Stokes Relations : (c) (a) (b) n1 n2 r2Ei rEi tEi t’tEi r’tEi trEi n1 Er=rEi Et=tEi (a) n1 n2 Ei rEi tEi (b) Principle of reversibility Define reflection (r) and transmission (t) coefficients for ray incident from medium 1 to 2 (Fig. (a)): If amplitude of incident beam is Ei, the reflected beam will have amplitude and refracted (transmitted) beam Situation in Fig. (b) is valid according to the principle of ray reversibility (12-14)

These are the Stokes relations Thus, two rays incident at interface from both media will result in reflected and transmitted rays as shown in Fig. (c). [Note reflection & transmission coefficient of ray incident from medium 2 are r’ and t’] Situations in Figs. (b) & (c) must be physically equivalent, thus, we have and Therefore, and These are the Stokes relations [We have only considered two-beam interference so far. We shall now proceed to multiple-beam interference.] (12-15) (12-16)

Multiple-beam Interference in a Parallel Plate : Internal reflection External reflection nf reflection & transmission coefficients: At external reflection: r & t At internal reflection: r’ & t’ Multiple parallel beams emerge from top and bottom of plate Multiple beams are coherent as they originate from one incident beam Multiple-beam interference observed when emerging beams are focused to point by lens

Consider superposition of reflected beams from top of plate: Multiple-beam Interference in a Parallel Plate : Consider superposition of reflected beams from top of plate: Phase difference between successive reflected beams is If incident ray is , successive reflected rays are: and so on … and the N-th reflected wave is When the waves above superposed, the resultant ER becomes Factoring, we have (12-17) (12-18)

Summation in square bracket is in form of geometric series of where Multiple-beam Interference in a Parallel Plate : Summation in square bracket is in form of geometric series of where since , series converges to sum thus, Using Stokes relations: we get simplifying Irradiance , and we have and, hence, (12-19) (12-20)

using identity: we obtain Multiple-beam Interference in a Parallel Plate : Therefore, using identity: we obtain and, in terms of irradiance, and the proportionality has been used. Similarly, the transmitted beams at the bottom of the plate will yield a resultant transmitted irradiance of (12-21) resultant reflected irradiance (12-22) where Ii  irradiance of incident beam (12-23)

Conservation of energy for non-absorbing films requires Multiple-beam Interference in a Parallel Plate : Some facts: Conservation of energy for non-absorbing films requires A minimum occurs in reflected irradiance (eqn. 12-22) when cos  = 1 or when  this is also the condition for a transmission maximum  1st reflected beam is exactly out-of-phase with the other reflected beams (2nd, 3rd, …) which are all in-phase with one another; perfect cancellation of 1st reflected beam with sum of the remaining ones Reflection maximum occurs when cos  = 1, or when  this is also the condition for a transmission minimum  thus resultant reflected & transmitted irradiances are: (12-24) (12-25) (13-19) (12-26)

Fabry-Perot Interferometer : Makes use of plane parallel plate to produce interference pattern by multiple beams of the transmitted light Applications: precision wavelength measurements, analysis of hyperfine spectral line structure, determination of refractive indices of gases, calibration of standard meter in terms of wavelength

two thick glass / quartz plates to enclose “plate” of air in-between Fabry-Perot Interferometer : Its construction: two thick glass / quartz plates to enclose “plate” of air in-between inner surfaces are very flat, highly reflecting (silver layer for VIS; or aluminium layer for  < 400 nm) outer surfaces inclined at very small angle relative to inner surfaces to eliminate unwanted fringe patterns formed because of multiple beams within glass plate spacing, or thickness, t of air layer  very important parameter it is called an etalon if the spacing is fixed How it works: A narrow, monochromatic beam of light from extended source point S makes angle (in air) t relative to optical axis of system This beam makes multiple reflections resulting in multiple coherent parallel beams to emerge that are brought to point P by converging lens L

