COHERENCE PROPERTIES IN THE IMAGE OF A PARTIALLY COHERENT OBJECT

(a) Introduction — Intuitive Ideas

In many cases, the coupling efficiency of an incoherent source into a multimode optical fibre is increased by imaging it onto the

entrance face of the fibre core. This is only advantageous if the fibre core radius is larger than that of the source. It is usual in such cases to assume that as long as the numerical aperture of the fibre is

filled with the incident light, all the bound modes of the fibre are excited (e.g. [18]). Under certain circumstances, this assumption may not be valid (cf. Chapter V), for example, when the primary source is not highly incoherent.

The degree of coherence in the image of a partially coherent source is an important parameter in many optical systems other than optical fibre excitation systems, e.g. microscopy [16]. The subject has received some attention in terms of the propagation of the coherence properties of a field through an optical system (e.g. [7]). Here it is the degree of spatial coherence in the image of a partially coherent object that is investigated [19].

In the image plane of a lens, the smallest areas of coherence, assuming a totally incoherent object, have radius of coherence FL =r^, given by the Airy disc radius (see [17], chapter VIII). This represents the minimum image coherence radius possible. When the object is

a diffraction limited system, "the coherence is propagated according to the laws of geometric optics" [20]. Thus, noting the relationship between image and object planes (conjugate planes), and using the

results in [20], the radius of coherence in the image of a partially coherent source is given by

R . = MR , MR > r ,

l o o A

MR < r ,

o A

3.1

where M is the transverse magnification of the system and R is the o

radius of coherence of the object. This law is drawn in figure 7.4.

The validity of this result requires examination, and so the intuitive ideas about partially coherent fields and radiance from Chapters II and III are used to provide an initial proof. A more rigorous Fourier optics proof is then developed. In both proofs the finite aperture of the lens must be taken into account as this

determines which of the components of the source field (object) reach the image plane.

(b) Geometric Optics Proof

Following the earlier definitions (Chapters II and III), the partially coherent source field (object) has a characteristic coherence angle defined by

3.2

(Throughout this analysis, the small angle approximation is made, i.e. sin 0 ~ 0 .)

Similarly, in the image plane

7.3 165

r

9*

Fig. 7.4: The image radius of coherence, R^ , as a function of the object radius of coherence, Rq, for a simple lens system. M is the transverse magnification and rA is the lens aperture Airy disc radius in the image plane.

(a) Qco < 0£o , where is the numerical aperture (NA) of the lens in the object space. The magnification of the lens then gives

0_co

M 3.4

Then (3.2), (3.3) and (3.4) combine to give

R. - MR

l <

3.5 o

as predicted (both 0 and 0^ are defined in the object space medium). 3.6

Then

0

Fig. 7.5: The geometry of the lens system. M = d^/dQ is the magnification. The axis of the system is taken to be the

transverse z-axis.

However, the lens angle in the image space, 0-^, corresponds to the Airy disc angular radius in the image plane, so that

R. = r . 3.8

l A

Hence the rather informal ideas about partial coherence, derived from just the directivity of the radiance distribution of a partially coherent field, exactly confirm the very intuitive result embodied in (3.1). This kind of approach is necessarily limited to the more incoherent situations when the radius of coherence is small compared to the source radius (cf. Chapter II).

In terms of the plane wave ensemble representation of

partially coherent fields, discussed in Chapters II and III, it can be seen that the lens aperture affects the coherence of the image by reducing the maximum angle of the plane wave spectrum, 6 , if

0 > ®Zo‘ T*ie ma9n:’Lfication again gives 0 ^ = 0 ^ / M and the same result is obtained.

(c) Fourier Theory

For a more formal examination, the methods of Fourier optics

(e.g. [21]) are employed. The standard quasimonochromatic approach is

used and polarisation effects are ignored so that the scalar quantity U,

the complex amplitude, may be used to represent the optical field. This

m ay be considered as one component of a more general vector field

representation. The optical system is assumed to be isoplanatic and

then, following Goodman's notation (see [21], pp.95-6) with subscripts o and i for object and image variables as before,

U . (x.,y . )

l l i

r f X~ ^

h (x. - x , y . - y ) M 1 U ___ o

y

1 1

where the object coordinates x ,y give

o o

x = - Mx ,

o o y o = - Myo 3.10

in the image plane. The lens point spread function is

h ( x . ,y . )

l i

-2iTi (x .E, + y . g) /Ad .

p(£,n) e i i i d^dr) ^ _3.11

where P is the pupil function and d_^ is defined in figure 7.5.

The mutual intensity function T 12 is defined for the points R 'R 0 2 in t^ie object by

r.

12 —ol -o2 o -ol o -o2 3.12

where ( ) indicates a time average and * the complex conjugate.

These ideas may be used to obtain the mutual intensity function in the image plane, where, adopting a convenient vector notation to indicate position in both object and image planes,

If the object is assumed to be unpolarised, homogeneous, isotropic and statistically stationary the mutual intensity function is a function only of the separation between the two points 1 and 2, i.e.

IR I, thus