Excerpt

## Table of contents

1. Introduction/Motivation

2. Laser medium

2.1 Requirements for lasing

2.2 Possibilities for energy levels

3. Laser resonator

4. He-Ne laser

5. Fabry-Perot interferometer

6. Dielectric mirrors

7. Basic optical formulas

7.1 Snell's law

7.2 Fresnel equations

7.3 Brewster angle

8. Propagation using ray transfer matrix analysis

9. Properties of M²

10. Coherence

11. Parameters of the practicum

12. Autocollimation of the laser beam

13. Transversal modes

14. Gain factor and Brewster angle

15. Axial modes

16. Holography

17. Gaussian laser profile

18. Identification of the transversal modes

19. Calculation of gain factor and Brewster angle

20. Fabry-Perot-Interferometer

21. Gaussian laser profile

22. More questions

23. Conclusion

24. Bibliography

## 1. Introduction/Motivation

Lasers are very commonly used in everyday life, e.g. in laser pointers, CD-Players or the lasers used in cash registers. But what makes lasers special and distinguishes them from other light sources? Which advantages do they have and what are their applications in physics?

In the experiment "Laser" in the Advanced Physical Practicum the task is to find out which mathematic coherences there are between parameters of the laser and to test how good they match the experimental observation.

In general, what makes a laser special?

Most important are two points: an unequaled high intensity as well as a very sharp frequency range of the emitted light. Additional laser light is very sharply focused, which leads to the already mentioned high intensity of laser radiation throughout a relatively low power output, as we talk in dimensions mostly smaller than 100 mW, which is really small compared to a light bulb with 60 Watt. Nevertheless the intensity of laser light is much higher, which makes them outstanding.

** Short overview of theory**

## 2. Laser medium

### 2.1 Requirements for lasing

A laser is an abbreviation for "light amplification by stimulated emission of radiation". The main operating principle of a laser is to amplify a parallel beam of monochromatic light. To fulfil this task, three essential components are needed.

At first, a gain medium is needed, in which photons can be emitted by niveau changes of atoms and molecules. This only works, if more particles are in an energetic higher state than in a lower, otherwise no photons will be emitted by the laser.

Because in the thermodynamic balanced case it's quite rare that higher energy levels are occupied according to Boltzmann, in a laser an optical pump is needed to guarantee an inverse population. This is possible by pumping energy in the gain medium. Further, this only can happen, if there are more than two energy levels, because by assuming only two energy levels and an inverse population, the Einstein coefficients will mismatch in a way that it will be more likely for an electron to lose energy by stimulated emission than to gain energy by photon absorption. Additionally, spontaneous emission will make it even worse.

Also a resonator will be needed to determine which of the generated photons can leave the laser in a way that most of the photons with odd frequencies leave as the laser beam, but most of the photons with the right amount of energy stay in the resonator to establish an even more stimulated emission. This is the reason why lasers have an outstanding sharp spectral range.

### 2.2 Possibilities for energy levels

Because a laser with two energy levels is not possible, the next idea will be one with three energy levels. Such a laser will be possible, if the pumping process goes from the lowest to the highest energy level, and the transfer from the highest level to the middle has to happen much faster than the one from the middle energy level to the lowest in order to make the stimulated emission caused by the pumping process disappear from the rate equations. It also appears within more detailed calculations that such a three-level laser can work if the right materials and intensities are chosen. Also a high light intensity inside the resonator is required, which means that the light intensity inside the resonator has to be even higher than outside of the resonator. In this case, inverse population is possible.

For a laser with four energy levels, the needed inverse population is satisfied automatically, which can be seen by calculating the rate equations. Here the pumping process has to go from the lowest energy level to the highest one.

## 3. Laser resonator

This part of a laser will be realized by two mirrors with matching parameters. In our case we will use a confocal construction, which means putting two spherical mirrors in a distance at which their focal points will match in one point. The resonator therefore is used to lengthen the way through the gain medium and to sort out frequencies, since only the frequencies which match the condition for constructive interference Abbildung in dieser Leseprobe nicht enthalten will be amplified, where N has to be a positive integer. All other frequencies will disappear due to destructive interference. Since we talk about the interference between the incoming and reflected wave, the result will be a standing wave. The quality of the resonator has to be especially good in case the amplification of the gain medium is low.

The laser resonator has to match the stability condition Abbildung in dieser Leseprobe nicht enthalten; Abbildung in dieser Leseprobe nicht enthalten(3.1)

L represents the length of the cavity and R is the radius of the mirrors.

We also have to differ between two kinds of modes.

The longitudinal mode is an oscillation lengthwise to the direction of light propagation, and due to the conditions at the edge of the resonator it is necessary that only special frequencies can be in the longitudinal mode.

The envelope of the amplification usually is a Gaussian curve, but since only the frequencies which matching the boundary conditions will be amplified, one will see peaks in the amplification. These peaks are the laser modes.

The envelope is generated by the Doppler broadening of the lines, which is a result of Browns movement of small particles. The spectral distance of the modes can be calculated using the formula Abbildung in dieser Leseprobe nicht enthalten. The longer the cavity is, the lower the spectral distance between two longitudinal modes will be.

If a wave doesn't fill the room perpendicular to the direction of light propagation, there will be a difference in the length of cavity and therefore the frequency will change. This leads to transversal modes. They can occur as TEM-Modes if using planar mirrors, otherwise hybrid modes will occur.

