11.2.1 The Sagnac Interferometer

The Sagnac, or cyclic, interferometer is illustrated in Figure 11.1. In this interferometer, the incident light beam is divided into two sub-beams by a beam splitter (BS). The reflected beam, on the incidence BS, is then sent into a path defined by the reflections on M₁, M₂, and M₃ mirrors. The transmitted beam, on the incidence BS, is sent into a path defined by the reflections on M₃, M₂, and M₁ mirrors. Both counter-propagating beams are recombined at the BS. The interference mechanics of the counter-propagating round trips can be described using Dirac’s notation via the probability amplitude:

where:

j refers to the reflection mode of the BS

j′ refers to the transmission mode of the BS

Assuming that

and

Then, Equation 11.2 reduces to

If j′ = 1 represents the BS in the transmission mode and j = 2 in the reflection mode, then Equation 11.5 can be written as (Duarte 2003)

which, for N = 2, can be expressed as

An alternative triangular Sagnac interferometer, with only two mirrors (M₁ and M₂), is shown in Figure 11.2.

11.2.2 The Mach-Xehnder Interferometer

The Mach–Zehnder interferometer is illustrated in Figure 11.3. In this interferometer, the incident light beam is divided into two sub-beams by a BS. The reflected beam, on the incidence BS, is then sent into a path defined by the reflection on M₁ toward the exit BS. The transmitted beam, on the incidence BS, is sent into a path defined by the reflection on M₂ toward the exit BS. Both counter-propagating beams are recombined at the exit BS. The interference mechanics of the counter-propagating beams can be described using Dirac’s notation via the probability amplitude

which can be abstracted to

If j′ = k′ = 1 represents the BS in the transmission mode and j = k = 2 in the reflection mode, then Equation 11.9 can be written as (Duarte 2003)

The same result can be obtained from

which leads to

However, since 〈1 | 1〉 and 〈2 | 2〉 illuminate x′ rather than x, the probability amplitude, for this geometry, reduces to that given in Equation 11.10.

A prismatic Mach–Zehnder interferometer is illustrated in Figure 11.4. In this prismatic version of the Mach–Zehnder, there is asymmetry in regard to the intra-interferometric beam dimensions. The P₁ – M₂ – P₂ beam is expanded relatively to the P₁ – M₁ – P₂ beam. Also, in this particular example (based on a prism with a magnification of k_1,1 ≈ 5), there is also a power asymmetry since the unexpanded beam propagating in the P₁ – M₁ – P₂ arm has about 30% of the incident power, whereas the expanded beam P₁ – M₂ – P₂ carries the remaining 70% of the incident power, for light polarized parallel to the plane of incidence. In this regard, it should be possible to design a prismatic Mach–Zehnder where the power density (W m⁻²) in each arm is balanced. Applications for this type of interferometer include imaging and microscopy. Additional Mach–Zehnder interferometric configurations include transmission gratings as BS (Steel 1967).

11.2.3 The Michelson Interferometer

The Michelson interferometer (Michelson 1927) is illustrated in Figure 11.5. In this interferometer, the incident light beam is divided into two sub-beams by a BS that serves as both input and output elements. The reflected beam, on the incidence BS, is then sent into a path defined by the reflection on M₁ and back toward the exit BS. The transmitted beam, on the incidence BS, is sent into a path defined by the reflection on M₂ and back toward the exit BS. Both beams are recombined interferometrically at the BS. For the Michelson interferometer, the interference can be characterized using a probability amplitude of the form:

which can be abstracted to

If j′ = 1 represents the function of the BS in the transmission mode and j = 2 in the reflection mode:

It is clear that substitution of the appropriate wave functions for the various terms in Equations 11.5, 11.9, and 11.14, and multiplication of these equations with their respective complex conjugates yield probability equations of an interferometric character. A variant of the Michelson interferometer uses retroreflectors (Steel 1967).

11.3.1 The Hanbury Brown–Twiss Interferometer

The Hanbury Brown–Twiss effect originates in interferometric measurements performed by an intensity interferometer used for astronomical observations (Hanbury Brown and Twiss, 1956). A diagram of the stellar intensity interferometer used to determine the diameter of stars is depicted in Figure 11.7. Feynman in one of his exercises to The Feynman Lectures on Physics (Feynman et al. 1965) explains that the electrical currents from the two detectors are mixed in a coincidence circuit where the currents become indistinguishable. Feynman then asks to show that the coincidence counting rate, in the Hanbury Brown–Twiss configuration, is proportional to an expression of the form:

where:

R₁ and R₂ are the distances from detector 1 and detector 2 to the source

Using the N-slit interferometric equation (Duarte 1991, 1993), that is, Equation 11.18, with N = 2, one immediately arrives at

and setting Ψ(r₁) = Ψ(r₂) = 1

Now, using (as suggested by Feynman) Ω₁ = kR₁ and Ω₂ = kR₂

which is the result given by Feynman in his problem book (Feynman 1965). From the measured signal distribution, and these equations, the angular spread of the emission can be determined, and knowing the distance from the source to the detector, it becomes possible to estimate the diameter of the aperture at the emission, in other words, the diameter of the star under observation.

Finally, since all the physics of the Hanbury Brown–Twiss interferometer follows from the generalized interferometric equation:

it can be easily seen that the experimental arrangement of the astronomical telescope should not be limited to just two parabolic mirrors and two detectors (N = 2), but can be extended to an array of N parabolic mirrors with their corresponding detectors.

11.3.2 The Fabry–Pérot interferometer

The second multiple-beam interferometer is the Fabry–Pérot interferometer depicted in Figure 11.8. This interferometer has already been introduced in Chapter 7 as an intracavity etalon. Generally, intracavity etalons are a solid slab of optical glass, or fused silica, with highly parallel surfaces coated to increase reflectivity (Figure 11.8b). These are also known as Fabry–Pérot etalons. Fabry–Pérot interferometers, however, are constituted by two separate slabs of optical flats with their inner surfaces coated as shown in Figure 11.8a. The space between the two coated surfaces is filled with air or other inert gas. The optical flats in a Fabry–Pérot interferometer are mounted on rigid metal bars, with a low thermal expansion coefficient, such as Invar. The plates can be moved, with micrometer precision or better, to vary the free spectral range (FSR). These interferometers are widely used to characterize and quantify the laser linewidth.

The mechanics of multiple-beam interferometry can be described in some detail, considering the multiple reflection, and refraction, of a beam incident on two parallel surfaces separated by a region of refractive index n as illustrated in Figure 11.9. In this configuration, at each point of reflection and refraction, a fraction of the beam, or a sub-beam, is transmitted toward the boundary region. Following propagation, these sub-beams interfere. In this regard, the physics is similar to that of the N-slit interferometer with the exception that each parallel beam has less intensity due to the increasing number of reflections. Here, for transmission, interference can be described using a series of probability amplitudes representing the events depicted in Figure 11.9

where:

j is at the reflection surface of incidence

k is immediately next to the surface of reflection

l is at the second surface of reflection

m is immediately next to the second surface of reflection as illustrated in Figure 11.9

The problem can be simplified considerably if the incident beam is considered as a narrow beam incident at a single point j. Propagation of the single beam then proceeds to l and is represented by the incidence amplitude A_i, which is a complex number, attenuated by the transmission factor t, so that the first three probability amplitudes can be represented by an expression of the form:

and Equation 11.23 reduces to

which, using the notation of Born and Wolf (1999), can be expressed as

defining

and

and taking the limit as p → ∞, Equation 11.25 reduces to (Born and Wolf 1999)

and multiplication with its complex conjugate yields an expression for the intensity

which is known as the Airy formula or Airy function. Plotting the ratio of the two intensities, as a function of

shows that the contrast of the fringes increases as the reflectivity increases (Born and Wolf 1999). From the geometry of Figure 11.9b, the path difference between the first reflected beam and the first beam that undergoes internal reflection, followed by refraction, is (see Problem 11.4)

which reduces to

Hence, using ΔL = mλ, the path difference can be expressed as

where:

n is the refractive index of the medium

d_e is the distance between the reflection surfaces

It also follows that the phase difference, using Equations 11.31 and 11.33, can be found to be

The device just described is an uncoated interferometer. If the two surfaces of the plate are coated with metal films of equal reflectivity, the phase term is modified so that (Born and Wolf 1999)

where:

ϕ represents a phase change

Under these circumstances, the multiple-beam interferometer is classified as a Fabry–Pérot etalon. An interferogram produced by the interaction of narrow-linewidth laser emission and a Fabry–Pérot etalon followed by a convex lens, of focal length f, is shown in Figure 11.10.

Using δ = 2πm in Equation 11.36, we get

Here, the brightest central ring corresponds to the maximum value of m₀, which is

which can be expressed as

where:

ε is the fractional fringe order

Using these equations, the diameter of the rings is given by (Born and Wolf 1999)

where:

p = 1, 2, 3... enumerates the successive rings from the center

11.3.3 Design of Fabry–Pérot Etalons

The FSR of a Fabry–Pérot interferometer, or a Fabry–Pérot etalon, corresponds to the difference in wavelength of two adjacent orders. From Equation 11.34, for a small angle of incidence,

and for an infinitesimal wavelength difference, corresponding to two adjacent orders, Δm becomes

Since Δm = 1, the wavelength difference corresponds to

which has the same form of Δλ ≈ λ² / Δx derived in Chapter 3 from interferometric principles while discussing Heisenberg’s uncertainty principle.