With nf = 1 for air, condition for bright fringe is thus Fabry-Perot Interferometer : Whether it’s a bright or dark fringe, it is determined by the path difference between successive parallel beams, With nf = 1 for air, condition for bright fringe is thus If t is fixed, eqn (13-1) is satisfied for certain angles t and concentric rings due to focusing of fringes of equal inclination are formed (13-1) Or if t is varied, a detector on the focussing plane will record the interference pattern as a function of time in an interferogram If source light has two wavelength components, output will be a superposition of double set of circular fringes

Fabry-Perot Interferometer : To see the component sets of circular fringes clearly, a prism is incorporated to disperse them Red Green Violet

Fringe Profiles : Airy Function It is the variation of irradiance in fringe pattern of the Fabry-Perot plotted as a function of the phase of path difference Sharpness of the fringes determines the resolving power of the instrument Irradiance of transmitted beams is (from 12-23) Using trigonometric identity: and simplifying, the transmittance T or Airy function is The factor in square bracket = coefficient of finesse F (13-2) (13-3)

And Airy’s formula for transmitted irradiance is then: Fringe Profiles : Airy Function And Airy’s formula for transmitted irradiance is then: As r changes from 0  1, F changes from 0   The coefficient of finesse also represents a measure of fringe contrast, written as the ratio: From (13-4), Tmax = 1 when sin (/2) = 0; and Tmin = 1/(1+F) when sin (/2) =  1; thus, substituting into (13-5) (13-4) (13-5) (13-6)

Dashed lines refers to comparable fringes from Michelson Fringe Profiles : Airy Function Plot of fringe profile for different values of r : Dashed lines refers to comparable fringes from Michelson Small resolving power Large resolving power T = Tmax = 1 at  = m(2) T = Tmin = 1/(1+F) at  = (m+½)(2) Note: Tmax = 1 independent of r Tmin  0 (but  0) as r  1

Resolving Power : When two fringe rings (e.g. from two set of circular fringes due to two component wavelengths of source) are closely spaced, the measured irradiance follows the dashed line dip crossover point If the wavelengths are very close, the fringes are very close, and it is may be impossible to distinguish two separate peaks in the measured irradiance. Minimum wavelength separation ()min that can be resolved by the instrument depends on the ability to detect the dip in the measured pattern between peaks. According to Rayleigh’s criterion, dip  20% of max. irradiance (applicable to profile of diffraction images) – not strictly applicable here Sufficient to require the crossover point be  ½ max. irradiance of either individual peak

Phase difference between two fringe maxima is: Resolving Power : Phase interval c between max & ½-max values of T is given by (from 13-4): Since c is small, Phase difference between two fringe maxima is: (13-7) (13-8) Corresponding minimum resolvable wavelength difference is determined as follows: The phase difference (as in multiple beam interference – (12-17)) is: For small wavelength intervals , we have

(where  may be either of the two wavelengths or their average Resolving Power : Combine with (13-8), we get At fringe maxima, or (where  may be either of the two wavelengths or their average Resolving power  is generally defined as: in Fabry-Perot interferometer: Desirable to have large resolving power – large values occur when order is large, near center of fringe pattern, cofficient finesse values are large corresponding to high reflectance To maximize m at pattern center, plate separation t (from 13-10)must be: (13-9) (13-10) (13-11) (13-12) (13-13)

Using eqns. (13-3), (13-10) to (13-12), we get Resolving Power (quantitative example) : Fabry-Perot interferometer has 1 cm spacing and reflection coefficient r = 0.95. For wavelength ~ 500 nm, find its maximum order of interference, its coefficient of finesse, its minimum resolvable wavelength interval, and its resolving power. Using eqns. (13-3), (13-10) to (13-12), we get Ǻ Good Fabry-Perot interferometers have  ~ 106 (i.e. over two orders of improvement over the performance of comparable prism and grating instruments)

Fabry-Perot rings obtained with the mercury green line, revealing fine structure.

