1. Course outline Maxwell Eqs., EM waves, wave-packets
Gaussian beams
Fourier optics, the lens, resolution
Geometrical optics, Snell’s law
Light-tissue interaction: scattering, absorption Fluorescence, photo dynamic therapy
Fundamentals of lasers
Lasers in medicine
Basics of light detection, cameras
Microscopy, contrast mechanism
Confocal microscopy
משואות מקסוול, גלים אלקטרומגנטים, חבילות גלים
קרניים גאוסיניות
אופטיקת פורייה, העדשה, הפרדה
אופטיקה גיאומטרית, חוק סנל
אינטראקציה אור-רקמה: פיזור, בליעה, פלואורסנציה, טיפול פוטו-דינמי
עקרונות לייזרים
לייזרים ברפואה
עקרונות גילוי אור, מצלמות
מיקרוסקופיה, ניגודיות
מיקרוסקופיה קונפוקלית

2. Fourier optics and imaging
Linear optical systems
Fresnel diffraction
Fraunhofer diffraction
The lens
Optical resolution

3. Light propagation - Intuition
Light from a point source propagates in spherical waves:
CCD array
Point source

4. The action of a lens
CCD array
clipping 
resolution
drop
Point source

5. Conventional microscopy
a Van-leeuwenhoek microscope

6. Can light interact with itself ?

7. Fourier optics
An arbitrary wave in free space may be analyzed as a superposition of plane waves.
If it is known how a linear optical system modifies plane waves, the principle of superposition can be used to determine the effect of the system on any arbitrary wave.

8. The linear-systems approach
The complex amplitudes in two planes normal to the optical (z) axis are regarded as the input and output of the system. A linear system may be characterized by:
its impulse response function - the response of the system to an impulse (a point) at the input.
its transfer function - the response to spatial harmonic functions.

9. Spatial harmonics
An arbitrary function f(t) may be analyzed as a sum of harmonic functions of different frequencies and complex amplitudes.
In two (spatial) dimensions:*
* Our definitions of temporal and spatial Fourier transforms differ in the sign of the exponent. The choice of this signs is arbitrary, as long as opposite signs are used in the Fourier and inverse Fourier transforms. With different signs in the spatial (2D) and temporal (1D) cases, the traveling wave exp[i(2t-kzz)] represents wave that moves in +z direction as time propagates.

10. A little on Fourier transform

11. Example 1
f(x,y)
|F(x,y)|2

12. Example 2
f(x,y)
|F(x,y)|2
Log (|F(x,y)|2)

13. Example 3
f(x,y)
|F(x,y)|2
Log (|F(x,y)|2)
P(x,y)

14. Example 4
f(x,y)
|F(x,y)|2
Log (|F(x,y)|2)

15. Spatial harmonics & Plane waves
A: Complex amplitude
Consider a plane wave:
(t=0, monochromatic)
with a wavevector
This wave propagates at angles:
x = 0 means that there is no component of k in the x-axis (kx=0).
At z=0:
We also know (previous slides) that
U is comprised of spatial harmonics:

16. Spatial harmonics & Plane waves
A single plane wave:
A single spatial harmonic:
(!)
→ Spatial frequency [cycles/mm]
→ Optical frequency [cycles/sec]
The angles of inclination of the wavevector are then directly proportional to the spatial frequencies of the corresponding harmonic function. Apparently, there is a one-to-one correspondence between the plane wave U(x,y,z) and the harmonic function f(x,y).

17. Spatial harmonics & Plane waves
plane wave U(x,y,z)  harmonic function f(x,y)
Given one, the other can be readily determined, provided the wavelength  is known:
- The harmonic function f(x,y) is obtained by sampling at the z0 plane, f(x,y) = U(x,y,z0).
- Given the harmonic function f(x,y), on the other hand, the wave U(x,y,z) is constructed by using the relation:
With:
A condition for the validity of this correspondence is that kz is real ( ). This condition implies that x < 1 and y < 1, so that the angles x and y exist.
The  signs represent waves traveling in the forward and backward directions. We shall be concerned with forward waves only.

18. Spatial-spectral analysis
With a single spatial frequency:
If the transmittance of the optical element f(x,y) is the sum of several harmonic functions of different spatial frequencies, the transmitted optical wave is also the sum of an equal number of plane waves dispersed into different directions. The amplitude of each wave is proportional to the amplitude of the corresponding harmonic component of f(x,y).

