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P. Piot, PHYS 630 – Fall 2008
Introduction• Course’s webpage:
http://nicadd.niu.edu/~piot/phys_630/• Grading:
– Homework 50 % (ok to work/discuss together)– MidTerm 20 % (class exam?)– Final 30 % (class exam?)
• Instructor:– Philippe Piot (NIU/FNAL & ANL) [[email protected]]
• Generally at NIU on Tues, Wed, and Th.• At FNAL or ANL the rest of the week.
• Handouts:– Slides and needed papers will be distributed weekly and made available
on the web…
• Fieldtrip:– Fermilab photoinjector laser system, or Argonne Terawatt laser
• Goals:– Introduce some advanced concepts of optics (especially nonlinear optics)– Provide practical example of application
P. Piot, PHYS 630 – Fall 2008
Notes on textbooks
• No textbook required, the lecture is a mixture of several texbooks(listed below) homework are homemade
• Many very good books you may want to check– J. Peatros, Physics of Light and Optics, a good (and free!) electronic
book (available http://optics.byu.edu/textbook.aspx)– B. Saleh, and M. Teich, Fundamentals of Photonics, Wiley-
Interscience, very complete, essential mathematical formalism– Y. B. Band, Light and Matter, Wiley and Sons (2006)– R. Guenther, Modern Optics, Wiley and Sons (1990): very good– H. Hecht, Optics, Wiley & Sons, good introduction [what you used
with Omar Chmaissem (up to chapter 8)] -- mathematical descriptionweak
P. Piot, PHYS 630 – Fall 2008
Ray optics: postulates• Postulates:
– Light travels in form of rays– Medium characterized by an index of refraction n defined as the ratio of
velocity of light in vacuum over velocity of light in medium
• Fermat’s principle: optical rays traveling betweentwo points A and B follow a path such that the timeof travel between two points is an extremum relativeto the neighboring paths.
• This extremum is usually a minimum: so light goesfrom A to B along the path of least time.
• This is the optics equivalent to the “least action principle”
P. Piot, PHYS 630 – Fall 2008
Ray Optics• Start with the wave (or Helmholtz) equation
• Take
• The wave eqn becomes
• Assume 1/kvac~0 then eqn reduces to
This means that diffraction effects
are ignored
P. Piot, PHYS 630 – Fall 2008
Ray Optics• Introducing the s unit vector in the di-
rection of , one finally obtain theeikonal equation:
• Taking the curl of the eikonal equation
• And integrating over an area
• Apply Stokes’ theorem
this equation implies Fermat’s principle
P. Piot, PHYS 630 – Fall 2008
Transfer matrix (ABCD) formalism• The propagation of light rays can be described piecewise via transfer
matrix,• In one plane: two scalars
define a ray: (x, x’)• In the paraxial approximation,
the ray are assumed to remainclose to the optical axis of thesystem and sin θ~θ
• Example: drift space: Drift length
P. Piot, PHYS 630 – Fall 2008
Transfer matrix (ABCD) formalism
P. Piot, PHYS 630 – Fall 2008
Transfer matrix (ABCD) formalism
P. Piot, PHYS 630 – Fall 2008
Image formation
• Imaging if
• With magnification
P. Piot, PHYS 630 – Fall 2008
Extension of ABCD formalism• ABCD is within one degree of freedom• Real system are at three dimensional
• Two dimension extension of ABCD formalism is straightforward:consider the transverse coordinate X=(x,x’,y,y’)
• Define a 4x4 transfer matrix (ABC…P)– This way we can treat the propagation of a ray in an optical
system in two dimension simultaneously– Can account for possible coupling or asymmetry, e.g., cylindrical
lenses (normal and tilted case)
• Six dimension not straightforward; see later.
P. Piot, PHYS 630 – Fall 2008
Maxwell’s equation I
• Maxwell’s equations in a medium (ε,µ) and charge/current density(ρ,J)
• Where
polarization
magnetization
electric field
induction
Electric displacement
Magnetic displacement
P. Piot, PHYS 630 – Fall 2008
Maxwell’s equation II
• Generally
• Consider “simple case” of homogenous,non conducting, non dissipating isotropicmedium then:
HB
rtvµ=
!
r D = "
r E + "
t #
r E + ...$
t "
v E
Tensor (or matrix)
Linear susceptibility
Linear opticsNL (2nd and 3rd order) optics
EEDvrr
!! =
"""
#
$
%%%
&
'
=
100
010
001
µ
BH
rv=
P. Piot, PHYS 630 – Fall 2008
Maxwell’s equation III
• If no source terms are present (assume no charge in the medium) thenMaxwell’s equations reduce to
• Note if medium is vacuum then:
0
0
µµ
!!
"
"
P. Piot, PHYS 630 – Fall 2008
Wave equation
• Take of Faraday’s law:
• Take of Ampere-Maxwell’s equation:
• Summary:
!
"#2r E + µ$%
t
2r E = 0
!
"#2r B + µ$%
t
2r B = 0
!
"2v
U #µ$%t
2r
U = 0
!
"2v
U #1
c$
t
2r
U = 0
In vacuum
P. Piot, PHYS 630 – Fall 2008
Helmholtz Equation
• Consider the function U to be complex and of theform:
• Then the wave equation reduces to
where
!
U(r r ,t) = U(
r r )exp 2"#t( )
!
"2U(
r r ) + k
2U(
r r ) = 0
!
k "2#$
c=%
cHelmholtz equation
