1. Fundamentals of biomedical optics and photonics
קורס
Fundamentals of biomedical optics and photonics
יסודות אופטיקה ופוטוניקה ביו-רפואית
מרצה: ד"ר דביר ילין, שעות קבלה: יום ג' 14:00-15:00
מתרגל: ליאור גולן, שעות קבלה יקבעו בהמשך.
ספר: Saleh & Teich, “Fundamentals of Photonics”, 2nd edition
היקף: נקודות זכות (שעתיים שיעור, שעתיים תרגול).
מבנה ציון: 80% בחינה, 20% תרגילים.
* ינתנו כחמישה תרגילים בסך הכל. כל התרגילים יבדקו, יוחזרו עם הערות, וינתנו ציונים בהתאם. תרגיל שלא יוגש במועד יקבל ציון אפס. לא יפורסמו פתרונות לתרגילים ולבחינות קודמות.
* הבחינה (כשלוש שעות) עם חומר פתוח מלבד ספרים ומחשבים ניידים, תכלול כארבע שאלות פתוחות, מתוכן אחת המבוססת על שאלה מתוך אחד התרגילים.

2. “… the market for biomedical optics doubled from 1985 to 1995, and then tripled from 1995 to 2005 to reach just over $6 billion… 5-fold increase is expected over the next five years.
… One example is the PillCam developed by Given Technology in Israel, which optics.org reported on a few months ago. The 11×31 mm capsule contains automatic lighting control, as well as tiny cameras at each end that capture four images per second for up to 10 hours. Although the technology is nothing new, the device enables clinicians to see parts of the body that cannot be reached by an endoscope.
“… to succeed in this market companies must be sure that their optics-based technology addresses a specific medical need. Indeed, analysis by medical device maker Johnson & Johnson reveals that some 87% of all innovations originate from the clinicians working in hospitals… companies must therefore attempt to focus on high-value solutions that address real-life problems…”
David Benaron, CEO of Spectros Corp. 2007

3. Photonics West conference in San Jose, CA (BIOS, LASE, MOEMS-MEMS, OPTO)
BIOS (biomedical optics) conference
2006: 65/260 pages
2007: 81/284 pages
2008: 81/308 pages
2009: 95/324 pages
2010: moving to a larger convention center in San Francisco.

4. Biomedical applications
Laser invention
Optical technologies
CCD technologies
Better light handling
Improved detection, imaging
Biomedical applications

5. מטרת הקורס
להעניק יסודות רחבים באופטיקה, בשיטות מדידה אופטיות, ובהדמיה, על מנת לתת כלים לסטודנט
להבין אופטיקה
להשתמש בשיטות מדידה והדמיה אופטיות בביו-רפואה
לשנות אמצעים אופטיים קיימים
לפתח טכנולוגיות חדשות למחקר ולפיתוח

6. Course outline Maxwell equations, wave equation
Electromagnetic waves, Gaussian beams
Fourier optics, the lens, resolution
Light-matter interaction: scattering, absorption
Fluorescence, photo dynamic therapy
Fundamentals of lasers
Lasers in medicine
Basics of light detection, cameras
Microscopy, contrast mechanism
Confocal microscopy, laser scanning microscopy
Nanoparticles in biomedical optics
Optical fibers and waveguides
Endoscopy
Advanced microscopy techniques, super resolution
משואות מקסוול, משוואות גלים
גלים אלקטרומגנטים, קרניים גאוסיניות
אופטיקת פורייה, העדשה, הפרדה
אינטראקציה אור-רקמה: פיזור, האטום, בליעה
פלואורסנציה, טיפול פוטו-דינמי
עקרונות לייזרים
לייזרים ברפואה
עקרונות גילוי אור, מצלמות
מיקרוסקופיה, ניגודיות
מיקרוסקופיה קונפוקלית, מיקרוסקופית לייזר סורק
ננו-חלקיקים באופטיקה ביו-רפואית
סיבים אופטים
אנדוסקופיה
מיקרוסקופיה מתקדמת, סופר-רזולוציה

7. Lectures 1-2 Maxwell equations The wave equation
Maxwell equations in medium
Helmholtz equation
Electromagnetic waves: plane, spherical, Gaussian beams
Properties of Gaussian beams

8. Maxwell equations
An electromagnetic field is described by two related vector fields that are functions of position and time:
Electric field:
Magnetic field:
In free space:
where 1. 2.
Electrical permittivity of free space in MKS units: 3. 4.
Magnetic permeability of free space

