Sub-Terahertz Dielectric Permittivity Tensor Discrepancies between Resonator Coupons and Quasi-Optical In-Situ Measurements

Slab

At sub-millimetre wavelengths, high-frequency circuit laminates stop behaving as uniform continuous mediums and act as heterogeneous composite structures. When signal frequencies enter the D-band (110 GHz to 170 GHz) and G-band (140 GHz to 220 GHz), reinforcement glass bundle dimensions approach the guided wavelength itself. Inside a PTFE composite, a 140 GHz signal has a guided wavelength of roughly 1.2 millimetres.

At that scale, electromagnetic fields interact directly with constituent boundaries ~ E-glass filaments, resin pockets, micro-glass spheres, and fluoropolymer matrices.

Dielectric permittivity at sub-terahertz frequencies is a second-order tensor matrix rather than a scalar value. Datasheets historically report scalar relative permittivity measured at 10 GHz with split-post dielectric resonators or clamped stripline resonators, but those figures mask directional variations across a pressed laminate panel. Matrix notation separates spatial permittivity into three orthogonal axes: x-axis in-plane transverse, y-axis in-plane longitudinal, and z-axis out-of-plane thickness.

The structural asymmetry of woven reinforcement fabrics leaves the in-plane components higher than the out-of-plane value.

Glass styles such as 1035, 1078, and 1067 introduce periodic spatial variations in dielectric constant across the panel. Pure E-glass has a dielectric constant near 6.6 at 140 GHz, whereas the surrounding fluoropolymer or hydrocarbon resin matrix sits between 2.0 and 2.4. Waves travelling parallel to the glass bundles see an effective permittivity tied to the volume fraction of glass along that axis.

Waves moving normal to the panel face pass through alternating thin layers of resin and glass cloth, following a series capacitive model that yields a lower numerical value than the in-plane parallel capacitive model.

Composite Glass Weave and Dielectric Anisotropy

Filament distribution inside thin core laminates governs the ratio between in-plane and out-of-plane tensor values. Unreinforced liquid crystal polymer films align along the extrusion axis during manufacturing, raising in-plane permittivity above out-of-plane permittivity. Dispersed micro-glass particle fills narrow this gap by randomizing filler orientation, and pressed PTFE substrates filled with random glass microfibres show less tensor divergence than glass cloth laminates.

In 1078 weave, the spatial pitch of the glass bundles produces dielectric hot spots spaced every 430 micrometres across the circuit layout.

Resin content shifts tensor anisotropy between lamination batches. High-resin cores (68 percent resin by weight) exhibit lower out-of-plane dielectric values than low-resin alternatives (45 percent resin by weight), where denser glass pulls the z-axis dielectric constant toward bulk glass values. Microsections show that lamination pressure drives matrix resin into the gaps between warp and weft yarns, leaving resin-rich pockets on outer core surfaces that alter local field confinement for surface microstrips.

Standard IPC-4101 slash sheets document scalar permittivity at 10 GHz using resonant methods that ignore out-of-plane tensor components operating in sub-terahertz transmission lines.

High-pressure lamination flattens surface glass bundles and alters the through-thickness fill fraction profile. Hydraulic press cycles above 25 bars compress round filaments into oval cross-sections, concentrating glass packing density near the laminate centreline. As a result of this distortion, the z-axis permittivity tensor component varies with depth through a single core layer, giving striplines on internal layers a different effective permittivity profile than microstrips routed on outer layers.

Matrix Formulations for Thin Substrate Permittivity

Representing the anisotropic permittivity tensor mathematically requires a diagonal matrix when principal material axes align with the circuit panel’s mechanical coordinates. The diagonal elements record relative permittivity along the orthogonal directions of the board.

Equation 1 defines the relative permittivity tensor matrix operating within an anisotropic laminate:

ε_r = , , ]

In isotropic homogeneous materials, all three diagonal elements are identical. In high-frequency glass-reinforced laminates, in-plane components exceed the out-of-plane component. Measurements of 1078 glass cloth PTFE laminates at 140 GHz show in-plane permittivity values of 3.32 along the warp yarn and 3.28 along the fill yarn, against an out-of-plane z-axis permittivity of 3.02.

Omitting this difference introduces phase velocity errors into planar transmission line models.

Phase velocity along a surface microstrip trace depends on the effective permittivity of the surrounding space. Electric field lines extend partly through air above the trace and partly through the substrate below. Inside the substrate, fields spread laterally across the trace width and vertically through the core thickness.

