REFLECTION RAY-TRACING:
reference & illumination angles are measured “the long way around” and
the object beam is in the +z direction
l1
l2 - direct
q ref
qobj
obj
q ill
(m = +1)
out
z
z
qout
l2 - Ø-conj
(m = –1)
ref
ill
ill
qout
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)
Robj
obj
R
out
R out
Rill
ref
ref
l2 - Ø-conj
ill
(m= –1)
R
out
ill
ill
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)cos2 q obj
cos2 q out
cos 2 q ill
R out
-
l2
R ill
= m
-
R obj
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.
cos 2 q ref
R ref
Off-Axis Reflection Holography
(direct, forward, m=+1 reconstruction)
Øtip1
1)EXPOSURE
at l1
t1
n1
n2
qref,ext
L1
qill,ext
qref,int
qobj,ext
qobj,int
OBJECT
Øtip2
2) RECONSTRUCTION
atl2
t2
qill,int
qout,int
L2
d
REFERENCE
OUTPUT
qout,ext
d
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
F tip1 =
qobj,int + q ref, int
q
+ q ill,int
, F tip2 = out,int
2
2
t1
t2
sin F tip1 =
sin F tip2
L1
L2
q obj,int - q ref,int ˆ
1
2
Ê
=
cosÁ 90∞ +
˜
L1 l1,int
2
Ë
¯
x-, z - grating representation (all m):
sin q obj,ext - sin q ref, ext
l1,ext
n1 ◊ t1
=
cos q obj,int - cos q ref,int
l1,ext
,
q
- q ill,int ˆ
1
2
Ê
=
cos Á 90∞ + out,int
˜
¯
L 2 l2, int
Ë
2
sin q out,ext - sin q ill, ext
1
=m
d
l2, ext
= n2 ◊ t2 ◊ m
‹ means that 1/R and cos2q/R still work!
cos qout,int - cos q ill,int
l2 ,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)
(
2 ◊ n1
1
=
cos q obj,int
L1 l1,ext
)
t
t1
= 2
L1 L 2
2 ◊ n2
1
=
cos q out,int ,
L2
l 2,ext
(
or pulling it together: n1 ◊ t1
cos qobj, int
cos q out,int
= n2 ◊t2
l1,ext
l2, ext
)
(±1, GHU)
RSHRINK: definitions of variables
Øtip1
1)EXPOSURE
atl1
t1
2
n1
n2
qref,ext
qill,ext
qref,int
qobj,ext
qobj,int
OBJECT
Øtip2
2)RECONSTRUCTION
atl2
t
L1
qout,int
L2
d
qill,int
OUTPUT
qout,ext
d
REFERENCE
ILLUMINATION
cosq x l "blue shift"
–refl = ––incidence
Reflection Gratings, forward reconstruction (m=1): RSHRINK pseudo-rules for TK-Solver+:
1)
Øtip1 = (qobj,int + qref,int)/2
"fringe tip angle during exposure"
2)
Øtip2 = (qout,int + qill,int)/2
"fringe tip angle during reconstruction"
3)
t2 = (1 – shrink/100) ◊ t1
"simple shrinkage definition"
4)
t1 ◊ tan(Øtip1) = t2 ◊ tan(Øtip2)
"fringe x-span constancy"
5)
diff1 = (qref,int – qobj,int )/2
"diffs are handy variables"
6)
diff2 = (qill,int – qout,int )/2
7)
(1/L1) = (2/l1int) ◊ cos(90°+diff1)
"fringe separation during exposure"
8)
(1/L2) = (2 /l2int) ◊ cos(90°+diff2)
"fringe separation during reconstruction"
9)
d ◊ cos(Øtip1) = L1
"change of fringe separation"
10) d ◊cos(Øtip2) = L2
internal vs. external
11) t1 ◊sin(Øtip1)/L1 = t2 ◊sin(Øtip1)/L2
12) sin(qobj,ext) = n1 ◊ sin(qobj,int)
"Snell's Law sequence"
13) sin(180° – qref,ext) = n1 ◊ sin(180° – qref,int)
"in case arcsin is limited to quads I & IV"
14) sin(180° – qill,ext) = n2 ◊sin(180° – qill,int)
15) sin(qout,ext) = n2 ◊sin(qout,int)
16)
l1ext = n1 ◊ l1int
"int vs. ext for wavelengths"
misc. cleanup
17) l2ext = n2 ◊ l2int
18) l1ext/d = sin(180° – qref,ext) – sin(qobj,ext)
"interference equation, x-component"
19) l2ext/d = sin(180° – qill,ext) – sin(qout,ext)
"diffraction equation, x-component"
20) (cos(qref,int) – cos(qobj,int)) ◊(1/l1int) ◊(t1/t2) =
(cos(qill,int) – cos(qout,int)) ◊ (1/l2int)
20)
qref,int = Øtip1 + diff1
21)
qill,int = Øtip2 + diff2
"z-component match"
"restatements, if needed"