9. The wave equation 1. 2. 3. 4. 3 1 Speed of light in free space:
For each component:
Similar procedure is followed for

10. Maxwell equations in medium
Assuming no free electric charges or currents.
Electric flux density:
Magnetic flux density:
In source-free media: 1. 2. + - 3. 4.
Nucleus
Electron cloud
In free space:

11. Electromagnetic waves in dielectric media - definitions
A dielectric medium is said to be linear if the vector field P(r,t) is linearly related to the vector field E(r,t). The principle of superposition then applies.
The medium is said to be nondispersive if its response is instantaneous, i.e., if P at time t is determined by E at the same time t and not by prior values of E. Nondispersiveness is clearly an idealization since all physical systems, no matter how rapidly they may respond, do have a response time that is finite.
The medium is said to be homogeneous if the relation between P and E is independent of the position r.
The medium is said to be isotropic if the relation between the vectors P and E is independent of the direction of the vector E, so that the medium exhibits the same behavior from all directions. The vectors P and E must then be parallel.

12. Induced polarization - + ! - + Or: where: In isotropic media:
Electric permittivity
Electric susceptibility
In isotropic media:
We will assume that P is linear with E, which is valid for low field intensities.

13. The refractive index (n)
Speed of light in free space:
The wave equation (in a medium):
where the speed of light in the medium is denoted c:
The ratio of the speed of light in free space to that in the medium, c0/c, is defined as the refractive index n:
For a nonmagnetic material, =0 and:

14. Boundary conditions
In a homogeneous medium, all components of the fields E, D, H, B are continuous functions of position. At the boundary between two dielectric media, in the absence of free electric charges and currents, the tangential components of the electric E and magnetic H fields, and the normal components of the electric D and magnetic B flux densities must be continuous.

15. Poynting vector
The flow of electromagnetic power is governed by the Poynting vector:
Which is orthogonal to both E and H.
The optical intensity I (power flow across a unit area normal to the vector S) is the magnitude of the time-averaged Poynting vector S.
The average is taken over times that are long in comparison with an optical cycle.

16. Monochromatic EM waves
For the case of monochromatic electromagnetic waves in an optical medium, all components of the fields are harmonic functions of time with the same frequency .
Angular frequency
frequency
Similarly:
“complex-amplitude” vectors

17. Maxwell equations in medium (Complex amplitude)
The Maxwell equations become (source-free medium, monochromatic): 1.
If the operations on the complex fields are linear, one may drop the symbol Re and operate directly with the complex functions. The real part of the final expression will represent the physical quantity in question. 2. 3. 4.
Also,

18. Intensity and power (Complex amplitude)
The last two terms on the right oscillate at optical frequencies and are therefore will be washed out by the averaging process, which is slow in comparison with an optical cycle:
Where the new vector
may be regarded as a complex Poynting vector.
The optical intensity is the magnitude of the vector S:

19. Linear, nondispersive, homogenous, and isotropic media1.
If we use the “material equations” for monochromatic waves: 2. 3. 4.
We obtain the Maxwell's equations which depend solely on the complex-amplitude vectors E and H: 1. 2. 3.
* linear, non-dispersive, homogenous, isotropic, source-free medium, monochromatic. 4.

20. Helmholtz equation Substitute the complex amplitude notation
into the wave equation
yields:
or:
Helmholtz equation
where represents the complex amplitude of any of the components of the electric and magnetic fields:

21. Elementary electromagnetic waves
Assumptions:
Medium: linear, homogenous, non-dispersive, isotropic.
Light: monochromatic
Plane waves
Spherical waves
Gaussian waves

22. Plane waves “wavelength” Solutions for the wave equation:
The real electric field:

23. Plane waves Proof: Plane waves satisfies the Helmholtz equation:
Implying that the length of the wave vector of the plane wave must be equal to the parameter k in Helmholtz equation:
Also:

24. Plane waves 1. 2. 3.
Substitute in Maxwell equations 1 & 2 yields (exercise): 4.
E is perpendicular to both k and H
H is perpendicular to both k and E
Transverse electromagnetic (TEM) wave:

25. Intensity of TEM waves
The ratio between the amplitudes of the electric and magnetic fields is known as the impedance of the medium:
the impedance of free space:
The complex Poynting vector:
Example: an intensity of 10 W/cm2 in free space corresponds to an electric field of 87 V/cm:

26. Spherical waves
Another simple solution (proof in exercise) of the Helmholtz equation is the scalar spherical wave:
Reminder: Laplacian in radial coordinates:
U is spherically symmetric:
An oscillating dipole radiates a wave with features that resemble the scalar solution. For points at distances from the origin much greater than a wavelength (r»λ or kr»2π), the complex-amplitude vectors may be approximated by:

