**3. Fundamentals of interferometry**

Interferometry is the technique of superposing two or more waves, to create an output wave that differs from the input waves [6]. An interferometer is an optical instrument that can measure small wavefront deformations with a high accuracy, of the order of a fraction of the wavelength. Two-beam interferometers produce an interferogram by superimposing two wavefronts, one of which is typically a flat reference wavefront and the other a distorted wavefront from the object, whose shape is to be measured. To study the main principles of interferometers, let us consider from Maxwell's equations that the electric field of a plane wave, with speed, , frequency, , and wavelength, , travelling in the *z*-direction, is given by

$$E(z,t) = \begin{pmatrix} E\_x \\ E\_y \end{pmatrix} e^{l(kz - av)}\tag{7}$$

where = 2 = 2/ is the circular frequency and = 2/ is the circular wave number. Assume that = 0, that is, the light is linearly polarized in the *x-*direction. At the location = 0, the electric field <sup>=</sup> cos. The intensity is given by the square of the amplitude, thus

$$E(z) = \langle E.E \rangle = (E\_{\times}^2) \langle \cos^2 \alpha t \rangle. \tag{8}$$

Consider a two-wave interferogram with flat wavefronts 1() and 2() reflected from the two mirrors of Michelson interferometer and combined at the detector. According to the principle of superposition, we can write

$$E(t) = E\_1(t) + E\_2(t). \tag{9}$$

Combining Eqs. (7)–(9), with some additional assumptions, gives finally

$$I = I\_1 + I\_2 + 2\sqrt{I\_1 I\_2} \cos\left(\frac{4\pi\Delta L}{\lambda}\right) \tag{10}$$

Eq. (10) is the essential equation of interference. Depending on the term 4/, the resultant intensity on a detector can have a minimum or a maximum, and it depends on the path difference or the wavelength. It is evident from Eq. (10) that the intensity has maxima for 4/ = 2, with = 0, ± 1, ± 2, ..., so that = /2 and minima for = ( + 0.5)/2. If the intensities *I*1 and *I*2 are equal, Eq. (10) reduces to

Several basic interferometric configurations are used in optical-testing procedures, but almost all of them are two-beam interferometers. In Section 3, we review fundamentals of interferometry with focus on two- and multiple-beam interferometers and its capability in featuring

Interferometry is the technique of superposing two or more waves, to create an output wave that differs from the input waves [6]. An interferometer is an optical instrument that can measure small wavefront deformations with a high accuracy, of the order of a fraction of the wavelength. Two-beam interferometers produce an interferogram by superimposing two wavefronts, one of which is typically a flat reference wavefront and the other a distorted wavefront from the object, whose shape is to be measured. To study the main principles of interferometers, let us consider from Maxwell's equations that the electric field of a plane wave, with speed, , frequency, , and wavelength, , travelling in the *z*-direction, is given by

> ( ) (,) - æ ö <sup>=</sup> ç ÷ ç ÷ è ø

*y E E zt e E*

where = 2 = 2/ is the circular frequency and = 2/ is the circular wave number.

= 0, the electric field <sup>=</sup> cos. The intensity is given by the square of the amplitude, thus

Consider a two-wave interferogram with flat wavefronts 1() and 2() reflected from the two mirrors of Michelson interferometer and combined at the detector. According to the

<sup>4</sup> 2 cosæ ö <sup>D</sup>

Eq. (10) is the essential equation of interference. Depending on the term 4/, the resultant intensity on a detector can have a minimum or a maximum, and it depends on the path

è ø

l

p

2 2 ( ) . ( ) cos . =á ñ= á ñ *E z EE E t <sup>x</sup>*

Combining Eqs. (7)–(9), with some additional assumptions, gives finally

1 2 12

=++ ç ÷

*<sup>L</sup> I I I II*

*x i kz t*

w

= 0, that is, the light is linearly polarized in the *x-*direction. At the location

w

(7)

(8)

(10)

1 2 *Et E t E t* ( ) ( ) ( ). = + (9)

the topography of surfaces.

88 Optical Interferometry

Assume that

**3. Fundamentals of interferometry**

principle of superposition, we can write

$$I = 2I\_{\rm i} \left( 1 + \cos \left( \frac{4 \pi \Delta L}{\lambda} \right) \right) = 4I\_{\rm i} \cos \left( \frac{2 \pi \Delta L}{\lambda} \right). \tag{11}$$

This means that the minimum intensity is zero and the maximum intensity is 4 <sup>1</sup>. Also, it is clear that if <sup>1</sup> or <sup>2</sup> are zero, the interference term vanishes and a constant intensity remains. The relative visibility, , of the interference can be defined as

$$V = \frac{I\_{\text{max}} - I\_{\text{min}}}{I\_{\text{max}} + I\_{\text{min}}} = \frac{2\sqrt{I\_1 I\_2}}{I\_1 + I\_2}. \tag{12}$$

If the magnitude of the optical path length between the two beams is greater than the temporal coherence length of the light source of the two beams, fringes will not be observed. As the OPD returns to zero, fringe visibility reaches a maximum. Temporal coherence <sup>=</sup> 2/ is inversely proportional to the spectral bandwidth of the light source of the two beams, where is the center wavelength and is the spectral bandwidth, measured at the full-width half maximum (FWHM) at short coherent length. It is worth mentioning that temporal coherence goes as the Fourier transform of the spectral distribution of the source. **Figure 8** shows two types of temporal coherence length: the left side is the long coherent length emitted from a

laser source (spectral bandwidth very small). The Fourier transform of a zero bandwidth source is a constant, so the temporal coherence is infinite. The right side is the short coherent length emitted from a femtosecond laser with FWHM around 30 μm.

**Figure 8.** Fringe visibility degradation due to temporal coherence can be improved by varying the OPD between the two beams.
