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Subsection 50.6.1 Interference Condition in Fabry-Perot Interferometer

In Fabry-Perot interfermoter, its an infinte collection of waves that interfere. When we have only two waves as in Subsection 50.3.1, we saw that the condition depended on phase difference \(\Delta_{12}\) between the two waves. There, we had concluded that if \(\Delta_{12}\) is an integral multiple of \(2\pi\text{,}\) then we will have constructive interference.

To figure out an expression of phase difference \(\Delta_{12}\text{,}\) we analyze the optical paths as we did in Subsection 50.3.1, of two succesive rays, shown as \(\text{R}_1\) and \(\text{R}_1\) in Figure 50.6.3.

To find an expression of \(\Delta_{12}\) we figure out the difference of optical path lengths of ABCDE and ABGF.

Now, from \(\triangle\text{ABX}\text{,}\) we can see that

Now, we note that BC, CD, and GF can be written using these since

Therefore, we get the following for \(\Delta\text{OPL}\)

Let \(\lambda\) be the wavelength in the medium of Fabry Perot. Now, we know that phase changes by \(2\pi\) radians when OPL changes by one wavelength, therefore, the phase difference between the two successive paths will be

Demanding this to be equal to \(2m\pi\) with \(m\) integers for constructive and equal to \(m^\prime\pi\) with \(m^\prime\) odd integers we get the interference conditions. Multiplying both sides by \(\lambda/2\pi\) we will write them as follows.

It is interesting to note in these equations that higher order of interference, corresponds to smaller angle.

That is, higher order interference are more towards the center of the Fabry Perot patterns. Similarly, in the same order, higher wavelength will be towards the center.

Subsection 50.6.2 Intensity of Trnasmitted Waves in Fabry Perot

A more complete understanding of the wave interference in Fabry Perot interferometer requires examining the actual electromagnetic waves as we did when we studied the Fresnel equations for reflection and transmission in Subsection 49.4.1. That work is a little bit beyond the scope of this book. Even so, it is important to discuss implications of the results since Fabry Perot is used in many applications.

Let \(R\) be the reflectance of the surfaces on the two sides of the cavity. Let \(I_t\) be the intensity transmitted and \(I_i\) the incident intensity. Let \(\Delta_{12}\) be the phase difference between two successive waves in direction \(\theta\) as given in Eq. (50.6.1). The, the transmittance can be shown to be

where \(F\) is called the \(finesse coefficient\) or quality factor. Finesse coefficient is a function of the reflectance.

Since \(0\le R \le 1.0\text{,}\) \(F\) is highly nonlinear. Typically, \(R\) is in the \(0.90\text{ to } 0.98\) range. That corresponds to \(F\) in range \(360 \text{ to } 9,800\text{.}\) With large finesse in Eq. (50.6.4), transittance becomes highly sensitive to \(\Delta\text{,}\) giving sharp peaks when plotted against \(\Delta_{12}\) or \(\theta\text{.}\)

The maxima and minima of transmittance in Eq. will correspond to the constructive and destructive interference, respectively. Therefore, for a fixed \(F\text{,}\) you can immediately see that \(T\) will me maximum when \(\sin(\Delta_{12}/2)=0\text{.}\) This immediately says that constructive interference will occur when

which is the result we have before. Additioanlly, Eq. also tells us how the maxima wil lbe distributed.

Figure 50.6.5 show plots of the transmittance versus the phase difference for two values of reflectance, \(R=0.5\) and \(R=0.8\text{.}\) You can see that peaks, which correspond the fringes, become narrower with increase in reflectance. It is not uncommon to find laser cavities \(R=0.9999\text{.}\)

I didn't plot that high a value here since you will just see a vertical in these plots if you used such values. A similar plot will result if we plot at a fixed angle but for various frequencies. This type of plot will give intensities of various frequencies in the direction specified by \(\theta\text{.}\) To do that we will write \(\Delta_{12}\) in terms of frequency by using \(\lambda f = v = c/n\text{.}\) We will also set \(n=1\) for air. This gives

Figure 50.6.6 shows a plot of transmittance versus frequency for a fixed \(\theta\text{.}\) The peaks in this plot correspond to different values of constructive interference order. We can see this by setting the right side of Eq. (50.6.5) with an index to \(f\) in the interference condition.

where \(m\) is an integer.

The peaks are called resonant frequencies of the cavity. The width of a peak is defined by the range of frequencies when transmittance is \(0.5\text{.}\) This width is called full width at half maximum (FWHM). Between successive peask, say between \(f_m\) and \(f_{m+1}\text{,}\) there is low signal. This range is call free spectral range.

