2. The laser

1. How much surface roughness must a target screen have in order to see a speckle pattern with a HeNe laser? (Assume that the optical axis of laser beam and visual field of view are normal at he target surface) is the more than one solution for the surface roughness?
   Answer:
   Destructive interference: $\Delta x=n\cdot \frac{\lambda }{4}\approx n\cdot 158\mathrm{n}\mathrm{m}$ with odd values of n.
2. There are many longitudinal oscillation modes in a resonator having a length $\ell$ , with wavelengths $\lambda =\frac{2\cdot \ell }{n}$ (where $n$ is an integer number).
   We consider a HeNe laser having 30 cm resonator length. Please calculate the frequency difference of of two neighbouring oscillation modes at the HeNe laser emission wavelength.
   We assume a natural linewidth (the ‘uncertainty’ of the laser light frequency) of 2 GHz. What is the consequence for the emitted frequency of the laser? Which role plays the length of the resonator?
   Antwort:
   $\Delta \nu ={\nu }_2-{\nu }_1=\frac{c}{{\lambda }_2}-\frac{c}{{\lambda }_1}=\frac{c\cdot n_2}{2\cdot l}-\frac{c\cdot n_1}{2\cdot l}=\frac{c\cdot \left(n_1+1\right)}{2\cdot l}-\frac{c\cdot n_1}{2\cdot l}=\frac{c}{2\cdot l}$
   Because of the natural linewidth several modes can be excited, if their frequencies are within the linewidth. Shortening of the resonator increases the frequency distance between modes and can result in a selection of single mode operation.
3. Snell‘s refraction law $\frac{sin\alpha }{sin\beta }=\frac{n_2}{n_1}$ is well-known. The incidence angle $\alpha$ becomes the Brewster angle ${\alpha }_B$ if the angle between refracted and reflected light beams is 90°. Please derive an equation which describes the dependence of the Brewster angle on the refractive index!
   Answer:
   Considering the angles, one finds: $\beta =90°-\alpha$.
   It follows: $\frac{sin{\alpha }_B}{sin\left(90°-{\alpha }_B\right)}=\frac{sin{\alpha }_B}{cos{\alpha }_B}=tan{\alpha }_B=\frac{n_2}{n_1}$
4. We consider a CW laser having 30 cm resonator length. Assuming that 10% of the light falling on the partially reflecting resonator mirror are transmitted, which distance must a photon propagate on average in order to initiate an induced emission?
   Answer:
   Photon nunbers are in a balance with a CW laser having a constant output power. Every photon which leaves the laser resonator through the partially transmitting mirror must be re-generated by a photon source (indiced emission). A 10% loss of laser light means that a photon will traverse the resonator 10 times forth and back before leaving it, whereby it generates one more photon through induced emission by reasons of photon balance. Hence, it follows a 6 m average distance of propagation.
5. Check out the laser applet at the end of the chapter. Please describe what the applet shows in physically correct terms.
   Answer:
   Keywords: Pump light pPump energy), atoms, ground state and excited state, photons, waves, resonator, resonance, mirror (partially transmittant), laser light, absorption, spontaneous and induced emission, population inversion.
6. Describe in your own words, why thermal light sources do not emit laser light.
   Answer:
   According to Boltzman‘s Law a population inversion of energy states is not possible.
7. Investigate in the literature the various laser types and their specific fields of application. Which properties are crucial for the various applications? Justify your results with arguments.
   Answer:
   Examples:
   • Carbon dioxide laser with high output power for material processing and environmental analysis, since many molecules have characteristic absorption lines in its spectral emission range
   • Ruby laser: a solid state laser and the first laser ever built
   • Neodymium-YAG laser (solid state laser; YAG stands for yttrium aluminium garnet; applications in land survey and photogrammetry, material processing, analytics and medicine
   • dye laser (liquid state laser): can be sensitively tuned over a broad wavelength range, extremly short pulses are possible; applications in e.g. aanalytics of chemical reactions
   • Semiconductior lasers (diode lasers): very common because of economic productions, small dimensions, high efficiency, direct pumping with electric current; e.g. in laser printers, optical disk drives (CD, DVD, BluRay), bar code scanner, laser pointer, optical data transmission
8. Try to identify alternative conventional light sources for the applications which you identified in the previous question. Discuss the pros and cons of lasers and conventional light sources.
   Answer:
   Should be discussed in the class room, or by teams of students.
9. Laser light can be dangerous! Whaat makes laser light more dangerous to the eye than the light of a tungsten lamp having the same optical power?
   Answer:
   Laser radiation is emitted with much smaller divergency than conventional light, and most laser types emit coherent radiation. Therefore the beam illuminates a spot on the eye's retina with a diameter of not more than a wavelength. This causes severe photochemical and thermal damage. Moreover, infrared and ultraviolet lasers are not visible to the eye!