## 4. He-Ne laser

This laser uses a mix of Helium and Neon gas in a thin glass capillary. Helium is used for pumping, while Neon is the gain medium. The relation between Helium and Neon determines the wavelength of the laser. The voltage, which is necessary for the ignition of the laser, is about 15 kV. To inverse the population, electrons bump inelastically with the helium atoms in order to excite them. The following factors can influence the observed bandwidth:

First, Doppler broadening causes an increase in bandwidth, as the center of mass of a mixture of to different gases is always in motion. Additionally, the light will be pressure broadened which is a result of the low gas pressure. This happens when inelastic bumps interrupt the stimulated emission.

An advantage of this kind of laser is a quite low bandwidth of gain, allowing more modes to oscillate at the same time within the resonator. Also with the He-Ne laser a variety of different wavelengths can be emitted, which depends on the partial pressures of both gases.

In the diagram the principle for the energy levels is shown. As can be seen in above, Helium will be excited in the 2s-state and then can bump with Neon, which will drop the energy levels from 5s to 4p, or from 4s to 3p. The last one is the visible transfer, which will be used in the practicum.

## 5. Fabry-Perot interferometer

This tool consists out of two mirrors, which reflect the main percentage of light intensity. A small percentage is being transmitted through one of the mirrors. In this way, some proportions of the incoming beam will pass the etalon without being reflected, others will leave after being reflected several times back and forth.

In the end, all beams coming out of the mirror will be collected by a lens and projected on a screen.

The finesse is an important quantity describing the quality of a Fabry-Perot etalon, as it describes how easy it will be to separate two nearby wavelengths.

The higher the finesse of a Fabry-Perot etalon is, the sharper the interference rings on the screen will be.

This construction has similar to a laser resonator certain wavelengths with maximal intensity, caused trough interference. The single maxima also aren't very sharp like a d-distribution, but have a Gaussian curve with its full width at half maximum (FWHM). The finesse now equals the relation of the spectral range between two maxima in the intensity to the FWHM of one maximum.

Abbildung in dieser Leseprobe nicht enthalten

Here is Dl the spectral range and dl the FWHM.

The Finesse is determined by the reflectivity of the two mirrors.

Abbildung in dieser Leseprobe nicht enthalten(5.1)

## 6. Dielectric mirrors

In order to achieve extremely high reflectivity of the mirrors, metallic mirrors will be unsuited. Instead, we use dielectric mirrors, which consist of many layers with different refractive indices. In this mirror, also known as the Bragg mirror, again interference is used in the way that the difference in the optical path length differs by one wavelength in the different layers. This can lead to very high reflectivity and therefore a high finesse of the Fabry-Perot etalon.

## 7. Basic optical formulas

### 7.1 Snell's law

Snell's law describes the angle of light reflection when hitting an interface of two materials with different refractive indices. It follows from the principle of Fermat, which states that small changes in the way of light should cause no difference in the optical path length, which is equivalent to the formulation that light takes the shortest possible way. If deriving the time light takes between two points over an interface and setting this expression equal to zero, Snell's law can be derived. Written in a formula, it says Abbildung in dieser Leseprobe nicht enthalten where a and b are the angles perpendicular to the plane of the interface.

### 7.2 Fresnel equations

The Fresnel equations can be derived by the boundary conditions and conservation of energy for the components of the electric and magnetic field:

a) for the parallel components: Abbildung in dieser Leseprobe nicht enthalten

b) for the perpendicular components: Abbildung in dieser Leseprobe nicht enthalten f is the angle between the incoming/reflected light ray and the surface and n is the relative refraction index which equals n2/n1. 1

### 7.3 Brewster angle

In the Brewster angle, no p-polarization passes the interface. The reflected light has linear polarization, because only the s-polarized parts are being reflected. If combining Snell's law with the condition that no light is reflected, one gets

Abbildung in dieser Leseprobe nicht enthalten. (7.3.1)

To determine the Brewster angle a polarization filter can be used, so that the reflected light is only s-polarized if light hits the interface under the Brewster angle. Therefore, if setting the polarization filter at zero degrees, no light will be transmitted. If now measuring the angle under which light hits the plate, the Brewster angle is being observed. For a glass plate this angle should be around 55-60 degrees.

## 8. Propagation using ray transfer matrix analysis

If calculating light transfer through different components, the calculations can be simplified if using paraxial approximations. One can use matrices to describe the light transfer by using the position and angle respectively slope of the incoming beam as components of the initial vector. By multiplying this expression with a ray transfer matrix, the outgoing light ray can be determined. Generally, this matrix form will look like

Abbildung in dieser Leseprobe nicht enthalten

Here, the index "i" stands for the incoming ray and "o" for the outgoing one. The expressions with a stroke stand for the slope of the rays.

Below some examples for such matrices are listed.

a) for translation Abbildung in dieser Leseprobe nicht enthalten

b) for a thin lens Abbildung in dieser Leseprobe nicht enthalten

c) for transmission through a dielectric with refractive index2 n Abbildung in dieser Leseprobe nicht enthalten

To get the matrix for different systems, the matrices have simply to be multiplied.

## 9. Properties of M²

M², also known as the beam quality factor represents how easy the laser beam can be focused. With increasing beam quality factor, it will be harder to focus the light. The optimum is a beam quality factor equal 1, which is satisfied for a Gaussian beam. In reality, this quantity will always be greater than one, leading to lower beam quality as it decreases by one over the square of the beam quality factor. For the determination of M², the middle section of a laser ray can be measured by calculating the variance of the intensity using the known expressions3

Abbildung in dieser Leseprobe nicht enthalten Abbildung in dieser Leseprobe nicht enthalten

The lower the variance and the standard derivation is, the better (more like a Gaussian beam) will be the laser.

**[...]**

- Quote paper
- Moritz Lehmann (Author)Niklas Stenger (Author), 2017, Laser. Advanced Physical Practicum, Munich, GRIN Verlag, https://www.grin.com/document/378240

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