声表面波耦合模理论分析——COM_Theory

∂u + = − j β u u + − j κ 12 u − exp( − j β G X ) ∂X ∂u − * + j κ 12 u + exp( + j β G X ) = + jβ u u − ∂X COM Equations for Forward & Backward Waves
βu: Wavenumber of Uncoupled Wave βG: Grating Vector (2π/p), p: Periodicity κ12 : Mutual Coupling Coefficient
Final COM Equations
∂u1 = − jβ u u1 − jκ u 2 ∂X ∂u 2 = − jβ u u 2 − jκ *u 1 ∂X
When Two Waveguides are Exchangable, κ is Real
General Solution
u 1 = A + exp( − j β + X ) + A − exp( − j β − X ) u 2 = rA + exp( − j β + X ) − rA − exp( − j β − X )
±
General Solution
U + ( X ) = A+ exp( − j θ p X ) + Γ− A− exp( + j θ p X ) U − ( X ) = Γ+ A+ exp( − j θ p X ) + A− exp( + j θ p X )
βp=θp+π/p: Wavenumber of Perturbed Wave θp = θu2 -|κ12|2 Γ+ =(θp-θu)/κ12 & Γ- =(θp-θu)/κ12*: Reflection
Floquet Floquet Theorem Theorem
u+ up Due to Periodicity, Eigen Modes in Infinite Periodic Gratings Satisfy Where β0 is Wavenumber of Grating Mode Define u± ( X ) = U ± ( X ) exp(m jβ 0 X ) Then We Obtain U ± ( X + p ) = U ± ( X ) ：Periodic Function
Contents Contents
• Colinear Coupling • Reflective Coupling • IDT Modeling • IDT Properties • SAW Device Simulation • Parameter Extraction • Parameter Extraction of Bidirectional IDTs • Parameter Extraction of Directional IDTs
Reflection Coefficient in dB
Reflection Phase in deg.
0
Ain Aref p (a) Γ+=Aref/Ain Ain Aref
(b) Γ-=Aref/Ain
p
Application of Boundary Condition
A
in
A
t
Ar L
U + ( X ) = A + exp( − j θ p X ) + Γ− A − exp( + j θ p X ) U − ( X ) = Γ + A + exp( − j θ p X ) + A − exp( + j θ p X )
U
2005-2006 International Distinguished Lecturer Program
Ken-ya Hashimoto Chiba University
Sponsored by The Institute of Electrical and Electronics Engineers (IEEE) Ultrasonics, Ferroelectrics and Frequency Control (UFFC) Society
Where β ± = β u ± κ
r =| κ | / κ
When κ is Real, Two Partial Waves Correspond to
Red：Symmetric Mode, Blue： Antisymmetric Mode
Application of Boundary Condition
= Reflectivity per Unit Length
For Derivation, Loss Less Condition was Applied
Define U±(X)=u±(X)exp(±jβGX/2). Since u±(X)=U±(X)exp( jβGX/2),
∂U + = − j θ u U + − j κ 12 U − ∂X ∂U − * = + j κ 12 U + + j θ uU − ∂X
u± ( X + p ) = u± ( X ) exp(m jβ 0 p )
Since U±(X) is Periodic Function
U± (X ) =
Where βG=2π/p: Grating Vector A±(n): Amplitude of n-th Partial Wave
n = −∞
u1 u2 ∆X u1 u2
u1 ( X + ∆X ) − u1 ( X ) + u2 ( X + ∆X ) − ( X ) + u2 ( X ) ∂X
[
∆X→0 gives
2 2
∆X
* * 1 * 2
=0
*
] = u ∂u + u ∂u + u ∂u + u ∂u = 0
Contents Contents
• Colinear Coupling • Reflective Coupling • IDT Modeling • IDT Properties • SAW Device Simulation • Parameter Extraction • Parameter Extraction of Bidirectional IDTs • Parameter Extraction of Directional IDTs
1 1 2 2
∂X
1
∂X
∂X
2
∂X
Substitution of COM Equations Gives
* 2 Im[βu ] u1 + u2 + Im[( κ − κ '*)u1 u2 ] = 0
(
2
2
)
To Satisfy for Arbitrary u1, u2 & X, Im[βu]=0 & κ’=κ*
Coefficient of Semi-Infinite Grating Looking toward ±X direction
⇒ κ12 is Real When Grating is Symmetric
A±: Amplitude of Partial Wave
Relative Wavenumber
π/|κ|
X
Multi-Strip-Coupler (MSC)
+ = + =
Velocity Difference in Short- & Open-Circuited Gratings BAW SAW Transversal Filter Using MSC
Contents Contents
• Colinear Coupling • Reflective Coupling • IDT Modeling • IDT Properties • SAW Device Simulation • Parameter Extraction • Parameter Extraction of Bidirectional IDTs • Parameter Extraction of Directional IDTs
Ai A1o A2o L Boundary Condition u1(0)=Ai & u2(0)=0 ⇒A+=A-=Ai/2
u 1 = Ai exp( − j β u X ) cos(| κ | X ) u 2 = − jrA i exp( − j β u X ) sin(| κ | X )
|u1| |u2| π/|κ|
SAW SAW Dispersion Dispersion in in Periodic Periodic Structures Structures
−βu +βu−2βｇ −β +β ω +β −β −βu+2βｇ u ｇ u ｇ +βu
−βu−βｇ
+βu+βｇ
−βｇ −βｇ/2
0
+βｇ/2
+βｇ
θp = θu2 -|κ12|2
θu +|κ12| 0 -|κ12| Re[θp] θu Stopband (Evanescent) |κ12| 0 Im[-θp]
|κ12| determines Both Stopband Width & Attenuation Constant
90 45 -5 0 -10 -45 -90 -15 -135 -20 -180 -25 -225 |κ12|p=0.02π -270 -30 -0.08-0.06-0.04-0.02 0 0.02 0.04 0.06 0.08 Relative frequency, θup/π
Coupling-Of-Modes Coupling-Of-Modes (COM) (COM) Theory Theory
Normal Mode Equation
∂u1 = − jβ u u 1 ∂X ∂u2 = − jβ u u 2 ∂X
Uncoupled
[Solution: u i ∝ exp( − jβ u X )]
β
Bragg Reflection
COM COM Analysis Analysis for for Periodic Periodic Structures Structures
Eigen Mode Equations [General Solution: u ± ∝ exp( m jβ u X ) ]
∑A
+∞
+∞
(n) ±
exp(m njβ G X )
u± ( X ) =
n = −∞
∑A
(n)
±
exp( m jβ n X )
Where βn=β0+nβG Incident Wave with β is Spatially Modulated, and Components with β+nβG are Generated.
Introduction to SAW Device Design and Simulation
Part 2: Coupling-Of-Modes Theory Ken-ya Hashimoto Chiba University
k.hashimoto@ http://www.em.eng.chiba-u.jp/~ken
βu: Wavevector at Uncoupled State
Coupling of Modes Equation
∂u1 = − j β u u 1 − jκ u 2 ∂X ∂u 2 = − j β u u 2 − j κ 'u 1 ∂X
Coupled
Loss-Less Condition (Unitary Condition)
0.06 0.04 0.02 0 -0.02 -0.04 -0.06
|κ12|p=0.02π Re(θp/p)
Re(θu/p)
Im(θp/p) -0.04 -0.02 0 0.02 Relative frequency, θup/π 0.04
Behavior Near Bragg Frequency
where θu=βu-βG/2 : Detuning Factor (θu=0 corresponds to Bragg Condition) Origin of Phase in κ12 Displacement of Reflection Center from Origin dr/pI=∠(κ12)/4π X=0 dr