z - Technion moodle
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Transcript z - Technion moodle
Course outline
1. Maxwell Eqs., EM waves, wave-packets
2. Gaussian beams
3. Fourier optics, the lens, resolution
4. Geometrical optics, Snell’s law
5. Light-tissue interaction: scattering, absorption
Fluorescence, photo dynamic therapy
חבילות גלים, גלים אלקטרומגנטים, משואות מקסוול.1
קרניים גאוסיניות.2
הפרדה, העדשה, אופטיקת פורייה.3
חוק סנל, אופטיקה גיאומטרית.4
, פלואורסנציה, בליעה, פיזור:רקמה- אינטראקציה אור.5
דינמי-טיפול פוטו
6. Fundamentals of lasers
עקרונות לייזרים.6
7. Lasers in medicine
לייזרים ברפואה.7
8. Basics of light detection, cameras
9. Microscopy, contrast mechanism
10.Confocal microscopy
מצלמות, עקרונות גילוי אור.8
ניגודיות, מיקרוסקופיה.9
מיקרוסקופיה קונפוקלית.10
Reminder
Plane waves
Solutions for the Helmholtz equation:
(proof in next slide)
The real electric field:
E r E0e
ik r
H r H 0eik r
E r , t Re E r eit
Re E0eik r eit
E0 is independent of r
infinite field!
r k:
r k:
“wavelength”
E r , t E0 cos k r t
E r , t0 E0 cos k r t0
E r , t0 constant
k
2
k
Gaussian beams – the paraxial wave
A0 ikr
e
Spherical wave: U r
r
Plane wave: E r E0 eik r
A paraxial wave is a plane wave traveling mainly 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 given by:
ikz
U r A r e
‘Carrier’ plane wave
Slowly varying complex amplitude (in space)
The paraxial Helmholtz equation
Substitute the paraxial wave
into the Helmholtz equation:
2U k 2U 0
2U 2U 2U
2
k
U r 0
2
2
2
x
y
z
2
ik z
2 A r ik z 2 A r ik z A r e
2
ik z
e
e
k
A
r
e
0
2
2
2
x
y
z
Paraxial wave
U r A r eikz
2 A r ik z 2 A r ik z 2 A r ik z
A r ik z
e
e
e
2ik
e
k 2 A r eik z k 2 A r eik z 0
2
2
2
x
y
z
z
2 A 2 A 2 A
A
2
ik
0
2
2
2
x
y
z
z
A A A
A
2
ik
0
2
2
2
x
y
z
z
2
2
2
Paraxial wave
U r A r eikz
We now assume that the variation of A(r) with z is
slow enough, so that:
2
A
2
k
A
z 2
2
A k A
z 2
z
These assumptions are
equivalent to assuming
that sin
and tan
2 A 2 A
A
2 2ik
0
2
x
y
z
Paraxial Helmholtz equation:
Transverse Laplacian:
A
2
T A 2ik
0
z
2
2
A
A
T2 2 2
x
y
Gaussian beams
A
A 2ik
0
z
Paraxial Helmholtz equation
2
T
One solution to the paraxial Helmholtz equation of the slowly varying complex
2
amplitude A, has the form:
ik
A1
Ar
e
q z
Where q z z iz0
and
2 q z
z0: “Rayleigh range”
2 x2 y2
q(z) can be separated into its real and imaginary parts:
1
1
i
q z R z W 2 z
Where
W(z): beam width
R(z): wavefront radius of curvature
Gaussian beams
The full Gaussian beam:
2
2
W0 W 2 z ikz ik 2 R z i z
U r A0
e
e
W z
With beam parameters:
z
W z W0 1
z0
2
A0 A1 iz0
z0 2
R z z 1
z
z
z tan 1
z0
W0
z0
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.
