S17_RefractiveIndex

Measuring the Refractive Index Using an Extended Cavity Diode Laser

Brendan King and Sarah Stangl

University of Minnesota

Abstract

In this experiment, the index of refraction of a thin glass slide was measured using an Extended Cavity Diode Laser (ECDL). This was done using saturated absorption spectroscopy of a Rubidium cell combined with a change in optical path length inside the extended cavity. Using Snell's Law and trigonometry, the index of refraction of a thin piece of BK7 glass was related to path length changes in the laser. Using a least squares fit, the index of refraction was measured to be n=1.52±0.032. This value agrees with the index of refraction of BK7 glass found literature, 1.51.

Introduction

The index of refraction of a material is a ratio of the speed of light in a vacuum to the speed of light in the material [1]. Snell observed that when light travels between materials of different indices of refraction, the angle of the incident light changes proportionally to the ratio of the indices of refraction. Several different methods can be used to measure the index of refraction of a material. One is to use Snell's Law directly by measuring the angles of incident and refraction of a light beam. Another is to measure the displacement of the beam. However, these methods are suited to measuring the index of refraction of thick materials because the displacements and angle changes are small when measuring thin materials [2]. A third method is to use the Brewster's Angle. This is the angle when the reflected polarized light is at a minimum and is related to the index of refraction [1]. However, this method is useful for single layer materials. Because these methods are not suited to measuring the index of refraction of a thin multi-layer film, we use a combination of saturated absorption spectroscopy and an Extended Cavity Diode Laser (ECDL).

Luetjen et al found that controlled changes in a diode laser cavity’s effective length combined with Doppler-free saturated absorption spectroscopy in order to provide a reference frequency were useful in determining the index of refraction of thin films [1]. This project seeks to validate the basis of Luetjan et al's experiment by calculating the index of refraction of a thin BK7 glass slide and in turn set up the basis for measuring thin multi-layer films.

In this experiment, the index of refraction is determined from analyzing the spectrum of a laser beam traveling through a sample. The sample is placed inside the extended cavity of a tunable ECDL and the angle the sample makes with the laser beam is altered. By Snell's Law, this changes the effective length of the cavity. By measuring the angle needed to change the effective length of the cavity by one mode spacing, the index of refraction can be determined using Snell's Law. However, these changes in the wavelength are on the order of a nanometer, so they lie outside the resolution of a typical spectrometer. Therefore, saturated absorption spectroscopy is used to resolve these small changes in the wavelength and to provide a reference frequency.

Theory

Laser Physics

A semi-conductor laser consists of a doped p-n junction in the gain medium where an electron is allowed to decay from the conduction band into the valence band releasing energy in the form of photons. This light is amplified when an emitted photon causes recombination and in turn an additional photon resulting in an increase of emitted photons and a coherent beam of light [3]. Only certain frequencies of this light are supported in a laser's cavity. Because light will constructively and destructively interfere, only frequencies that have a 2π-multiple phase change inside the cavity will be amplified [1]. The spacing between these supported frequencies (modes) is described by,

. (1)

where nL is the effective length of the cavity and c is the speed of light.

These wavelengths are maintained if the gain of the light through one round trip of the cavity is equal to or greater than the lasing threshold (unity). If the gain of the light is less than this threshold, then the light decreases in intensity as photons are lost due to radiation lost through the output coupler, the ratio between the absorption and emission rate, and the probability of the occupancy of the conduction and valence bands [3].

The gain curve of a diode laser is Lorentzian. Therefore the gain of the resonant frequency is larger than each successive mode. However, the bandwidth of a typical laser is about 1 GHz so only the resonant frequency is supported.

The line-width of each mode experiences broadening. Natural broadening is a result of the transition time of an electron from a higher energy level to a lower energy level. The transition time is finite due to the Heisenberg Uncertainty Principle,

. (2)

Therefore, the longer the transition time, the larger the broadening of the line-width [4]. Typical values of natural broadening correspond to a line-width of a few KHz [4].

Tunable Extended Cavity Diode Laser

An extended cavity diode laser with a diffraction grating in place of the end mirror is shown in the Littrow configuration of Figure 1. In this set-up, the diffraction grating sends the first-order diffraction of the beam back into the cavity to be amplified while the zeroth-order diffraction is reflected as the output beam. Rotating this grating allows the lasing frequency of the laser to be selected, in other words tuned [5]. This tuning can be achieved by mounting the diffraction grating on a Piezoelectric Transducer (PZT) that can be controlled via an applied voltage.

Figure 1: A schematic showing the set-up of a Littrow configuration. Original Figure.

Since the PZT is controlled via voltage, it can also be set to sweep across a range of frequencies when a changing voltage is applied. This is used in conjunction with a rubidium cell and some non-linear optics to get saturated absorption spectroscopy.

Saturated Absorption Spectroscopy

Using saturated absorption spectroscopy (SAS), the laser is locked to a specific atomic transition. This method involves a set-up as shown in Figure 2. Two counter propagating laser beams (the pump and probe beams) derived from the same source are crossed inside a rubidium cell. The pump beam is about ten times as intense as the probe beam. The pump beams saturates, 'bleaches', the gas in the cell and the counter propagation of the beams allow interaction between the molecules with zero velocity with respect to the path of the beams [6].

Figure 2: A diagram showing the counter propagation and crossing of

the pump and probe beam inside a gas cell. Original Figure.

