1. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 1 Chap 6 Frequency analysis of optical imaging systems

2. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 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. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 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. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 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. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 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. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 6 This notational simplification yields a convolution equation (called amplitude convolution integral)

7. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 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. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 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. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 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. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 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. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 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. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 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. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 13 From the convolution equation, it is found that where the frequency spectra of the input and output are respectively expressed as and The Fourier transform of a object. The Fourier transform of an image. and the amplitude transfer function (ATF ) is the Fourier transform of space-invariant amplitude impulse response.

14. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 14 Since the impulse response is a scaled Fourier transform of the pupil function, we have For notational convenience, we set the constant and ignore the negative signs in the arguments of equal to unity applications of interest to us here have pupil functions that are symmetrical in x and y). Thus (almost all

15. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 15 6.2.2 Examples of amplitude transfer

16. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 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. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 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. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 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. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 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.