Файл: Ersoy O.K. Diffraction, Fourier optics, and imaging (Wiley, 2006)(ISBN 0471238163)(427s) PEo .pdf

where the last factor is the first two terms of the Taylor series expansion of e jyðx;yÞ.

Hence, no image is observable. We note that Eq. (9.5-3) is true because the first term of unity due to undiffracted light is in phase quadrature with jyðx; yÞ generated by the diffracted light. In order to circumvent this problem, a phase plate yielding p=2 or 3p=2 phase shift with the undiffracted light can be used. This is usually achieved by using a glass substrate with a transparent dielectric dot giving p=2 or 3p=2 phase shift, typically by controlling thickness at the focal point of the imaging system. The undiffracted light passes through the focal point whereas the diffracted light from the specimen is spread away from the focal point since it has high spatial frequencies. On the imaging plane, the intensity can now be written as

with 3p=2 phase shift. Equations (9.5-4) and (9.5-5) are referred to as positive and negative phase contrast, respectively. In either case, the phase variation yðx; yÞ is observable as an image since it is converted into intensity. An example of imaging with phase contrast microscope in comparison to a regular microscope is shown in Figure 9.7.

Figure 9.7. (a) Image of a specimen with a regular microscope, (b) image of the same specimen with a phase contrast microscope [Nikon].

Figure 9.9. A confocal optical microscope system.

can be written as

where Pðx1; y1Þ is the pupil function of the lens system.

In confocal microscopy, since one object point is imaged at a time, Uðx1; y1Þ can be written as

Hence,

¼ h x0 ; y0 M M

The intensity of the point image becomes

Assuming a circularly symmetric pupil function of radius a, Pðx1; y1Þ can be written as

146 FOURIER TRANSFORMS AND IMAGING WITH COHERENT OPTICAL SYSTEMS

where

PðrÞ is the cylinder function studied in Example 2.6. The impulse response function becomes the Hankel transform of PðrÞ, and is given by

In practice, there are typically two lenses used, as seen in Figure 9.9. They are called objective and collector lenses. Suppose that the two lenses have impulse response functions h1 and h2, respectively. By repeated application of Eq. (9.7-1), it can be shown that the image intensity can be written as

If the lenses are equal, with impulse response h given by Eq. (9.7-7), the output intensity can be written as

EXAMPLE 9.1 Determine the transfer function of the two-lens scanning confocal microscope.

Solution: Since ht ¼ h1h2, we have

Htðfx; fyÞ ¼ H1ðfx; fyÞ H2ðfx; fyÞ

where, for two equal lenses,

H1ðfx; fyÞ ¼ H2ðfx; fyÞ ¼ Pðld0x1; ld0y1Þ

If the pupil function P is circularly symmetric, Htðfx; fyÞ is also symmetric, and can be written as

Htðfx; fyÞ ¼ HtðrÞ

where

q r ¼ fx2 þ fy2

Hence,

HtðrÞ ¼ H1ðrÞ H2ðrÞ

where

(

1 0 r 1

PðrÞ ¼

0otherwise

9.7OPERATOR ALGEBRA FOR COMPLEX OPTICAL SYSTEMS

The results derived in Sections 9.1–9.5 can be combined together in an operator algebra to analyze complex optical systems with coherent illumination [Goodman]. For simple results to be obtained, it is necessary to assume that the results are valid within the paraxial approximations involved in Fresnel diffraction and geometrical optics.

An operator algebra is discussed below in which the apertures are assumed to be separable in rectangular coordinates so that 2-D systems are discussed in terms of 1-D systems.

The operators involve fundamental operations that occur within a complex optical system. If U(x) is the current field, OðuÞ½U1ðxÞ& will denote the transformation