Gaussian beams - properties
Intensity
I r U r
I 0 A0
2
2
W0
U r A0
e
W z
2
W0
I , z I0
e
W z
22
W
2
2
W 2 z
e
z
W z W0 1
z0
z
ikz ik
2
z0 2
R z z 1
z
At any z, I is a Gaussian function of . On the beam axis:
z tan 1
2
W0
I0
I 0, z I 0
2
W
z
1 z z0
I
(Lorentzian)
W0
z
z0
z0
1/2
0
z0
z
- Maximum at z=0
- Half peak value at z = ± z0
z=0
z=z0
1
1
z=2z0
1
2
2 R z
i z
Gaussian beams - properties
Beam width
W0
U r A0
e
W z
2
W 2 z
e
z
W z W0 1
z0
2
z0 2
R z z 1
z
z
W z W0 1
z0
1 e line 1 e2 in intensity
2W0
2
z tan 1
W0
z0
z
z0
ikz ik
2
2 R z
i z
Gaussian beams - properties
Beam divergence
z
z0
z
W z W0 1
z0
W
W z 0 z
z0
2
2
W 2 z
e
z
W z W0 1
z0
ikz ik
2
2 R z
2
z0 2
R z z 1
z
W02
z0
W0 W0
0
2
z0 W0 W0
Thus the total angle is given by
W0
U r A0
e
W z
4
20
2W0
z tan 1
W0
z
z0
z0
2W0 2 0
4
i z
Gaussian beams - properties
Depth of focus
z0
W0
W02
2 z0 2
W0
U r A0
e
W z
2
W 2 z
e
z
W z W0 1
z0
2
z0 2
R z z 1
z
z tan 1
W0
z
z0
z0
z0: Rayleigh range
The total depth of focus is often defined as twice the Rayleigh range.
ikz ik
2
2 R z
i z
Gaussian beams - properties
Phase
W0
U r A0
e
W z
W0
U r A0
e
W z
2
W 2 z
e
ikz ik
2
2 R z
i z
z
0, z kz tan
z0
1
2
W 2 z
e
z
W z W0 1
z0
ikz ik
2
2 R z
2
z0 2
R z z 1
z
z tan 1
W0
z
z0
z0
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.
i z
Gaussian beams - properties
Wavefront
2
z
k
, z kz tan 1
z0 2R z
W0
U r A0
e
W z
2
W 2 z
e
z
W z W0 1
z0
ikz ik
2
2 R z
i z
2
z0 2
R z z 1
z
z tan 1
W0
z
z0
z0
~ spherical wave
~ plane wave
z0 2
R z z 1
z
z
R z
z
Gaussian beams - properties
Propagation
W0
U r A0
e
W z
2
W 2 z
e
z
W z W0 1
z0
W
W0
R
ikz ik
2
2 R z
i z
2
z0 2
R z z 1
z
z tan 1
z
W0
z
z0
z0
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
W0
W
1 W R
2
located to the right at a distance
z
R
1 R W
2 2
2
Gaussian beams - properties
Propagating through lens
The complex amplitude induced by a thin lens of focal length f is proportional to exp(-ik2/2f).
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 change (W'=W), but the wavefront
does.
W0
U r A0
e
W z
2
W 2 z
e
ikz ik
2
2 R z
i z
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 (ignore sign):
kz k
2
2R
k
2
2f
kz k
2
2R '
Where
1 1 1
R' R f
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'.
Gaussian beams - properties
Propagating through lens
The magnification factor M 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.
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:
W0 '
z'
W0
z
1 z0 f
2
f
z
1 f z0
M z 0
f
0
f
f
f
W0
W
z0
W02 0 W0
f
0
f
2
1
1 z0 f
z
2
W0
f
0
f
z0
f
W0
Gaussian beams - properties
The ABCD law
Reminder:
A
Ar 1 e
q z
ik
2
2 q z
, where q z z iz0
1
1
i
or:
q z R z W 2 z
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:
Aq1 B
q2
Cq1 D
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 = q1 + d.
*Generality of the ABCD law
The ABCD law applies to thin optical components as well as to propagation in a
homogeneous medium. 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.