1. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 1 Chap 5 Wave-optics Analysis of Coherent Systems

2. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 2 Outline 5.1 A thin lens as a phase transformation 5.2 Lens as a Fourier transform 5.3 Image formation ： monochromatic illumination

3. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 3 5.1 A thin lens as a phase transformation

4. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 4 The total phase delay suffered by the wave at coordinates in passing through the lens may be written Where n is the refractive index of the lens material. is the phase delay introduced by the lens. is the phase delay introduced by the remaining region of free space between the two planes.

5. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 5 Equivalently the lens may be represented by a multiplicative phase transformation of the form The complex field across a plane immediately behind the lens is then related to the complex field incident on a plane immediately in front of the lens by

6. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 6

7. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 7 Total thickness function as the sum of three individual thickness functions The thickness function is given by

8. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 8 The second component of the thickness function comes from a region of glasses of constant thickness. The third component is given by

9. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 9 The total thickness is seen to be where

10. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 10 We consider only values of x and y sufficiently small to allow the following approximations to be accurate With the help of these approximation, the thickness function becomes

11. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 11 Substitution into yields the following approximation to the lens transformations ： Note ： For a thin lens

12. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 12 The physical properties of the lens (that is, n, and ) can be combined in a single number f called the focal length, which is defined by

13. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 13 Neglecting the constant phase factor, which we shall drop hereafter, the phase transformation may now be written This equation will serve as our basic representation of the effects of a thin lens on an incident disturbance.

14. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 14 5.2 Lens as a fourier transform

15. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 15 Pupil function ：

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

17. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 17 is similar to where is called Fraunhofer (or far-field) diffraction formula (or FT of )

18. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 18 Recall Fresnel diffraction formula

19. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 19 is neglected.

20. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 20 Lens as a Fourier transformer ：

21. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 21 5.3 Image formation ： monochromatic illumination

22. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 22 By the shifting property, we see

23. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 23

24. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 24 To find the impulse response (or called point spread function) h, let and substituting it to gives

25. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 25 After passage through the lens (focal length f), the filed distribution becomes

26. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 26 Finally, using the Fresnel diffraction equation to account for propagation over distance, we have

27. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 27 Where constant phase factors have been dropped. Combining ， and again neglecting a pure phase factor, yields the formidable result

28. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 28 Let the magnification (Ref. geometrical optics), we see

29. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 29 As it currently stands, the impulse response is that of a linear space-variant system, so the object and image are related by a superposition integral but not by a convolution integral. This space-variant attribute is a direct result of the magnification and image inversion that occur in the imaging operation. To reduce the object-image relation to a convolution equation, we must normalize object-plane variables be introduced ：

30. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 30 in which case the impulse response of reduces to

31. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 31 which depends only on the differences of coordinates A final set of coordinate normalizations simplifies the results even further. Let

32. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 32 Then the object-image relationship becomes or Where is the geometrical-optics prediction of the image. (convolution form)

33. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 33 There are two main conclusions from the analysis and discussion above. The ideal image produced by a diffraction-limited optical system (i.e. a system that is free from aberrations) is a scaled and inverted version of the object. The effect of diffraction is to convolve that ideal image with the Fraunhofer diffraction pattern of the lens pupil.