Ray 4. Diagrams and Phase Space.
2 1 3 6
Window: Defines Object SIZE
4 5 6
rw 2 2
p w
3
rp
5 2 4 1
L
Pupil: Defines SOLID ANGLE for all points of the object
Consider the ray tracing diagram in a field-free region with two circular defining apertures.
r 1 2 3 3 2
r
6
5 1 4 5 6 4
In this work, it is assumed that the solid angle from all radial positions in the object are the same, which is satisfied in the paraxial approximation i.e. where the distance, L, between window and pupil are sufficiently large. If this, in fact, is not the case then the central part of the object will be brighter than the perimeter, i.e. vignetting occurs.
Vignetting in light optics
Collimation
2 1 3
Note: in the small angle approximation sin tan
6
rw 2
4 5 6
w
3
rp
5 2 4 1
2
p
L
r 1 2 3
p rp /L = Pencil Angle B rw /L = Beam Angle
2 3
rArea
= 2rp x 2B = 4 rp x rw/L 6
Area = 2rw x 2p = 4 rw x rp/L
4 5 6
5
AREA is Conserved.
1
4
The area can be generalised as pr, where p = mv, is the range of transverse momenta and r is the range of transverse positions in an object/image/beam. This pr product moves us back to phase space, and hence the area is still conserved when energy is taken into account. Liouvilles theorem Helmholtz-Lagrange Relation
pr = mv r
1 h11 = 2 h2 2
Note here is the PENCIL angle, as the angle must always be finite. (See Fig 2.)
Again, refers to pencil angle not beam angle. Hence once it is defined in object space, it is also defined in image space, via Helmholtz-Lagrange. It is possible to control the beam angle; in fact it can be made equal to zero in image space by placing the entrance (object) pupil at the focal length of the lens. (See diagram over). This is useful as it minimises the trajectorys angular range in the lens and so minimises aberrations.
Apertures and a Lens
V1 r11 = V2 r2 2
Vignetting
Only use two apertures in a lens, else some trajectories from the edges of the object get cut vignetting.
Real and Virtual Apertures
Note, apertures not in the same object /image space.
A virtual window has the same effect as a real window (defining aperture) via Helmholtz Lagrange.
Lamberts law and the Langmuir Equation
Assume Lamberts Law for a planar emitter:
J0 cos = cos i () =
i() is the current density at in the direction to the surface normal, and J0 is the total current density (Am-2) emitted into a hemisphere.
Lamberts law assumes a perfectly diffuse flat surface that emits equally well in all directions
The current emitted over the entire cathode area, r12, into an annular ring at an angle 1 in a unit sphere (see Figure 1) is:
(1 cos1 )(r12 )2 sin1d1 dI (1 ) =
Apply Abbes sine theorem to find corresponding current element in image space:
E2 2 2 dI ( 2 ) = 1 E M r1 2 sin 2 cos 2 d 2 1
( )
Simplifies to:
E2 2 dI ( 2 ) = 1 E r2 2 sin 2 cos 2 d 2 1 r22 = M 2r12
( )
Integrating these expressions from i = 0 to i, i.e. over the angular range of the entrance and exit solid angles, then:
1
I1 =
0
dI (1 )d1 = 1 r12 2 sin 1 cos 1d1 = 1 r12 sin 2 1
(a)
( ) 0
2
1
( )
Similarly:
E2 2 r2 sin 2 2 I 2 = dI ( 2 )d 2 = 1 E 1 (b) 0
( )
Taking the ratio of these expressions gives:
2 J2 I1 r2 E2 sin 2 2 = 2 = J 1 r1 I 2 E1 sin 2 1
The maximum possible value of 1 from a planar cathode is 90 and the initial energy of the electron will be taken as kT. The final image is at a potential of 2, with respect to the cathode at ground potential, and the therefore final energy of the electron is E 2 = e 2 + kT. Consequently, the maximum current density in the final image is given by:
(e 2 + kT ) 2 e 2 2 e 2 2 sin 2 = J1 + 1 sin 2 J1 J 2 = J1 sin 2 kT kT kT
as e 2 >> kT . This is referred to as Langmuirs Equation.
Brightness
As in light optics, we can define the brightness, , to be the electron intensity emitted per unit area of the surface into a unit solid angle: dI = dAd From equations (a) and (b):
1 = I1 2 r12 sin 2 1
If the intervening medium is non-absorbing, then current will be conserved: I = I1 = I2, and applying Abbes sine theorem:
I1 E1r12 sin 2 1
=
I2
2 E2 r2 sin 2 2
1 2 = Hence: E1 E2
i.e. the brightness-to-energy ratio is conserved from the object to the image.
Grid
Electron Sources
A schematic diagram of a simple planar triode electron gun. Electrons emitted by the hot filament are accelerated towards the positively biased anode potential through the grid aperture. In this idealised schematic, the thickness of the anode and grid apertures are negligible in comparison to their hole sizes.
Filament
Anode
a
b
a ~ 0.5 1mm, b ~1 - 3mm c ~ 0.25mm = Filament Grid separation.
The simple triode gun acts like a two element immersion lens, with an image of the source about 0-1mm to the right of the anode aperture. This image, formed in a potential region defined by VAnode, acts as the object for the following lens system. Note: Local heating of elements by electron bombardment, hence surface changes over time and out-gassing. Hot filament can droop under gravity! Slow evaporation of the filament over time. Screen for escaping laterally electronsdont let them reach the target or the detector. Electrons impacting on insulating ceramics can charge them up and distort local potentials.
More Complex Sources
Oxide Cathode (large area, low temperature) Photoelectron from laser ionisation or synchrotron radiation. Spin Polarised electrons (laser + GaAs surface) Positrons (i.e. radioactive sources) Discharge sources for ions, Highly charged ions to name b...