Post on 09-Sep-2018
EE 485, Winter 2004, Lih Y. Lin
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Chapter 3 Beam Optics
- An important paraxial wave solution that satisfies Helmholtz equation is Gaussian beam. Example: Laser.
3.1 The Gaussian Beam A. Complex Amplitude
phase Radial )(2
exp
phase alLongitudin tanexp
factor Amplitude )(
exp)(
)(
2
0
1
2
20
0
−×
−−×
−=
−
zRkj
zzkzj
zWzWWArU
ρ
ρ�
(3.1-7)
2
00 1)(
+=
zzWzW (3.1-8)
+=2
01)(zzzzR (3.1-9)
λ
π 20
0Wz ≡ (3.1-11)
→ Knowing 0W and λ (or 0z ), a Gaussian beam is determined! Ref: Verdeyen, “Laser Electronics,” Chapter 3, Prentice-Hall B. Properties of Gaussian Beam
Intensity and power
EE 485, Winter 2004, Lih Y. Lin
2
−
=)(
2exp)(
),( 2
22
00 zWzW
WIzI ρρ (3.1-12)
( )2002
1 WIP π= (3.1-14)
Beam radius and divergence
)( when ),0(1),( 2 zWzIe
zI == ρρ
2
00 1)(
+=
zzWzW : Beam radius (3.1-17)
0W : Waist radius 02W : Spot size 00 2)( WzW =
0
0 Wπλθ = : Divergence angle (=
2θ in Fig. 3.2 shown above)
(3.1-19) → Highly directional beam requires short λ and large 0W . Depth of focus (Confocal parameter)
EE 485, Winter 2004, Lih Y. Lin
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λ
π 20
022 Wz = (3.1-21)
Longitudinal phase
−= −
0
1tan)(zzkzzϕ
→ Second term: phase retardation
Phase velocity c
zz
z
cz
vp >
−
=
=−
−
0
1
1
tan2
1πλω
ϕ !
Wavefront bending Surface of constant phase velocity:
≡+=+ −
0
12
tan ,22 z
zqR
z ξπλξλρ
→ Parabolic surface with radius of curvature R
EE 485, Winter 2004, Lih Y. Lin
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3.2 Transmission through Optical Components
Gaussian beam remains a Gaussian beam after transmitting through a set of circularly symmetrical optical components aligned with the beam axis. Only the beam waist and curvature are altered.
A. Transmission through a Thin Lens
WW
RRR=
−=
''
11'
1 (3.2-2)
22
0
'1
'
+
=
RW
WW
λπ
(3.2-3)
EE 485, Winter 2004, Lih Y. Lin
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2
2
'1
''
+=
WR
Rz
πλ
(3.2-4)
Waist radius 00 ' MWW = (3.2-5) Waist locatioin )()'( 2 fzMfz −=− (3.2-6) Depth of focus )2('2 0
20 zMz = (3.2-7)
Divergence M
00
2'2 θθ = (3.2-8)
Magnification 21 r
MM r
+= (3.2-9)
fz
zrfz
fM r −≡
−≡ 0 , (3.2-9a)
Example: A planar wave transmitting through a thin lens is focused at distance fz =' .
B. Beam Shaping
Waist of incident Gaussian beam is at lens location. fR −='
2
0
00
1
'
+
=
fz
WW (3.2-13)
→ To focus into a small spot, we need large incident beam width, short focal length, short wavelength.
EE 485, Winter 2004, Lih Y. Lin
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2
0
1
'
+
=
zf
fz (3.2-14)
F number of a lens
DfF ≡#
02WD = : Diameter of the lens
Focal spot size #04'2 FW λπ
= (3.2-17)
C. Reflection from a Spherical Mirror Same as transmission through a thin lens, 2
Rf −= 12 WW = (3.2-19)
RRR211
12
+= (3.2-20)
D. Transmission through an Arbitrary Optical System The ABCD Law
DCqBAqq
++=
1
12 (3.2-9)
EE 485, Winter 2004, Lih Y. Lin
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Like in the case of ray-transfer matrix, the ABCD matrix of a cascade of optical components (or systems) is a product of the ABCD matrices of the individual components (or systems). 3.3 Hermite-Gaussian Beams
Modulated version of Gaussian beam → Intensity distribution not Gaussian, but same wavefronts and angular divergence as the Gaussian beam.