Vertical field lines couple to the z-axis permittivity element, while fringing fields extending sideways from the trace edges interact primarily with the in-plane transverse element.

Phase velocity shifts in sub-terahertz circuits scale directly with this combined interaction. A design calibrated strictly to the out-of-plane tensor element underestimates capacitance from lateral fringing fields, leaving microstrips with higher effective permittivity than single-axis z-axis resonator tests predict. Striplines confine fields entirely within the dielectric, but fringing fields remain sensitive to the ratio between in-plane and out-of-plane tensor elements.

Designing phase-matched interconnect lines at sub-terahertz frequencies requires using the full tensor matrix across the operational bandwidth.

Sub-terahertz substrate studies leave open the question of how local thermal expansion in the resin alters this tensor ratio during high-temperature operation.

Beam

Free-space quasi-optical benches evaluate unclad dielectric panels by focusing electromagnetic energy through free-space optical trains. Off-axis parabolic mirrors or hyper-hemispherical dielectric lenses direct a Gaussian beam waist onto the test specimen, while horn antennas coupled to vector network analyzer frequency extenders launch fundamental waveguide modes into space. At 140 GHz, the Gaussian spot size at the focal plane spans 8 to 15 millimetres in diameter depending on numerical aperture, averaging dielectric properties across hundreds of glass weave periods.

Resonator coupons rely instead on field confinement inside metallic cavities or along etched planar transmission lines. Split-cylinder resonators operating under IPC-TM-650 Method 2.5.5.13 excite TE011 cavity modes within a cylindrical volume clamped around an unclad laminate sample. Balanced circular disk resonators and ring resonators use copper features etched directly onto the test laminate.

These resonator techniques extract permittivity from measured resonant frequencies and quality factors. Because their electric fields concentrate within small volumes, results are sensitive to local copper roughness and glass filament density.

Resonator Coupon Field Confinement Mechanics

Planar resonator coupons concentrate energy into microscopic boundary zones. A microstrip ring resonator for 140 GHz operation uses a trace width of roughly 110 micrometres on a 100-micrometre substrate, packing fields tightly into the dielectric beneath the trace. In such volumes, local resin pockets, glass yarn crossings, or copper surface teeth exert heavy influence on the resonant frequency, so the extracted permittivity reflects the local microstructure under that coupon trace rather than a bulk average.

Split-cylinder resonators test unclad substrate sheets without etched copper conductors. The cylindrical cavity splits along its mid-plane to receive the panel, and the TE011 mode sets up circular electric field lines parallel to the sheet. This field layout interacts exclusively with in-plane tensor elements ε_xx and ε_yy, with no coupling to the out-of-plane z-axis element ε_zz.

Permittivity derived from split-cylinder measurements therefore overstates the dielectric constant that microstrip z-axis fields encounter.

Balanced circular disk resonators use metal cladding on both sides of a circular dielectric disk. Exciting the TM010 mode generates electric field lines oriented strictly along the z-axis normal to the disk, coupling directly to the out-of-plane element ε_zz. Bonded copper foil, however, adds conductor loss and surface profile capacitance to the measurement.

Separating copper roughness losses from dielectric dissipation at sub-terahertz frequencies requires mathematical models of metal profile reactance.

Whose Calibration Standard Governs Dielectric Tensor Verification?

Quasi-optical free-space systems measure reflection and transmission coefficients (S11 and S21) across broad bands without contacting the material. Time-domain gating isolates primary reflections at the sample surface from fixture reflections. Calibration protocols like Thru-Reflect-Line or Line-Reflect-Match set reference planes on the material face.

Using polarization control plates, these optical setups rotate the incident electric field vector relative to the material axis, extracting both in-plane tensor components in a single sweep.

Ellipsometric quasi-optical setups illuminate substrate samples at oblique incidence angles. By measuring how reflected polarization changes at Brewster’s angle, they separate in-plane and out-of-plane permittivity. Because the optical beam covers a bulk volume containing tens of thousands of glass yarn intersections, microscopic spatial variations average out into a baseline for the material class.

Free-space quasi-optical ellipsometry at 140 GHz isolates pure substrate permittivity tensor components without introducing copper foil profile reactance or edge fringing distortions.

Table 1 details the operational capabilities, field alignments, and extraction sensitivities across primary sub-terahertz dielectric test methods.

Choosing a test method sets the baseline data for high-frequency stackup design dockets.

Resonator Method Alignment ~ Match the fixture’s electric field orientation to the primary field mode of the target PCB transmission line.

