Download Lecture 23 MOEMS 5

EEL6935 Advanced MEMS (Spring 2005)
Instructor: Dr. Huikai Xie
Lecture 23
Optical MEMS (5)
„
Agenda:
Ê
Microlenses
– Diffractive
Ê
Microgratings
Ê
Example Devices
Reference: S. Sinzinger and J. Jahns, Chapter 6 in Microoptics, Wiley-VCH, 2003
EEL6935 Advanced MEMS
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4/6/2005
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Limitation of Refractive Optical Elements
• Refractive optical elements require considerable thickness.
• For example, microlenses need 10~100µm-thick
microstructures.
• Even more challenging if aspherical lenses or complex phase
profiles are needed.
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Reduced Phase Thickness
• But we know that a light wave repeats itself after any multiple
wavelengths or a phase lag of multiple 2 π.
U ( x, φ ) = U 0 ( x)eiφ = U 0 ( x)e (
i φ + 2 Nπ )
Thus, only a maximum phase lag of 2 π
is necessary.
The phase difference between path 1
and path 2 is given by
∆φ = ( nt − t )
tmax =
EEL6935 Advanced MEMS
2π
Path 1
Path 2
n
t
λ0
λ0
( n − 1)
For example, for λ0=633nm and n=1.5,
the thickness needed is only 1.27µm.
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Blazing
Therefore, the thickness of a prism or lens can be
significantly reduced.
High lateral resolution is required, which is in fact
photolithography can provide.
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Phase Quantization
Step 1
Continuous phase Æ mapping of the phase into [0,2π] periodically
Step 2
Continuous phase in each interval [0,2π] Æ an integer number of
discrete values
e
iφQ ( x )
=
q =+∞
∑e
iq
 φ ( x)

2π
− q
rect 
N
π
N
2
/


(1)
q =−∞
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Phase Quantization
The rect function can be expanded into a Fourier series
 φ ( x)
 +∞  sinc ( J / N ) 2π i ( J / N ) q iJ φ ( x ) 
rect 
e
e
− q = ∑ 

N
 2π / N
 J =−∞ 

(2)
Substituting (2) into (1) yields
e

1  i Nm +1)φ ( x ) 

= ∑ sinc  m +  e (

N

m 

m i ( Nm +1)φ ( x )

 1  +∞  ( −1) e

= sinc   ∑ 
Nm + 1

 N  m =−∞ 
iφQ ( x )
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Phase Quantization
If φ(x) is the phase of g(x) and φQ(x) is the quantized phase of φ(x),
then he reconstructed signal GQ(ν)
GQ
1
( v ) = s in c   ∑
 N  m
 ( − 1 )m 

G m ( v )
 N m + 1 
where
G m (v ) =
∫e
i 2 π m vx
e
i ( N m + 1 )φ ( x )
dx
(2)
Superposition of a number
of “ghost” signals
• The m=0 term recovers the undistorted signal.
• The m ≠0 terms generate ghost images which contribute to
background noise if not separated.
• Diffraction Efficiency
 1 
η = s in c 2 

 N 
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Alternative Quantization Schemes for Microlenses
Fresnel zones
• Concentric rings
decrease with
increasing radii
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Superzones
• Phase depth
increases
toward edges
Constant pixel
size quantization
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Diffractive Optical Components
1x2
Beamsplitter
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1xN
Beamsplitter
(e.g.,
Dammann
Grating)
Beam
Deflector
Diffractive
Lens
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Fabrication of Diffractive Optics
Multi-mask Processing
Linear Mask Sequence
Logarithmic Mask Sequence
• N-1 processing steps
• Arbitrary step heights
• (log2N) processing steps
• Dependent etching depths
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Fabrication of Diffractive Optics
Fabrication Errors
• Ideal Process
• Lateral
misalignment
of two masks
• Error in
etching
depth
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Fabrication of Diffractive Optics
Fabrication Errors
Resulting profile
Phase error
• Over-etching
• Under-etching
• Partially
isotropic
etching
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Diffractive Lenses
ƒ Fresnel zone plates (FZPs)
rj2 + f 2 = ( f + jλ )
2
rj = 2 j λ f + ( j λ )
2
For paraxial approximation,
i.e., f>>jmaxλ,
• Binary phase or amplitude rings
• Optical path difference from adjacent zones
is Nλ, where N is an integer and λ is the
wavelength.
• Size is comparable to the optical fiber, good
for integration
rj ≈ 2 j λ f
and
• Flat and thin, ideal for microfabrication
f =
r12 − λ 2 r12
≈
2λ
2λ
• Easily designed to different wavelengths
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Microgratings
1st order (m = -1)
θ
0th order (m = 0)
φ
a
1st order (m = 1)
a
α
Two types of reflective grating
• Grating Equation:
a (sin θ − sin φ ) = mλ
(m = 0,±1,±2, …)
• Resolution Power: R = λ /(∆λ ) min = mN
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(m = 1)
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DOE Example Devices -1
• Polysilicon microstructure
• Latched up vertically
• 280µm in lens diameter
Lin et al, IEEE Photonics Technology Letters, 1994
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DOE Example Devices -2
XYZ-Adjustable Microlens
Fan et al. (UCLA), Transducers’97
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DOE Example Devices -3
32-level Fresnel Zone Plate
Pawlowski, 1993
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DOE Example Devices -4
Kinoform Microlens Array (Fused Silica)
•
•
•
•
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Eight levels
Fused silica
0.2mm by 0.2mm
Diffraction efficiency: 85%
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Soldatenkov et al, LFNM 2003
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DOE Example Devices -5
Diffraction Grating for Spectroscopic Applications
Kiang et al. (UC-Berkeley), Transducers’97
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DOE Example Devices -6
Force Sensor Based on Diffractive Gratings
Zhang et al (Stanford), Sensors &
Actuators, 2004
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