-slit experiment using a source with central wavelength $\lambda$ and spectral width $\Delta\lambda\ll\lambda$, the fringe contrast is lost beyond a certain order $m_{\rm max}$ (measured from the central maximum). Which expression gives the approximate maximum order $m_{\rm max}$ of well‑contrasted fringes?