The size (spatial coherence) and monochromaticity (temporal coherence) of the light

source must satisfy certain minimum requirements that depend on the geometry of

the system, as described by Hansen (1955, 1984) and by Birch (1979). It is interesting

to know that if the optical element under test has very steep reflections, the state of

polarization of the light may change in the reflection, introducing changes in the

contrast (Ferguson, 1982). However, in most of the cases, the important factor in the

contrast is the coherence of the light source.

**2.3.1. Spatial Coherence**

The light source for interferometry must satisfy some minimum requirements of

spatial as well as temporal coherence, depending on the interferometer configuration

and the specific application and needs. As described in Chapter 1, Section 1.1.2.,

some gas or vapor lamps can be used in conjunction with a small pinhole to

illuminate an interferometer. These lamps with the pinhole do not have perfect

spatial and temporal coherence. A gas laser, however, has perfect spatial coherence

and can have almost perfect temporal coherence. We might think at first that this is

the ideal light source for interferometry, but this is not always the case. The

coherence length is, in general, so large that many unwanted reflections from other

surfaces in the optical system may produce a lot of spurious fringes in addition to the

speckle noise that make it difficult to analyze the interferogram. On the contrary, the

perfect spatial coherence produce scattering spherical waves from many unavoidable

small pieces of dust or scratches on the optical elements, which in turn produce many

spurious rings of fringes. This problem has been studied in detail by Schwider (1999).

The conclusion is that, quite frequently, it is a better option to use a gas or vapor

light source instead of a laser. However, if the optical path difference is large, it is

unavoidable to use a gas laser. In this section we will study the coherence requirements

for the light source.

There are two cases for which the collimated wavefront has ray lights spread over

a solid angle with diameter 2*θ*, and hence the final accuracy of the interferometry or

the contrast will be reduced:

(1) The collimator has spherical aberration, in which

where *TA* is the maximum value of the transverse spherical aberration of the

collimator at its best focus position. This aberration might limit the accuracy of

the interferometer unless the OPD remains constant with changes in the angle *θ*.

Otherwise, given the maximum value of *θ*, the maximum change in the OPD should

be smaller than the desired accuracy.

(2) The light source is not a mathematical point but has a small diameter 2a; then

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where *f* is the focal length of the collimator.

Fringes with high contrast are obtained, using an extended thermal source, only if

the OPDs for the two paths from any point of the source with different value of *θ*

differ by an amount smaller than λ/4 according to the Rayleigh criterion. On the

contrary, radiometric considerations usually require as large a source as possible that

will not degrade the contrast of the fringes.

When the beam splitter is a glass plate and is not compensated by another identical

glass plate, we may show that the maximum light source size has an elliptical shape.

This is the reason why the fringes are elliptical in an uncompensated Michelson

interferometer. The shape and the size of the ellipse not only are functions of *φ*_{0}, *θ*,

and *ψ* but also depend very critically on *t*_{0}.

The simpler case of a glass plate with its normal along the optical axis can be

analyzed with more detail as will be shown. The OPD is given by Eq. (2.7). As shown

in Figure 2.12, the value of the OPD changes with the value of *θ* depending on the

value of *t*_{0}. The maximum allowed value of the angular semidiameter *θ* of the light

source as seen from the collimator is that which gives a variation of the OPD equal to

λ/4. On the contrary, the maximum allowed value of the angle *θ* due to spherical

aberration of the collimator is that which gives a variation of the OPD equal to the

accuracy desired from the interferometer.

When testing small optics using a nonmonochromatic light source, the optical

path difference can be adjusted to be zero. Then, it is convenient to choose

so that OPD(0^{o}) = 0, but this situation will require an even smaller light source. It

should be pointed out that when testing large optics, the value of *t*_{0} cannot be changed

at will, because in general it will be very large.

If an extended quasi-monochromatic light is used, a good condition in order to

make the optical path difference insensitive to the angle *θ* is

yielding

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It is interesting to see that this equation is equivalent to the condition that the

apparent distance of the image of the collimator (or the light source in a Michelson

interferometer) to the observer is the same for both arms of the interferometer. This

condition seems reasonable if we consider that then the angular size of the two

images of the light source is the same as pointed out by Steel (1962) and Slevogt

(1954).

When the light source is extended and the interferometer is compensated in this

manner, the fringes are localized at a certain plane in space. To find this plane, the

system may be unfolded as studied by Hansen (1942, 1955). For an interferometer

with plane mirrors, this location for the fringes is near the plane mirrors because of

the way the image of the light source moves when one of the mirrors is tilted in order

to obtain the fringes as shown in Figure 2.5. Thus, the viewing system must be

focused near the mirrors to see the fringes.

