REFLECTION RAY-TRACING:
reference & illumination angles are measured “the long way around” and
the object beam is in the +z direction
l2 - direct
l1
q
(m = +1)
q
ref
out
z
q
z
qout
obj
obj
ill
l2 - Ø-conj
(m = –1)
ill
ref
ill
q
out
m = +1 for “direct” reconstruction
(illum & ref from same side)
–1 for “phase conjugation”
(illum & ref from opposite sides)
z
out
q ill
distances = radii of curvature (negative => real image)
l2 - direct
l1
(m= +1)
out
z
Robj
obj
R
z
R out
Rill
ref
l2 - Ø-conj
ref
ill
(m= –1)
R
ill
ill
z
out
R out
(negative)
HORIZONTAL FOCUS (out of the plane of the page)-
1
Rout
-
l2
1
1
R ill
Robj
= m
-
l1
1
R ref
VERTICAL FOCUS (in of the plane of the page)2
cos q out
R out
-
l2
2
cos q ill
2
cos q obj
R ill
R obj
= m
-
2
cos q ref
l1
All angles on this page are the usual external angles.
The usual time that internal angles (the q ' ) are used is in the “fringe-tip and -separation” calculations,
and the “z-equation” for the allowed angles (if you use that approach).
The angles and wavelengths are first determined by those calculations, and are then plugged in to these
focus equations to solve the imaging questions. Recall that negative radii of curvature mean converging
waves for all directions of travel!
R ref
Off-Axis Reflection Holography
(direct, forward, m=+1 reconstruction)
Øtip1
1) EXPOSURE
at l1
t1
n1
n2
qref,ext
qill,ext
qref,int
qobj,ext
qobj,int
OBJECT
Øtip2
2) RECONSTRUCTION
at l2
t2
L1
OUTPUT
qill,int
qout,ext
qout,int
L2
d
d
REFERENCE
ILLUMINATION
recall: Snell' s Law: sin q xxx, ext = ni ⋅ sin q xxx, int
also: next⋅lext = nint⋅lint
tilted - stacked - mirror representation:
t1 ⋅ tan F tip1 = t2 ⋅ tan F tip2
q
+ q ref, int
q
+q
ill,int
F tip1 = obj,int
, F tip2 = out,int
2
2
t1
t
sinF tip1 = 2 sin F tip2
L1
L2
q obj,int - q ref,int ˆ
q
-q
Ê
Ê
ˆ
1 = 2
1 = 2
ill,int ˜
˜ ,
cosÁ 90° +
cos ÁË 90°+ out,int
¯
Ë
¯
L
l
L
l
2
2
1
1,int
2
2, int
x-, z - grating representation (all m):
sin q out,ext - sin q ill, ext
l 2, ext
n2 ⋅ t2
cos qout,int - cos q ill,int
l 2 ,ext
=1 = m
d
sin qobj,ext - sin qref, ext
l 1,ext
‹ means that 1/R and cos2q/R still work!
= m ⋅ n ⋅ t cos q obj,int cos q ref,int
1 1
l 1,ext
(± 1, Goodman - Heisenberg Uncertainty)
Special Case: On-Axis Reflection “Denisyuk” Holography
(direct, forward, m=+1 reconstruction)
qref, ext = 180° - q obj,ext , so F tip1 = F tip2 = 90° (conformal fringes)
so that: q out,ext = 180° - q ill,ext (mirror reflection)
1 = 2 ⋅ n1
cos q obj,int
L 1 l1,ext
(
)
t1 = t2
L1 L 2
1 = 2 ⋅ n2
cos (q out,int ) ,
L 2 l 2,ext
cos q out,int
cos qobj, int
or pulling it together: n2 ⋅ t2
= n1 ⋅ t1
( ±1, GHU)
l 2, ext
l 1,ext