PHYS 20312 Wave Optics Section 3 Michelson Interferometer (University of Manchester)

THIRD YEAR LABORATORY SPECTROMETRY WITH A MICHELSON INTERFEROMETER

1

Aims

To use Michelson interferometry to determine the spectra of a variety of sources of light

2

Objectives

1. To calibrate the interferometer using a HeNe & diode laser, and the mercury green line and measure the wavelength of yellow light.

2. To measure the separation of the mercury yellow doublet.

3. To investigate the coherence length and spectra of high intensity LED’s.

3

Introduction

Spectrometry involves the Fourier transform of a signal in order to extract its frequency content. In the case of an ordinary prism or grating spectrometer, as used in optical spectrometry, the incoming signal is a fluctuating electromagnetic wave. At the entry slit of the spectrometer the signal may be described by E (t ) , the electric field strength as a function of time, φ (v ) , the Fourier transform of E (t ) describes the frequency content, where φ (v )dv is the amplitude observed when the signal has been filtered to allow only the frequency range v → v + dv to pass. The power transmitted within this frequency range is φ (v )φ * (v ) the square modulus of φ (v ) .

The usual grating spectrometer sends signals of different frequencies in different directions so that there is a lateral dispersion of the signal, and a slit of suitable width will isolate the frequency range v → v + dv . Then, when the slit is moved, the detector traces out the curve of I (v ) against v the spectrum of the incoming signal.

4

Theory

The Michelson interferometer behaves quite differently. If monochromatic light enters the instrument, the beam is split into two, with amplitudes A1 and A2, where A1 A1* + A2 A2* = AA* = I where I is the intensity of the incoming radiation. The intensity of any radiation is of course proportional to the square modulus of the amplitude, which in this case we take to be the electric vector E 0 e 2πivt at the beam-splitter.

Consider the moment when t = 0 . Then at the beam-splitter, just before the light is split, I = A02 . At an infinitesimal distance after the beam-splitter, the amplitudes of the beam are A0 iφ1 A e and 0 e iφ 2 , the phase changes on reflection and transmission.

2 2

-1-

Mirror

Mirror Beam splitter

Screen/ detector Figure 1:Experimental arrangement of a Michelson interferometer At the mirrors the amplitudes are accompanied by a phase change

2π

λ

A0 iφ1 i e e 2

2πd1

λ

and

A0 iφ2 i e e 2

2πd 2

λ

because a path change d is

d.

When the beams recombine at the beam-splitter the resultant amplitude is

A0 iφ1 i e e 2

2π 2 d1

λ

e

iφ2

i A + 0 eiφ2 e 2

2π 2 d 2

λ

e

iφ1

2π 2 d 2π 2 d 2 i  A0 i (φ1 +φ2 )  i λ 1 = e +e λ  e 2  

which takes account of the extra phase-shifts on reflection and refraction. Where d1 and d2 are the path lengths of the two beams.

This outgoing amplitude, multiplied by its complex conjugate, is the intensity observed at the detector.

A02 I (∆ ) = [1 + cos(2πv∆ )] 2

(1)

Where ∆ = 2(d1 − d 2 ) is the path-difference travelled by the two beams, and , ν = λ−1 is the wave-number (often measured in cm-1).

Question 1. Show that equation (1) is valid. Remember that half the light intensity that recombines on the beam splitter is sent to the detector.

Equation (1) shows that when the path-difference is changed steadily the intensity at the detector changes sinusoidally. The period depends on the rate at which the path-difference is changing and on the wavelength (λ) of the radiation. The intensity is a maximum at pathdifferences 0, λ 2 , λ , 3λ 2 ... . We usually write: I (∆ ) =

I (0 ) [1 + cos(2πv∆ )] 2

-2-

If, instead of monochromatic radiation, we have a continuous spectrum entering the interferometer, the intensity in the wave-number interval v → v + δv can be described by I (v )dv . Then the output at the detector is: I (v ) [1 + cos(2πv∆ )]dv −∞ 2

I (∆ ) = ∫

∞

the integral comprising all possible wavelengths (and hence wave-numbers) in the incoming beam. This simplifies to: I (∆ ) =

I (0 ) ∞ I (v ) +∫ cos(2πv∆ )dv −∞ 2 2

This is the clue to Fourier spectroscopy: the left hand side, I (∆ ) is equal to a constant I (0) 2 plus the Fourier transform of I (v ) . Consequently, if we take the Fourier transform of the emerging signal, recorded as a function of a path difference ∆ , we get a constant plus a function proportional to the spectral content of the beam.

Question 2. Calculate the Fourier transform for a number of different I (ν ) , for example choose I (ν ) to be a Gaussian function, and then choose I (ν ) to be a ‘top hat’ function. This will give you an idea what you expect to see for I (∆ ) in these cases.