27. Gaussian beams – the paraxial wave
Spherical wave:
Plane wave:
A paraxial wave is a plane wave traveling along the z direction (e-ikz, with k=2π/λ), modulated by a complex envelope that is a slowly varying function of position, so that its complex amplitude is:
Paraxial waves:
‘Carrier’ plane wave
Slowly varying complex amplitude (in space)

28. The paraxial Helmholtz equation
Substitute the paraxial wave into the Helmholtz equation:
Paraxial wave

29. Paraxial wave
We now assume that the variation of A(r) with z is slow enough, so that:
These assumptions are equivalent to assuming that
and
Transverse Laplacian:
Paraxial Helmholtz equation:

30. Gaussian beams Paraxial Helmholtz equation
Substitute the paraxial wave into the paraxial Helmholtz equation yields a solution of the form:
Where
z0: Rayleigh range
and
q(z) can be separated into its real and imaginary parts:
Where W(z): beam width
R(z): wavefront radius of curvature

31. Gaussian beams The full Gaussian beam: With beam parameters:
A0 and z0 are two independent parameters which are determined from the boundary conditions. All other parameters are related to z0 and  by these equations.

32. Gaussian beams - properties
Intensity
At any z, I is a Gaussian function of . On the beam axis:
- Maximum at z=0
- Half peak value at z=±z0
z=0
z=z0
z=2z0 1 1 1

33. Gaussian beams - properties
Beam width

34. Gaussian beams - properties
Beam divergence
Thus the total angle is given by

35. Gaussian beams - properties
Depth of focus
z0: Rayleigh range

36. Gaussian beams - properties
Phase
A. Ruffin et al., PRL (1999)
The total accumulated excess retardation as the wave travels from - to  is .
This phenomenon is known as the Gouy effect.

37. Gaussian beams - properties
Wavefront
~ spherical wave

38. Gaussian beams - properties
Wavefront
Plane wave
Spherical wave
Gaussian beam

39. Gaussian beams - properties
Propagation
Consider a Gaussian beam whose width W and radius of curvature R are known at a particular point on the beam axis.
The beam waist radius is given by
located to the left at a distance

40. Propagating through lens
Gaussian beams - properties
Propagating through lens
The complex amplitude induced by a thin lens of focal length f is proportional to exp(ik2/2f). When a Gaussian beam passes through such a component, its complex amplitude is multiplied by this phase factor. As a result, the beam width does not altered (W'=W), but the wavefront does.
Consider a Gaussian beam centered at z=0, with waist radius W0, transmitted through a thin lens located at position z. The phase of the emerging wave therefore becomes:
Where
 The transmitted wave is a Gaussian beam with width W'=W and radius of curvature R'. The sign of R is positive since the wavefront of the incident beam is diverging whereas the opposite is true of R'.

41. Propagating through lens
Gaussian beams - properties
Propagating through lens
The magnification factor M evidently plays an important role. The waist radius is magnified by M, the depth of focus is magnified by M2, and the divergence angle is minified by M.

42. Gaussian beams - properties
Beam focusing
For a lens placed at the waist of a Gaussian beam (z=0), the transmitted beam is then focused to a waist radius W0’ at a distance z' given by:

43. Gaussian beams - properties
The ABCD low
Reminder: where or:
The ABCD Law
The q-parameters, q1 and q2, of the incident and transmitted Gaussian beams at the input and output planes of a paraxial optical system described by the (A,B,C,D) matrix are related by:
Example: transmission Through Free Space
When the optical system is a distance d of free space (or of any homogeneous medium), the ray-transfer matrix components are (A,B,C,D)=(1,d,0,1) so q2 = q2 + d.
*Generality of the ABCD law
The ABCD law applies to thin optical components as well as to propagation in a homogeneous medium. All of the paraxial optical systems of interest are combinations of propagation in homogeneous media and thin optical components such as thin lenses and mirrors. It is therefore apparent that the ABCD law is applicable to all of these systems. Furthermore, since an inhomogeneous continuously varying medium may be regarded as a cascade of incremental thin elements, the ABCD law applies to these systems as well, provided that all rays (wavefront normals) remain paraxial.

44. Summary Maxwell equation:
linear, non-dispersive, homogenous, isotropic, source-free medium, monochromatic light E k H
Helmholtz equation
Plane waves:
Spherical waves:
Paraxial Helmholtz equation:
The Gaussian beam: