S17_StellarInterferometer

A Daylight Experiment for Teaching Stellar Interferometry

Adam Hines & Nick Skuza

Abstract

A quasi-monochromatic, spatially incoherent light source was used to simulate star light. A telescope fitted with masks containing pairs of pinholes was used to observe the simulated star, producing interference patterns. Increasing the pinhole separation produced a change in the degree of coherence of the light and fringe visibility of the interference patterns. Relating pinhole separation to fringe visibility using the van Cittert-Zernike theorem, the diameter of the light source was determined to be 2.33 ± 0.18 mm. This diameter agrees with the actual diameter of 2.31 mm with an error of 1.12% or 0.14σ, though the data does not fit to the predicted equation.

Introduction

Traditional methods of directly observing stars have several limitations including atmospheric effects, low apparent magnitudes, and an inability to resolve stars that subtend small angles, like most stars in our galaxy [1]. As an alternative to traditional methods, the method of stellar interferometry is better suited to resolve the disks of objects that subtend small angles [1]. Interferometers do this by effectively extending the baseline objective of a telescope from the size of a single mirror to the distance between two telescopes which can be several kilometers in practice. The first star to be studied using a stellar interferometer was Betelgeuse in 1920 by Albert A. Michelson [1]. Michelson’s interferometer design was able to measure the diameter of the Orion star Betelgeuse as 2.58 AU [2]. This experiment scales down Michelson's original experiment. Light from an LED source that simulates star light is passed through pairs of pinholes, producing interference patterns that are observed by a telescope. By scaling this method down to a lab setting we hope to determine the accuracy of stellar interferometry.

Theory

The coherency of light is the degree to which it has the same wavelength and a constant difference in phase [1,3]. There are two types of coherences: temporal and spatial. Temporal coherence refers to a wave’s ability to produce an interference pattern with a spatially shifted version of itself [4]. For a light source with a frequency bandwidth Δυ, a coherence time, Δt, can be defined as the reciprocal of Δυ [5]. The coherence time is the time interval within which a wave of light has a predictable phase. Similarly, the coherence length is the distance over which a light wave has a predictable phase and is given by cΔt [5]. A light source with a coherence time and length that are greater than those of another source is said to have a greater degree of temporal coherence than the second source [5].

Spatial coherence refers to the correlation between the phases emitted at two different points on the light source [5]. If every point on a source emits waves at the same phase, the source is said to be spatially coherent. Similarly, if every point of the source emits at distinct phases, the source is said to be spatially incoherent. Note, however, that if a spatially incoherent light source is placed a distance L from two pinholes, separated by a distance d, the light observed at the pinholes appears highly correlated if L is much larger than d [5]. This phenomenon is because the emitted light spreads out as it travels and as L increases, the pinholes subtend an increasingly small angle as seen from the source.

Both types of coherence are quantified by the complex degree of coherence, γ₁₂(τ), which is found by normalizing the mutual coherence function of light [3,5]. γ₁₂(τ) finds the correlation between he phases at two different points in space and therefore will vary depending on the two points measurements are taken at. For an extended light source, γ₁₂(τ) for two distinct points in space P₁ and P₂ can be calculated using the van Cittert-Zernike theorem, which is given by [3]:

I(S) is the intensity at any given point on the surface of the light source, υ is the light frequency, and k is equal to 2π divided by the average wavelength [3]. I(P₁) and I(P₂) are the intensities at the points at which the light source is being observed. R₁ and R₂ are the distances from a certain point on the source of light to P₁ and P₂. To solve this integral, one would define the geometry of the light source, make a first order approximation assuming the distance from the source to the detector is much greater than both the source's diameter and the pinhole separation, assume the source emits light at a constant intensity across its surface, and then integrate over a circular light source [3]. Doing so produces the following expression for the complex degree of coherence [3]:

where J₁ denotes the first order bessel function of the first kind, a is the diameter of the source, d is the separation between points P₁ and P₂, L is the distance between the source and the detector, and λ is the wavelength of light [3].

