Wave Optics flash cards

1. 1.State Huygens' principle.

   Every point on a wavefront acts as a source of secondary spherical wavelets that spread out in all directions with the speed of the wave. The new wavefront at a later time is the forward envelope (tangential surface) of all these secondary wavelets.

   Hint: Each point = new source of secondary wavelets.

2. 2.What is a wavefront?

   A wavefront is the locus of all points of a medium vibrating in the same phase. The direction of propagation (a ray) is always perpendicular to the wavefront.

   Hint: Surface of constant phase; ray $\perp$ wavefront.

3. 3.Name the shapes of wavefronts from (i) a point source, (ii) a line source, (iii) a source at infinity.

   (i) Point source $\to$ spherical wavefront. (ii) Line source $\to$ cylindrical wavefront. (iii) Source at infinity (parallel rays) $\to$ plane wavefront.

   Hint: Point, line, infinity.

4. 4.How does Huygens' principle explain the laws of reflection and refraction?

   By constructing secondary wavelets from a plane wavefront hitting a surface, geometry gives $i=r$ (reflection) and $\dfrac{\sin i}{\sin r}=\dfrac{v_1}{v_2}=n_{21}$ (Snell's law). Refraction arises because the wave speed changes in the second medium.

   Hint: Geometry of secondary wavelets on the interface.

5. 5.When light passes from a rarer to a denser medium, what happens to its speed, wavelength and frequency?

   Frequency stays the same (set by the source). Speed decreases ($v=c/n$) and wavelength decreases ($\lambda_{med}=\lambda_{vac}/n$).

   Hint: Frequency unchanged; $v$ and $\lambda$ drop by factor $n$.

6. 6.State the principle of superposition of waves.

   When two or more waves overlap at a point, the resultant displacement is the vector sum of the individual displacements: $y=y_1+y_2+\dots$. This is the basis of interference and diffraction.

   Hint: Resultant = sum of individual displacements.

7. 7.What is interference of light?

   The modification in the distribution of light energy due to the superposition of two coherent light waves. It produces alternating regions of maximum (bright) and minimum (dark) intensity, redistributing — not destroying — energy.

   Hint: Superposition of coherent waves; energy redistributed.

8. 8.Distinguish constructive and destructive interference in terms of phase and path difference.

   Constructive: waves in phase, $\Delta\phi=2n\pi$, path diff $=n\lambda$ $\to$ maximum intensity. Destructive: waves out of phase, $\Delta\phi=(2n+1)\pi$, path diff $=(2n+1)\lambda/2$ $\to$ minimum intensity.

   Hint: Even vs odd multiples of $\lambda/2$.

9. 9.What are coherent sources?

   Two sources that emit light waves of the same frequency with a constant (time-independent) phase difference. They are essential for a stable, observable interference pattern.

   Hint: Same frequency, constant phase relation.

10. 10.Why can't two independent light bulbs (or sodium lamps) produce interference?

    Light is emitted by billions of atoms in random, rapidly changing bursts, so the phase difference between two independent sources fluctuates randomly (~$10^{-9}$ s). The pattern shifts too fast to be seen — the sources are incoherent.

    Hint: Random, rapidly changing phase difference.

11. 11.How are two coherent sources obtained in practice?

    By deriving both from a single source: (i) division of wavefront (Young's double slit, Fresnel biprism); (ii) division of amplitude (thin films, Newton's rings, Michelson interferometer).

    Hint: Split one source — wavefront or amplitude division.

12. 12.Describe the setup of Young's double slit experiment (YDSE).

    Monochromatic light illuminates a single slit $S$, whose wavefront reaches two close slits $S_1$ and $S_2$ (coherent sources). Waves from $S_1,S_2$ overlap on a screen a distance $D$ away, producing bright and dark fringes.

    Hint: One slit feeds two slits; overlap on distant screen.

13. 13.Write the expression for path difference at a point on the screen in YDSE.

    $\Delta x=\dfrac{y\,d}{D}$, where $d$ = slit separation, $D$ = slit-to-screen distance, $y$ = distance of the point from the central axis (valid for $d\ll D$).

    Hint: $\Delta x=yd/D$.

14. 14.State the conditions for bright and dark fringes in YDSE (path difference form).

    Bright (maxima): $\Delta x=n\lambda$, $n=0,1,2,\dots$ Dark (minima): $\Delta x=(2n-1)\dfrac{\lambda}{2}$, $n=1,2,3,\dots$

    Hint: Bright: $n\lambda$; Dark: odd $\times\lambda/2$.

15. 15.Give the positions of the $n$-th bright and dark fringes on the screen.

    Bright: $y_n=\dfrac{n\lambda D}{d}$. Dark: $y_n=\dfrac{(2n-1)\lambda D}{2d}$.

    Hint: Multiply fringe order by $\lambda D/d$.

16. 16.Define fringe width and give its formula.

    Fringe width $\beta$ is the distance between two consecutive bright (or dark) fringes: $\beta=\dfrac{\lambda D}{d}$. It is the same for bright and dark fringes.

    Hint: $\beta=\lambda D/d$.

17. 17.How does fringe width depend on $\lambda$, $D$ and $d$?

    $\beta=\dfrac{\lambda D}{d}$: $\beta$ increases with wavelength $\lambda$ and screen distance $D$, and decreases as slit separation $d$ increases.

    Hint: $\beta\propto\lambda$, $\propto D$, $\propto 1/d$.

18. 18.In YDSE, what is the angular fringe width?

    Angular fringe width $\theta=\dfrac{\beta}{D}=\dfrac{\lambda}{d}$. Unlike linear width, it does not depend on the screen distance $D$.

    Hint: $\theta=\lambda/d$.

19. 19.Write the intensity distribution in YDSE for two equal sources.

    $I=I_0\cos^2\!\left(\dfrac{\phi}{2}\right)=4I_1\cos^2\!\left(\dfrac{\phi}{2}\right)$, where $I_1$ is the intensity of each slit and $\phi$ is the phase difference. Max $I=4I_1$, min $I=0$.

    Hint: $I\propto\cos^2(\phi/2)$.

20. 20.Relate phase difference $\phi$ to path difference $\Delta x$.

    $\phi=\dfrac{2\pi}{\lambda}\,\Delta x$. A path difference of one full wavelength corresponds to a phase difference of $2\pi$.

    Hint: $\phi=(2\pi/\lambda)\Delta x$.

21. 21.For two waves of intensities $I_1$ and $I_2$, write the resultant intensity in interference.

    $I=I_1+I_2+2\sqrt{I_1 I_2}\cos\phi$. Max: $I_1+I_2+2\sqrt{I_1I_2}=(\sqrt{I_1}+\sqrt{I_2})^2$; Min: $(\sqrt{I_1}-\sqrt{I_2})^2$.

    Hint: Cross term $2\sqrt{I_1I_2}\cos\phi$.

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