Subsection 50.6.3 Applications of Fabry-Perot Interferometer

Fabry-Perot has been valuable tool for high-resolution spectroscopy. In spectroscopy, you are interested in separating light of different wavelengths (or equivalently, of different frequencies). If a source of light emits light of many frequencies, and you want to know how much of light in which frequency, you would use a spectroscope. A simple prism can also separate light light into component frequencies, but it is not as resolving as a high power Fabry-Perot instrument. In the next chapter we will study diffraction gratings, which can also achieve good resolution. But, it is hard to beat the resolving power of a Fabry-Perot instrument.

By looking at the properties that go into the interference condition in the Fabry-Perot interferometer, you can identify kinds of properties that you can use the interferometer for - it relates refractive index \(n\) of the medium, the length \(d\) of the cavity, and wavelength \(\lambda\) of light.

Depending upon our particular purpose for using a Fabry-Perot interferometer, we may vary one of these parameters and study the change with a detector.

When the cavity plates are fixed to a definite separation \(d\text{,}\) the arrangement is also called an etalon. For fixed \(n\) and \(d\text{,}\) the interference condition will depend upon the wavelength. So, if light source consists of two wavelengths, say a red light of \(650\text{ nm}\) and a blue light of \(450\text{ nm}\text{,}\) two separate rings for each interference order \(m\) are illustrated in Figure 50.6.4. Of course, to separate such a widely different wavelength, you don't need Fabry-Perot.

A famous high resolution use of Fabry-Perot took place in the separation of sodium yellow doublet of wavelengths \(588.9950\text{ nm}\) and \(589.5924\text{ nm}\text{.}\) You can see an image shown in Figure 50.6.7 of the screen of sodium doublet in an experiment by visiting Hyperphysics website. The two lines are repeated for different orders.

An instrument used for resolving spectral lines is rated by its resolving power. It is proportional to the inverse of minimum separation in wavelengths \(\Delta\lambda_\text{min}\) that the instrument can resolve .

To make resolving power a dimensionless quantity, we multiply the right side by average wavelength. Thus,

How do we tell if two nearby wavelengths are separated? This is the same type of question as we had when we were studying telescopes. There, the question was how to tell if we could tell whether there were two stars or one star on the CCD. A commonly used criteria, called Raleigh criterion is that if the maximum of peak of one will be at less than half-width from the other peak, then we cannot be sure there are two peaks.

Suppose, we have a light that is mixture of two wavelengths \(\lambda_1\) and \(\lambda_2 \gt \lambda_1\text{.}\) When this light is sent through a Fabry-Perot, they form peaks at \(\lambda_1\) and \(\lambda_2\) with half-width at half maximum of each to be \(w_1\) and \(w_2\) respectively. Raleigh criterion says that to be resolvable, we must have

If \(w_2 \ge w_1\text{,}\) then, we just need to check \(\lambda_1 \lt \lambda_2 - w_2\text{.}\) Figure 50.6.8(a) illustrates the idea behind what we would consider barely separated by Raleigh criterion. Figure (b) is visibly separated as you can see the dip between the peaks.

We can use Eq. (50.6.4) to cast the resolving power in terms of finesse coefficient \(F\) of the instrument. For nearly normal incidence, resolving power can be shown to be

where the quantity between brackets, viz. \(\pi \sqrt{F}/2 \text{,}\) is called finesse of the instrument. We will denote finesse by \({\cal{F}}\text{.}\)

It is useful to put finesse and the resolving power in one place with the reflectance.

where I have omitted subscripts for simplicity. You can use this formula to find \({\cal{F}}\) you need for the resolution of light required, which can then be used to find the minimum reflectance needed on the cavity mirrors.

In the next chapter you will study another device called the diffraction gratings which are also used for the separation of light of different wavelengths. It turns out that it is much harder to construct a diffraction grating with as high a resolving power as a Fabry-Perot interferometer. For instance, let \(n = 1\text{,}\) \(d = 1\text{ cm}\text{,}\) \(\lambda = 650\text{ nm}\text{,}\) and reflectance \(R = 0.98\text{,}\) then the resolving power would be \(4.8 \times 10^6\text{.}\) To get this kind of resolving power from a diffraction grating (which will be discussed in the next chapter) that has slit separation of \(1000\text{ nm}\text{,}\) you will need diffraction from \(1,560,000\text{ slits}\text{,}\) or \(1.56\)-meter wide diffraction grating!