This seems to be a complicated way of doing spectroscopy compared with the usual process of recording spectra which involves rotating a prism or grating inside a spectrometer, but it has one enormous advantage: the amount of radiation that traverses the instrument is hundreds of times greater than can be transmitted by a grating spectrometer with its narrow entry and exit slits. Furthermore, because there is only one detector, looking at all the radiation (or at least, half of it!) all the time, the signal/noise ratio is much improved.

This type of spectrometry, where all frequencies are recorded at the same time and analysed later, is called “multiplex” spectrometry.

5

Practical details

The Michelson interferometer is a two-beam interferometer and when properly adjusted gives fringes of constant inclination. These fringes are apparently at infinity and can be focussed on to a screen with a simple lens, where they may form a set of concentric rings. Why do you see a series of rings? Each ring corresponds to light transmitted through the instrument at a different angle to the optic axis (which is the axis of the focusing lens), and the angles for which transmission is possible are given by the equation: 2nd cos θ = mλ where n is the refractive index (usually ~1 in air), 2d is the path-difference between the two beams passing through the instrument, θ is the angle between the transmitted light and the axis of the interferometer, λ is the wavelength and m is an integer when the transmitted intensity is a maximum. This is the most important formula in physical optics: you should learn it by heart.

-3-

For a given wavelength, there are various angles θ corresponding to different values of m. We are concerned here with the light that comes through parallel to the axis, which is focused to a point, or at least a small circle. Then θ = 0 and 2d = mλ .

Arrange a mercury lamp at the entry port, with a ground glass screen or a piece of thin tissue to give a bright but fairly uniform light source. If the interferometer is not in adjustment, a needle or sharp point silhouetted against the source will show two images, one from each mirror. Adjust the tiltable mirror until the two images coincide. As they do, you should see fine fringes crossing the light source. Continue the adjustment delicately, to make the fringes as large as possible. Eventually they should appear circular or elliptical, and the instrument is properly adjusted when the fringe pattern is stationary as you move your eye across the aperture. If now you replace your eye with a lens, you should be able to put an image of the fringe pattern on a white card. If necessary, remove the ground glass screen to increase the brightness of the image. You can use a green filter in front of the mercury lamp to isolate the 5460.74Ǻ line in the mercury spectrum. This will increase the fringe contrast.

The central spot of the fringe pattern may not be illuminated – it depends on the pathdifference as you set up the instrument. This merely means that the order at the centre is not integer. At the centre of the pattern you require a small hole to isolate the radiation coming through near θ = 0 . This will be received by the detector and constitutes the signal I (∆ ) which you are going to Fourier-transform. The path-difference can be changed steadily with the synchronous motor, and the signal from the photodiode is recorded at the same time, as a series of “samples”. The references below will show how the spectrum can be reconstructed from a knowledge of these “samples”, using the Fast Fourier Transform method, a computation technique which is of great importance in experimental physics and which you should understand.

The sampling theorem which you should look up, is important1 and should be well understood, particularly where it describes “Nyquist frequency” and the phenomenon of “aliasing” artefacts.

You should take precautions against scattered light entering the detector, using metal tubes or baffles which you can make from black card. Be aware that the photo-diode has a different response from your eye. Lasers and LEDs are also available to assist you with the calibration of this instrument.

Use your Michelson Interferometer to establish fringes from the Helium Neon laser. You should investigate the coherence length of the laser using your interferometer (you will need to fully understand the meaning of coherence length). Discuss with your demonstrator what you expect the coherence length to be, and justify why you think this is so. The Helium Neon laser can be used to calibrate the encoder counts from the translational stage attached to the mirror. Its wavelength is well defined (λ=632.816nm) and so too is the distance between the fringes.

Replace the Helium Neon laser with the laser diode. Redo your experiment, and note any differences which are observed between the two types of laser. Explain where you think these differences come from.

1

It is important is all spectroscopy and indeed all experimental physics: it shows you how to make maximum use of the experimental time available, to collect useful information and avoid useless information.

-4-

You have been given a range of LEDs, these LED’s produce light of different colours, and indeed one of these is a so-called ‘white light’ LED. From your results with these different LED’s, determine the bandwidth of the radiation that they emit. Can you infer how the socalled white LED produces white light and how it operates?

6. References [1] W.H. Steel. “Interferometry” Cambridge University Press, 1967. Chapter 10.

[2] J.F. James & R.S. Sternberg. “The Design of Optical Spectrometers”, Chapman & Hall, 1969. Chapter 8.

[3] E.Hecht. “Optics” Benjamin Cummings, 2003. Revision of Fourier Optics, coherence and interferometry.

[4] The fast fourier transform (such as that used in MatLab) is described in detail in ‘Numerical Recipes in C or Fortran’, W.H.Press, CUP, 2007.

Use Matlab to analyse the data.

You need to be careful to ensure that you know what the programme is doing. Use test functions to evaluate what the output is compared to what you expect (see question 2 above). Once you are confident that you understand how the programme works, you can start to analyse the real data from the experiment, Last Modified: Phil Dawson, August 2013.

-5-