Finesse another commonly used figure of merit The phase separation between adjacent transmission peaks is called the free spectral range (FSR) of the cavity, fsr Thus: The half-width at half maximum (HWHM) 1/2 of the transmittance peaks can be found by setting T=1/2

Trigonometric identities and a small angle approximation can be used to verify that, at half-maxima, Thus, with Cavities with more highly reflecting mirrors have higher values for the finesse and so narrower transmission peaks than do cavities with less highly reflecting mirrors. The finesse of a cavity is the ratio of the free spectral range of the cavity to the full-width at half-maximum (FWHM) of the cavity transmittance peaks:

The transmittance may be regarded as a function of the round-trip phase shift  or any of the factors upon which  depends, such as the mirror spacing (cavity length) d, the frequency ν (and so wavelength ) of the input field, or the refractive index n of the medium in the space between the mirrors. In different modes of operations, one of these quantities is typically varied while the others are held constant. Although the values of the free spectral range and the FWHM of the transmittance peaks depend on the chosen independent variable, the ratio of these quantities (i.e. the finesse) dependes only on the reflectivities of the mirrors and so is the useful figure of merit for the Fabry-Perot cavity.

Example: Estimate the coefficient of finesse F, the finesse F, and the mirror reflectivity r for a Fabry-Perot cavity with the transmittance curve shown in the figure: Using: and noting from the figure that: The coefficient of finesse is found to be: The finesse is: Or alternatively:

The mirror reflection coefficient can be obtained from: Rearranging gives: Taking the positive root of this quadratic reveals:

Scanning Fabry-Perot Interferometer The Fabry-Perot cavity is commonly used as a scanning interferometer i.e. the irradiance transmitted through the Fabry-Perot is measured as a function of the length of the cavity. Transmittance T as a function of the change in cavity length Δd, for a monochromatic input field Piezoelectric spacer used to control the mirror separation d.

The transmittance is maximum whenever: Rearrangement gives the condition for a maximum as: Accordingly, the free spectral range in this mode of operation is: The cavity length change required to move from one transmittance peak to another is thus a measure of the wavelength of the source. In practice, however, this relation, by itself, is not used to experimentally, determine the wavelength of the source because the length change cannot be measured with the desired accuracy. Instead, it can be used to calibrate the length change of the cavity in order to determine the difference in wavelength of two closely spaced wavelength components in th einput to the Fabry-Perot cavity.

Example of a record that would result when light of two different but closely spaced wavelengths 1 and 2 are simultaneously input into a Fabry-Perot cavity of nominal length d = 5 cm. Suppose that 1 and 2 are known to be very near a nominal wavelength  = 500 nm. If it is known that the adjacent peaks have the same mode number m, then

The wavelength difference is, then, Thus, When it is true that the absolute length of the cavity is unlikely to be known to a high degree of accuracy, one can use the nominal length, d ≈ d1 , of the cavity in this expression. Similarly, one typically replaces the wavelength 1 by its nominal value .. From the figure: So that for d = 5 cm, That is, the Fabry-Perot interferometer easily resolves a fractional difference in wavelength of less than one part in a million.

Mode Suppression with an Etalon Many laser systems permit multimode operation i.e. steady-state output with frequencies corresponding to many different cavity resonances Transmittance for laser cavity of length l A cavity mode of a given frequency will be present in the laser output only if it is amplified by the laser gain medium satisfies also the low loss condition imposed by the laser cavity

In some applications, it is preferable for the laser to have an output at only a single cavity resonant frequency Such a single-mode laser has a longer coherence length than the multimode laser To suppress all but one single laser mode :  insert a Fabry-Perot etalon of length d into the laser cavity of length l > d A cavity mode of a given frequency will be present in the laser output only if it is amplified by the laser gain medium satisfies also the low loss condition imposed by both the laser cavity and the etalon

Example: A certain argon-ion laser can support steady-state lasing over a range of frequencies of 6 GHz. That is, the gain bandwidth of the argon-ion laser is about 6 GHz. If the length of the laser cavity is l = 1m, estimate the number of longitudinal cavity modes that might be present in the laser output. Also find the minimum length d of an etalon that could be used to limit this laser to single-mode operation. Solution: The longitudinal cavity modes are separated by a free spectral range of the laser cavity: Therefore, the number of lasing modes would be given by: To ensure single-mode operation, the free spectral range must exceed the gain bandwidth:

Transverse Electro-Magnetic Modes 06/12/2018