As it will be described later in this chapter, to test an optical system, one of the

plane mirrors is replaced by the system to be tested, plus some auxiliary optics to

send back a collimated beam to the interferometer, just like one single mirror would

do. The returning collimated beam has to have the same diameter. Thus, it is easy to

conclude that this whole system, including the element under test has the following

general characteristics:

- It is afocal.
- Its magnification is either one or minus one. If it is minus one, the returning

wavefront will be rotated with respect to the incident wavefront. - The system is symmetric and hence it is always free of coma.
- Entrance and the exit pupil are symmetrically placed with respect to the

system and have the same diameter.

With these properties we see that since the system is reflective (it is retroreflector

only if the magnification is minus one), the entrance and exit pupil are at the same

plane. An important conclusion is that the fringes should be observed at this entrance

and exit pupil plane. This problem has been studied with detail by Schwider and

Falkenstorfer (1995).

It should be noticed that the entrance pupil of the whole system is not necessarily

the same as the pupil of the lens under test. However, when testing a lens, the fringes

are to be observed at the pupil, which ideally should be the same. This does not

happen with a single mirror; therefore, the mirror should be as close as possible to the

lens. This is the reason why a convex mirror with the longest possible radius of

curvature is desirable (Steel, 1966) when testing telescope objectives. On the contrary,

the entrance pupil of a microscope objective is at infinity; hence, the exit pupil

is at the back focus. Dyson (1959) described an optical system such as the one to be

described in Chapter 12, which images the mirror surface on the back focus of the

microscope objective, where the fringes are desired.

The limitation on the size of a pinhole source was examined in a slightly different

manner by Guild (1920–1921) as explained below. Imagine that the small source is

greatly enlarged to form an extended source. Then fit an eyepiece in front of lens L_{2}

(see Fig. 2.1) to form a telescope. Under these conditions, equal inclination fringes in

the form of concentric rings (like the ones normally observed in the Michelson

interferometer) are observed. If the mirrors are exactly perpendicular to their optical

axes, the rings will be exactly centered. The ideal size of the source is that which

allows only the central spot on the fringe system to be observed. The size of

the central spot increases when the OPD (*θ*) reduces its dependence on *θ* by one

of the adjustments described above, making possible the use of a larger source,

although the effective size of the spot is then limited by the pupil of the observing eye

or the camera.

In all the foregoing considerations, the two interfering wavefronts are assumed to

have the same orientation, that is, without any rotations or reversals with respect to

each other. In other words, if one of the beams is rotated or reversed, the other should

also be rotated or reversed. A wavefront can be rotated 180^{o} by means of a cube

corner prism or a cat’s-eye retroreflector formed by a convergent lens and a flat

mirror at its focus. The wavefront can be reversed upon reflection on a system of two

mutually perpendicular flat surfaces, e.g., in a Porro prism. Murty (1964) showed that

if one of the wavefronts is rotated or reversed with respect to the other, then, to have

fringes with good contrast and without phase shifts, the pinhole diameter 2α should

satisfy the condition

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so that diametrically opposite points over the wavefront are coherent to each other.

Here, *f *and *D* are the collimator’s focal length and diameter, respectively. Then 2α

is extremely small and therefore an impractical size for some sources. However,

there is no problem if a gas laser is used, because its radiance and spatial

coherence are extremely high. This subject will be examined with more detail

in Chapter 5.

When testing an optical element as it will be described in the following section,

the wavefront is sometimes inverted (up-down) or reversed (left-right) or rotated

(both), which is equivalent to a rotation of one of the wavefronts by 180^{o}. Then, the

spatial coherence requirements increase. If a laser is used, no problem arises. If a gas

or vapor source is used, the reference wavefront has also to be inversed, reversed, or

rotated, like the wavefront under test.

When there is no alternative but to use a gas laser source, due to a large optical

path difference, speckle noise and spurious fringes may be reduced by artificially

reducing the spatial coherence of the light a little. This is possible by placing a small

rotating ground glass disc on the plane of the pinhole as described by Murty and

Malacara (1965), Schwider and Falkenstorfer (1995), and Schwider (1999).

The size (spatial coherence) and monochromaticity (temporal coherence) of the light

source must satisfy certain minimum requirements that depend on the geometry of

the system, as described by Hansen (1955, 1984) and by Birch (1979). It is interesting

to know that if the optical element under test has very steep reflections, the state of

polarization of the light may change in the reflection, introducing changes in the

contrast (Ferguson, 1982). However, in most of the cases, the important factor in the

contrast is the coherence of the light source.

**2.3.1. Spatial Coherence**

The light source for interferometry must satisfy some minimum requirements of

spatial as well as temporal coherence, depending on the interferometer configuration

and the specific application and needs. As described in Chapter 1, Section 1.1.2.,

some gas or vapor lamps can be used in conjunction with a small pinhole to

illuminate an interferometer. These lamps with the pinhole do not have perfect

spatial and temporal coherence. A gas laser, however, has perfect spatial coherence

and can have almost perfect temporal coherence. We might think at first that this is

the ideal light source for interferometry, but this is not always the case. The

coherence length is, in...

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