Consider a gas with two energy levels. The pump beam promotes half of the electrons of the gas to higher energy. The pump beam exhibits an absorption profile as seen in the upper image of Figure 3. When the laser is tuned to the resonance of the gas, the beams interact with the same molecules. When the probe beam passes through, it can interact with an excited electron and cause it to undergo stimulated emission. The result is an emission feature as seen in the lower image of Figure 3. This feature is called the 'Lamb Dip' [6]. When this feature appears, the laser is lasing at a half integer multiple of the frequency of this atomic transition. Therefore, this technique provides a reference frequency.

Figure 3: The spectrum as a result of only feeding the pump beam

into the Rb cell is shown in the upper figure. The spectrum

with both beams is shown in the lower figure. Image taken from [6].

Experimental Set-up and Methods

This experiment uses the change in the optical path length (OPL) of the laser cavity to measure the index of refraction. A schematic and actual image of the set-up are shown in Figure 4 and Figure 5. A sample is placed in the extended cavity of the laser between the collimating lense and the diffraction grating with a known angle θ. Using Snell's Law, the change in OPL of the glass is,

. (3)

The diffraction grating is tuned in order to support the same frequency for the resonance of the rubidium gas cell for saturated absorption spectroscopy. The frequency peak shifts to the side as it appears at different lengths of the PZT, a different scanning voltage.

Figure 4: A picture of the actual set-up of the experiment. Original Figure. Figure 5: A schematic of the set-up. Original Figure.

The glass slide is rotated to θ’=θ+Δ, the path length becomes,

. (4)

This changes the position of the resonant peak in the PZT scan.

When the slide has been rotated by angle such that the change in OPL corresponds to half the resonate frequency, the resonance peak, will appear in the scan position in the scan which leads to the relation,

(5)

where l is the change outside the glass, d is the thickness of the glass, and n is the index of refraction. For the derivation of Equation 5 see [6]. A schematic of the geometry of the glass is seen in Figure 6.

Figure 6: A diagram showing the geometry of light being refracted in a material. Original Figure.

This experiment was conducted by placing a BK7 glass slide in the extended cavity diode laser. The slide was rotated until the resonance peak indicating the atomic transitions of rubidium were visible and as seen in Figure 7. The angle of the slide with respect to the vertical was recorded. Then the slide was rotated until the frequency peak indicating the atomic transitions disappeared and reappeared in the same position in the PZT scan. This angle of rotation corresponded to a change in OPL of a half-integer multiple of the resonant frequency of Rb. The angle change was recorded and the process was repeated multiple times. The initial angle, the change in angle, and the number of changes in the angle (the number of times the frequency reference disappeared and reappeared in the scan) were all recorded.

Figure 7: An image showing the absorption profile for the

S to P orbital transition of Rb. Original Figure.

Results and Data Analysis

The recorded data is shown in Figure 8 and Figure 9. These plots have been generated for two trials with two different starting angles. They have been plotting as the number of half-integer wavelength changes from the initial angle times half the wavelength versus the OPL change calculated using Equation 5. These have been plotted against each other because according to Equation 5 used with θ held constant, Δ as the total displacement from the original angle, and k as the number of half-integer wavelength changes, the slope will be one. The index of refraction was altered using a least squares fit such that the slope was 1. This resulted in the index of refraction of BK7 of n=1.52±0.032.

Figure 8: The data for an initial angle of 20.32 degrees with respect to the horizon. Original Figure.

Figure 9: The data for an initial angle of 15 degrees with respect to the horizon. Original Figure.

Conclusions

The index of refraction of a glass sample was measured using an extended cavity diode laser and saturated absorption spectroscopy. The sample was inserted in the extended cavity and the laser was tuned to the resonant frequency of Rubidium gas. Then the glass slide was rotated so that the frequency of the laser was the resonant frequency of Rb. The slide was rotated so that the frequency of the laser was a half-integer multiple of the resonant frequency of Rb. After recording the angle changes and analysis, the index of refraction of BK7 was determined to be n=1.52±0.032. The value quoted in literature is 1.51 [10]. The quoted value falls within error of the determined value. This validates the technique of using an extended cavity diode laser and saturated absorption spectroscopy to measure the index of refraction of thin materials.

Future Work

The results validated the technique of SAS an ECDL for measuring the index of refraction of a thin material. Future work on this project could be to measure the index of refraction of multi-layer samples. In the work conducted by Luetjen et al, the index of refraction of a thin film of fluid trapped between two glass slides was measured. This technique has been validated and so can be extended in the future to include the measurement of samples such as those explored in Luetjan et al.

Acknowledgements

We would like to thank our adviser Kurt Wick for providing support and insight through our experiment. Also, we would like to thank Professor Crowell for providing us with insight into saturated absorption spectroscopy.

References

[1] B. E. A. Saleh and M. C. Teich. Fundamental of Photonics. 1st ed. Wiley-Interscience, 1991.

[2] C. Luetjen, J. Hallsted, and M. Kleinert. “Measuring the Refractive Index of Thin Films Using an Extended Cavity Diode Laser”. In: American Journal of Physics 81 (2013), pp. 929–935.

[3] B. Van Zeghbroeck. Principles of Semiconductor Devices. 2011.

[4] Optical Cavity and Laser Modes. University of Babylon. url: http://www.uobabylon. edu.iq/eprints/publication_2_14877_1775.pdf.

[5] G. P. Nyamuda. “Design and Developement of an Extended Cavity Diode laser for Laser Cooling and Spectroscopy Applications”. MA thesis. Stellenbosch University, 2006.

[6] T. Reiger and T. Volz. Doppler-Free Saturation Spectroscopy. Max-Planck Institute fur Quantenoptik, Garching.

[7] Mikhail Polyanskiy. Refractive Index Database. url: https : / / refractiveindex . info/.