Chap 6 Frequency analysis of optical imaging systems

1. Chap 6 Frequency analysis of optical imaging systems

2. Outline • 6.1 Generalized treatment of imaging systems • 6.2 Frequency response for diffraction-limited coherent image • 6.3 Frequency response for diffraction-limited incoherent image

3. 6.1.1 A Generalized Model • To specify the properties of the lens system, we adopt the point of view that all image

4. 6.1.2 Effects of diffraction on the image • Diffraction effect plays a role only during passage of light from the object to the entrance pupil, or alternatively and equivalently, form the exit pupil to the image. • There are two points of view that regard image resolution as limited by • (1) The finite entrance pupil • (2) The finite exit pupil

5. According to the Abbe theory, only a certain portion of the diffracted components generated by a complicated object are intercepted by this finite pupil. The components not intercepted are precisely those generated by the high-frequency component of the object amplitude transmittance.

6. This notational simplification yields a convolution equation (called amplitude convolution integral)

7. where the ideal image (i.e., the geometrical-optics prediction of the image) for a perfect imaging system is expressed as and the impulse response (called amplitude impulse response) is given by where the pupil function P is unity inside and outside the projected aperture.

8. Thus, for a diffraction-limited system, we can regard the image as being a convolution of the image predicted by geometrical optics with an impulse response that is the Fraunhofer diffraction pattern of the exit pupil (i.e., the Fourier transform of the exit pupil)

9. 6.1.3 Polychromatic illumination: the coherent and incoherent cases • When the object illumination is coherent, the various impulse responses in the image plane vary in unison, and they must be added on a complex amplitude basis. Therefore, a coherent imaging system is linear in complex amplitude. • It follows that an incoherent imaging system is linear in intensity and the impulse response of such a system (called intensity impulse response) is the squared magnitude of the amplitude impulse response.

10. An incoherent imaging system is linear in intensity and the impulse response of such a system (called intensity impulse response) is the squared magnitude of the amplitude impulse response. Thus, for incoherent illumination, the image intensity is found as a convolution of the intensity impulse response with the ideal image intensity .

11. 6.2 Frequency response for diffraction-limited • As emphasized previously, a coherent imaging is linear in complex amplitude. This implies, of course, that such a system provides a highly nonlinear intensity mapping. If frequency analysis is to be applied in its usual form, it must be applied to the linear amplitude.

12. 6.2.1 The amplitude transfer function • In a coherent system, a space-invariant form of the amplitude mapping is given from the manipulation of convolution. One can anticipate, then the transfer-function concepts can be applied to the system, provided it is the convolution is done on an amplitude basis.

13. From the convolution equation, it is found that where the frequency spectra of the input and output are respectively expressed as The Fourier transform of a object. and The Fourier transform of an image. and the amplitude transfer function (ATF) is the Fourier transform of space-invariant amplitude impulse response.

14. Since the impulse response is a scaled Fourier transform of the pupil function, we have For notational convenience, we set the constant equal to unity and ignore the negative signs in the arguments of (almost all applications of interest to us here have pupil functions that are symmetrical in x and y). Thus

15. 6.2.2 Examples of amplitude transfer

16. 6.3.1 The optical transfer function (OTF) • Recall the imaging systems that use incoherent illumination have been seen to obey the intensity convolution integral • Such systems should therefore be frequency-analysis as linear mappings of intensity distributions.

17. Application of the convolution theorem to the above equation then yields the frequency-domain relation where the normalized frequency spectra of and are respectively defined by and

18. And the normalized transfer function of the system is similarly defined by By international agreement, the function is known as the optical transfer function (OTF) of the system and it is also the normalized autocorrelation of the amplitude transfer function. Its modulus is known as the modulation transfer function (MTF). Where PTF is phase transfer function.

19. 6.3.2 General properties of the OTF • The most important of these properties are as follows: • Property 1 follow directly by substitution of • The proof of Property 2 is left as an exercise for the reader, it being no more than a statement that the Fourier transform of a real function has Hermitian symmetry. • To proof Property 3 we use Schwarz’s inequality.