In this experiment, we image interference patterns produced by light from a source passing through pairs of pinholes. From these images, the fringe visibility of the interference patterns is calculated. Fringe visibility is defined as [3]:

where I_max and I_min are the maximum intensity of the central bright fringe and the minimum intensity of the adjacent dark fringes, respectively. Furthermore, when the intensity across the surface of the light source is uniform, visibility is related to the complex degree of coherence in the following manner [3]:

Thus, the fringe visibility is given by [3]:

Note that in this experiment, the two points P₁ and P₂ are two pinholes that produce interference patterns. Therefore, equation 6 relates the visibility of the interference patterns to the pinhole separation.

_{Experimental Methods}

Figure 1 shows the experimental setup. The goal of the LED setup was to simulate starlight which has been defined as a light source that is circular, spatially incoherent at its surface, and emits at a uniform intensity across its surface [1]. The LED source was chosen to be a Dolan-Jenner Model 190 Fiber-Lite, which consists of LED’s coupled to polymer optic fiber to produce spatially incoherent light with a uniform intensity across its surface. The light was sent through a biconvex lens which focused the light through an iris with an adjustable diameter. A filter was used to make the source quasi-monochromatic.

The telescope used in this experiment was a Powerseeker™ 127 mm aperture Refractor Telescope with a 1000 mm focal length and a 4 mm eyepiece. A Canon Rebel XSI camera was mounted to the eyepiece of the telescope, where it took pictures of the interference patterns. To produce the interference pattern the light was first passed through a specially designed mask and cover plate that were fitted to the front of the telescope. By rotating the plate, one pair of pinholes with a known separation could be selected to allow light to pass through. The plate and mask are shown in Figure 2. Images of the interference patterns were taken using 1.5 minute long exposures. The image processing software ImageJ was used to find the I_max and Imin for a given interference pattern. Using those intensities, the fringe visibilities were calculated. Data was then fit using equations 4 and 6.

Results

Figure 3 shows two of the images taken and their corresponding intensity profile. Figure 4 shows a graph of visibility versus pinhole separation. The fit parameter b is equal to πa / Lλ. The value of b found from the least squares fit produces a source diameter a of 2.33 ± 0.18 mm. The reduced χ² is 54.13, indicative of a poor fit. Furthermore, Figure 5 shows the χ_i graph, where it can be seen that χ_i values range from 2.5 to 10.5.

Conclusion

The linear diameter of a simulated star was calculated by observing how its degree of coherence decreases when viewed from increasingly separated pinholes in front of a telescope. The diameter of the simulated star was determined to be 2.33 ± 0.18 mm, which agrees with the actual value of 2.31 mm with an error of 1.12%. It was found that instabilities and vibrations in the telescope did not significantly affect the data. In addition, the spacing between the interference fringes decreased as the pinhole separation increased, reducing the camera’s ability to resolve the intensity extrema for larger separations, introducing additional uncertainties. Data analysis revealed the data does not fit to the prediction equation derived from the van Cittert-Zernike theorem. It is postulated that this poor fit is a result of either the light source not having a sufficiently uniform intensity across its surface or the telescope not having an adequate mirror to resolve the simulated star from a distance of 56 m.

References

[1] M. A. Illarramendi, R. Hueso, J. Zubia, G. Aldabaldetreku, G. Durana and A. Sánchez-Lavega Am. J. Phys. 82, 649 (2014); http://dx.doi.org/10.1119/1.4869280

[2] A. A. Michelson, F.G. Pease. ApJ, 53, 249-259 (1921); adsabs.harvard.edu/abs/1921APJ….53..249M

[3] M. Born and E. Wolf, Principles of Optics, 7th ed. (Pergamon Press, London, 2002).

[4] R. D. Guenther, Modern Optics. (Wiley, 1990).

[5] E. Hecht and A. Zajac, Optics, 4th ed. (Addison-Wesley, Reading, MA, 2003).