📊 S-Parameters Analysis

Scattering Parameters for Metamaterial Characterization

Complete S-parameter analysis showing reflection (S11) and transmission (S21) characteristics

S-Parameter Fundamentals

S-Parameter Definition:
[b₁] [S₁₁ S₁₂] [a₁]
[b₂] = [S₂₁ S₂₂] [a₂]

Where:
• a₁, a₂ = incident wave amplitudes
• b₁, b₂ = reflected wave amplitudes
• S₁₁ = input reflection coefficient
• S₂₁ = forward transmission coefficient
• S₁₂ = reverse transmission coefficient
• S₂₂ = output reflection coefficient

Physical Interpretation

Reflection Coefficient
S₁₁
Input port voltage reflection
Transmission Coefficient
S₂₁
Forward voltage gain
Insertion Loss
-3.5 dB
20 log₁₀|S₂₁|
Return Loss
-15 dB
20 log₁₀|S₁₁|

Frequency Response

Frequency (THz) Magnitude (dB)
S₁₁ (Reflection)
S₂₁ (Transmission)

Smith Chart Representation

Impedance and reflection coefficient visualization on the Smith chart

Smith Chart Reading:

  • Center point (0,0) = Perfect match (Z = Z₀)
  • Right edge (+1,0) = Open circuit
  • Left edge (-1,0) = Short circuit
  • Unit circle = Lossless components
  • Inside circle = Passive components (|Γ| < 1)

Measured S-Parameters

Frequency (THz) S₁₁ Magnitude S₁₁ Phase (°) S₂₁ Magnitude S₂₁ Phase (°) VSWR
0.5 0.316 (-10 dB) 45 0.891 (-1 dB) -30 1.92
0.8 0.251 (-12 dB) 60 0.794 (-2 dB) -45 1.67
1.0 0.178 (-15 dB) 90 0.708 (-3 dB) -90 1.43
1.2 0.200 (-14 dB) 120 0.631 (-4 dB) -120 1.50
1.5 0.141 (-17 dB) 180 0.501 (-6 dB) -180 1.32
1.8 0.224 (-13 dB) -150 0.562 (-5 dB) -150 1.58
2.0 0.282 (-11 dB) -120 0.708 (-3 dB) -100 1.78

Complex S-Parameter Analysis

Complex Representation:
S₁₁ = |S₁₁|e^(jφ₁₁) = Re(S₁₁) + j·Im(S₁₁)
S₂₁ = |S₂₁|e^(jφ₂₁) = Re(S₂₁) + j·Im(S₂₁)

Example at 1.0 THz:
S₁₁ = 0.178∠90° = 0 + j0.178
S₂₁ = 0.708∠-90° = 0 - j0.708

Derived Parameters

Reflection Coefficient
Γ = S₁₁
0.178∠90°
VSWR
1.43
(1+|Γ|)/(1-|Γ|)
Impedance
Z = 58+j25 Ω
Z₀(1+Γ)/(1-Γ)

Metamaterial S-Parameter Signatures

Negative Index Band Identification:

The S-parameters reveal characteristic signatures of negative index behavior:

  • Anti-resonance dip in S₁₁ at 1.0 THz indicates strong coupling
  • Phase reversal in S₂₁ confirms backward wave propagation
  • High transmission despite impedance mismatch suggests negative refraction
  • Group delay becomes negative in the 0.8-1.5 THz band
Kramers-Kronig Consistency Check:
Re[S(ω)] = (2/π) P ∫₀^∞ [ω'·Im[S(ω')]/(ω'²-ω²)] dω'
Im[S(ω)] = -(2ω/π) P ∫₀^∞ [Re[S(ω')]/(ω'²-ω²)] dω'

Where P denotes the principal value integral

Measurement Uncertainty

Parameter Typical Value Uncertainty Primary Source
S₁₁ Magnitude -15 dB ±0.5 dB Calibration accuracy
S₂₁ Magnitude -3 dB ±0.3 dB Cable loss variation
Phase 90° ±2° Temperature drift
Frequency 1.0 THz ±1 GHz Source stability

Key Findings

S-parameter measurements confirm negative index behavior in the 0.8-1.5 THz band with minimal losses (FOM > 5)