19. Spatial-spectral analysis
Mathematically, if f(x,y) is a superposition integral of harmonic functions,
with frequencies x and y, and amplitudes F(x,y), the transmitted wave U(x,y,z) is the superposition of plane waves,
with complex envelopes F(x ,y) and
For any z:

20. Spatial-spectral analysis
A thin optical element of complex amplitude transmittance f(x,y) decomposes an incident plane wave into many plane waves. Each wave travels at angles x = sin-1(x) and y = sin-1(y) and has a complex envelope F(x,y).
This process of "spatial spectral analysis" is analogous to the angular dispersion of different temporal-frequency components (wavelengths) by a prism. Free-space propagation serves as "spatial prism“, sensitive to the spatial rather than temporal frequencies of the waves.

21. The transfer function input plane output plane
We regard f(x,y) and g(x,y) as the input and output of a linear system. The system is linear since the Helmholtz equation, which U(x,y,z) must satisfy, is linear.

22. Linear systems Impulse response Transfer function
Shift-invariant system: Invariance of free space to displacement of the coordinate system.
A linear shift-invariant system is characterized by its impulse response function h(x,y) or by its transfer function H(x,y).
Impulse response
Transfer function

23. Transfer function of free space
consider a single harmonic input function,
which corresponds to a plane wave:
where:
After propagating a distance d:
Thus transfer function of free space:

24. Transfer function of free space
The (complex) transfer function of free space:
sphere

25. Fourier optics and imaging
Linear optical systems
Fresnel diffraction
Fraunhofer diffraction
The lens
Optical resolution

26. Fresnel approximation
If the input function f(x,y) contains only spatial frequencies that are much smaller than the cutoff frequency -1, so that
the plane wave components of the propagating light then make small angles xx and yy , corresponding to paraxial rays. * *
parabola
sphere

27. Input-output relation (Fresnel)
Given the input function f(x,y), the output function g(x,y) may be determined as follows:
1. we determine the Fourier transform
2. the product H(x,y) F(x,y) gives the complex envelopes of the plane wave components in the output plane.
3. the complex amplitude in the output plane is the sum of the contributions of these plane waves:
Using the Fresnel approximation for H(x,y), we have:

28. Impulse response of free space
The impulse response function h(x,y) of the system of free-space propagation is the response g(x,y) when the input f(x,y) is a point at the origin (0,0).
which is the inverse Fourier transform of (exercise):
Thus, each point in the input plane generates a paraboloidal wave; all such waves are superimposed at the output plane.

29. Free space propagation as a convolution (Fresnel)
Knowing h, we can regard f(x,y) as a superposition of different points (delta functions), each producing a paraboloidal wave.
The wave originating at the point (x’,y’) has an amplitude f(x’,y’) and is centered about (x’,y’) so that it generates a wave with amplitude f(x’,y’)h(x-x’,y-y’) at the point (x,y) in the output plane. The sum of these contributions is the two-dimensional convolution:
which in the Fresnel approximation becomes:
Unlike previous derivation of g from f, here no Fourier transform is involved.

30. Convolution

31. Fresnel approximation: summary
Within the Fresnel approximation, there are two approaches for determining the complex amplitude g(x,y) in the output plane, given the complex amplitude f(x,y) in the input plane:
1. space-domain approach in which the input wave is expanded in terms of paraboloidal elementary waves.
2. frequency-domain approach in which the input wave is expanded as a sum of plane waves:

32. Fourier optics and imaging
Linear optical systems
Fresnel diffraction
Fraunhofer diffraction
The lens
Optical resolution

33. Fraunhofer approximation
Start with the space-domain approach of Fresnel approximation:
If f(x,y) is confined to a small area of radius b, and if the distance d is sufficiently large so that b2/d is small, then the phase factor is negligible:

34. Fraunhofer approximation
In the Fraunhofer approximation, the complex amplitude g(x,y) of a wave of wavelength  in the z=d plane is proportional to the Fourier transform F(x,y) of the complex amplitude f(x,y) in the z=0 plane, evaluated at the spatial frequencies x =x/d and y =y/d.
The approximation is valid if f(x,y) at the input plane is confined to a circle of radius b satisfying:
“Fraunhofer region”
Fresnel region 2b
Fraunhofer region

35. Fraunhofer – FT in the far-field
One slit. The width of the silts is varied.
Two slits. The width of the silts is constant and the distance between them is varied.