Renaming the wavelength difference as the FSR, we can write

which in the frequency domain becomes

Here, the approximations δλ ≈ (λ₁ − λ₂) and λ² ≈ λ₁λ₂ are justified since λ >> Δλ.

The FSR corresponds to the separation of the rings in Figure 11.10 and a measure of the width of the rings determines the linewidth of the emission being observed. The minimum resolvable linewidth is given by

where:

ℱ is the effective finesse.

Thus, a Fabry–Pérot etalon with an FSR_e = 7.49 GHz and F = 50 provides discrimination down to Δν_FRS ≈ 150 MHz. The finesse is a function of the flatness of the surfaces (often in the λ/100−λ/50 range), the dimensions of the aperture, and the reflectivity of the surfaces. The effective finesse is given by (Meaburn 1976)

where:

ℱ_R, ℱ_F, and ℱ_A are the reflective, flatness, and aperture finesses, respectively

The reflective finesse is given by (Born and Wolf 1999)

11.4.1 Example

In Figure 11.12, the double-slit interferogram produced with narrow-linewidth emission from the 3s₂ - 2p₁₀ transition of a He–Ne laser, at λ ≈ 543.3 nm, is displayed. The visibility of this interferogram is calculated, using Equation 11.50, to be V ≈ 0.95. In Figure 11.13, the double-slit interferogram produced, under identical geometrical conditions, but with the emission from an electrically excited coherent organic semiconductor interferometric emitter (Duarte et al. 2005), at λ ≈ 540 nm,

is displayed. Here, the visibility is lower (V ≈ 0.90). A comparison between the two interferograms reveals that the second interferogram has slightly broader spatial features relative to the interferogram produced with illumination from the 3s₂ - 2p₁₀ transition of the He–Ne laser. The differences in spatial distributions between these two interferograms have been used to estimate the linewidth of the emission from the interferometric emitter (Duarte 2008).

11.5.1 Fabry–Pérot Wavelength Meters

For a multiple-beam interferometer, the transmission intensity is given by Equation 11.30, where, in reference to Figure 11.9, the phase term δ can be expressed by Equation 11.36, which depends on the reciprocal of the wavelength. Thus, recording of the transmission interferometric signal by a photodiode array, or charge-coupled device (CCD), yields information on the wavelength of the radiation being measured.

Wavelength meters based on Fabry–Pérot interferometers generally involve configurations with multiple etalons in parallel (Byer et al. 1977; Fischer et al. 1981; Konishi et al. 1981). In the multiple-etalon configuration depicted in Figure 11.15, each etalon has a different FSR that is compatible with the FSR and the finesse of the next etalon. For instance, Fischer et al. (1981) used three etalons with FSRs of 1000, 67, and 3.3 GHz. Briefly, the methodology of this measurement consists in the application of Equations 11.38 and 11.39 to determine λ with reduced uncertainty at each etalon. Using this approach, Fischer et al. (1981) reported the measurements of laser frequencies with an accuracy of 60 MHz.

A simple interferometric configuration that is widely used in the measurement of laser wavelengths is that of the Fizeau, or optical wedge, two-beam interferometer first introduced by Snyder (1977). In this configuration, the incident laser beam propagates on an axis that is at an angle to the optical axis of the digital detector (see Figure 11.16). Also, the incident beam illuminates a wide area of the interferometer using some beam expansion method in conjunction with a spatial filter. The interference of the two beams is recorded by a digital detector with a 25 μm resolution or better. The method employs two predetermined parameters: the angle of the wedge and the wedge separation at a reference position as shown in Figure 11.16b.

The spacing of the Fizeau fringes Δz provides an approximate value for the wavelength according to

Assuming a very small wedge angle,

and

so that (Demtröder 2008)

thus allowing the determination of the order of interference at a minimum. The method then compares, using a computer program, the periodicity of the interference pattern with that predetermined from two-beam interference intensity functions. As indicated in Equation 11.18, these interferometric functions depend on the cosine of a phase term that, in this case, depends on Δz, which in turn is a function of the wavelength and the geometry of the wedge. Thus, accurate calibration of the spacing of the wedge allows a determination of the wavelength compatible with the accuracy to which the angle α is known. This static two-beam interferometric approach offers a wide wavelength range and an accuracy better than two parts in 10⁶ (Gardner 1985) in very compact configurations.