36. Periodic objects - gratings
G: lines per meter 2b
1/(2d2)
1/(42d2) x
-dG
dG

38. Sarcomere contractions
Sarcomeres are multi-protein complexes composed of different filament systems.

39. Sarcomere contractions
-dG
dG x d
Helium-Neon laser  = 632 nm

40. Fourier optics and imaging
Linear optical systems
Fresnel diffraction
Fraunhofer diffraction
The lens
Optical resolution

41. Angular spectrum - definition
Definition: The angular spectrum A(x,y,z) of a wave U(x,y,z) emerging from an object:
AU FT pair:
Thus A(x,y,z) is simply the equivalent for the Fourier transform F of the object f(x,y):
Angular spectrum
Fourier transform of the object
Complex wave
Object complex transmission

42. Propagation of angular spectrum
Substitute into Helmholtz equation
And executing the derivatives of the x and y coordinates gives:
= 0
Which yields a differential equation:
with a solution:
Fresnel approximation:
Propagation of the angular spectrum (Fresnel approximation)

43. Phase transformation with a thin lens
From straight-forward geometrical considerations: t0 n R1 R2
Lens thickness
Paraxial approximation
R >> lens diameter:
Lensmaker’s equation for a thin lens in air:
The phase transformation  (x,y) by the lens is given by (k=2/):
glass
air
The pupil function:
1, inside the aperture
0, otherwise
P(x,y) =
Therefore, the phase transformation of a perfect thin lens is:

44. Fraunhofer diffraction by a lens
The field after a (thin) lens (neglecting the exp(iknt0) term): f
Using the Fresnel integral:
Lens  Fraunhofer diffraction of U(x,y,0-) multiplied by the pupil function

45. Fourier transform with a lensd1 f
Assume U(x,y,0-) has an extent less than P(x,y):
Reminders:
Propagation of the angular spectrum from -d1 to 0-:
The field at the focal plane of the lens is the 2D Fourier transform of the field at z = -f

46. Fourier transform with a lens
Fourier transform property of a lens:
The complex amplitude of light at a point (x,y) in the back focal plane of a lens of focal length f is proportional to the Fourier transform of the complex amplitude in the front focal plane evaluated at the frequencies x/λf, y/λf. This relation is valid in the Fresnel approximation. Without the lens, the Fourier transformation is obtained only in the Fraunhofer approximation, which is more restrictive.

47. Image formation with a lens
Assume a positive, aberration-free thin lens and monochromatic light.
Free space propagation as a convolution (Fresnel): d1 di
To find h, we replace f(x’,y’)  U(x,y,-d1) (x-x1,y-y1,-d1):

48. Image formation with a lens=0 d1 di
The lens law
In this case, the impulse response becomes
Magnification
Neglect P(x,y):
Imaging by a lens:
Inversion
Magnification

49. Imaging examples F F f f f f f f f f f f f f f f f1 f2=2f f1 f1 f2 f2
The lens law F f f f f F
Magnification F F f f f f f f F f f f f F f1
f2=2f F F f1 f1 f2 f2

50. The 4-f system

51. Imaging examples f f f f
Object
Mask
Image (inverted)

52. Fourier optics and imaging
Linear optical systems
Fresnel diffraction
Fraunhofer diffraction
The lens
Optical resolution

53. Point-spread function (PSF)
PSF ↔ Impulse response
Diffraction limited (also “Fourier limited”) system:
Perfect spherical wave
Point object
Image

54. Point-spread function (PSF)
PSF ↔ Impulse response d1 di
Consider:
An impulse at x1=y1=0
An imaging condition:
Circular pupil function:
(D - lens diameter)
2D Fourier transform of the scaled pupil
x’=x/λdi
(neglect constants)

55. PSF yi  xi
“Airy disk”

56. Airy disk of microscope objectives
In photography: “angular aperture” or “f-number” D f n θ

57. Gaussian beams - properties
Reminder
Gaussian beams - properties
Beam divergence
NA for Gaussian beams
Thus the total angle is given by

58. Resolution
The final image is a convolution of the perfect image with the system’s impulse response:

59. Convolution

60. 2D convolution examples

61. Quantifying resolution The Rayleigh criterion

62. Rayleigh criterion
“Two point sources are just resolved if they have an angular separation equal to the angular radius of the Airy disk.”
For an ideal lens:
For a microscope objective:

63. Effect of noise on resolution

64. Summary
An arbitrary wave may be analyzed as a superposition of plane waves.
U(x,y,0)=f(x,y)e-ikz can be represented as a combination of spatial harmonics:
Fresnel approximation
Propagation of the angular spectrum
Fraunhofer approximation
At its focal plane a lens performs a Fourier transform of the incoming field
The 4-f system allows Fourier domain image manipulations
The PSF of a lens is limited by its pupil function