Sample Cladding State ~ Use unclad dielectric panels to extract intrinsic material tensors, and clad coupons to verify composite line impedance.

Spot Size Ratio ~ Ensure the test field volume spans at least ten unit cells of the fabric weave to prevent local sampling bias.

Roughness Decoupling ~ Use electro-thermal matrix extraction software when separating conductor profile reactance from dielectric permittivity in clad resonator tests.

Free-space optical measurements isolate dielectric properties directly, but etched copper profiles on finished planar circuits introduce phase velocity shifts that optical benches do not capture.

Variance

Discrepancies between resonator coupon extractions and free-space measurements stem from microstructural effects at the copper-dielectric interface. Comparing dielectric values from an etched microstrip ring resonator against those from a quasi-optical free-space bench reveals a consistent offset: the resonator coupon yields higher effective relative permittivity. At 140 GHz, this gap ranges from 0.15 to 0.45 permittivity units on standard ultra-low-loss glass-reinforced hydrocarbon substrates ~ a difference driven by interfacial physics rather than test noise.

Copper foil treatment creates a boundary where metal teeth embed into resin during lamination. Standard electrodeposited foil has high roughness, with peak-to-valley profile heights (R_z) exceeding 3.0 micrometres. Very low profile (VLP) and ultra-low profile (ULP) foils drop this to between 0.6 and 1.2 micrometres, while rolled annealed copper reaches profiles below 0.4 micrometres.

At 140 GHz, conductor skin depth drops below 180 nanometres, confining current paths to the rough surface profile of the foil.

Copper Profile Reactance and Interface Capacitance

The boundary between rough copper and substrate forms a localized slow-wave structure that acts as added profile capacitance. Fields penetrating the valleys between copper teeth concentrate locally, slowing guided wave phase velocity. When numerical extraction algorithms process this delay assuming a smooth conductor boundary, they assign the extra phase delay to an inflated substrate dielectric constant.

Modified Huray cannonball models represent surface roughness as stacks of conductive micro-spheres embedded in the dielectric matrix. These spheres alter interface boundary conditions, adding profile capacitance directly to parallel-plate capacitance and shifting phase velocity. The magnitude of this shift scales inversely with substrate thickness: thin cores such as 50-micrometre or 75-micrometre dielectrics are far more sensitive to profile reactance than 200-micrometre cores.

A microstrip trace on a 50-micrometre core with a 1.5-micrometre copper surface profile shows an apparent permittivity shift of +0.32 in high-frequency resonator calculations. The molecular polarizability of the dielectric remains unchanged; instead, conductor micro-geometry introduces a reactive impedance layer that mimics a higher bulk permittivity.

Boundary Layers and Local Resin Fill Shift

Lamination fluid dynamics create a resin boundary layer between copper foil and internal glass fabric. Because glass filaments cannot press fully flush against solid copper during pressing, matrix resin flows around yarns to wet the foil, leaving a glass-free resin layer 5 to 15 micrometres thick beside the metal. This boundary layer has a lower permittivity than the core bulk, where glass bundles are packed closely.

Planar resonator microstrips concentrate peak electric fields within this interface resin layer, with field strength decaying exponentially into the substrate. Because peak fields reside inside the resin zone, the effective permittivity seen by the trace shifts downward toward the resin value. Concurrently, copper roughness embedded in that resin layer creates an opposing upward shift.

The apparent permittivity extracted from a resonator coupon reflects resin dilution competing against copper profile capacitance.

Quasi-optical free-space measurements transmit electromagnetic beams through the full cross-section of unclad samples, bypassing foil and boundary layers to extract the intrinsic volume-averaged permittivity tensor of the bulk dielectric. Using unclad quasi-optical numbers in planar circuit equations without correcting for copper profile reactance yields transmission lines with lower phase velocities and lower characteristic impedances than specified.

Volumetric Integration against Local Field Peaks

Spatial integration scales differ between coupon resonators and free-space optical systems. Quasi-optical Gaussian beams cover hundreds of glass weave periods, averaging local variations into a single macroscopic tensor. Microstrip resonator traces have conductor widths comparable to or smaller than individual glass filaments and weave gaps.

A 110-micrometre microstrip routed parallel to warp yarns can land directly on a glass bundle intersection or sit entirely over a resin window between bundles.

Routing a transmission line over a glass bundle raises local permittivity, while placing it over a resin window lowers it. Microstrip coupon resonators record the specific local environment beneath that trace. Testing multiple coupons across a panel reveals permittivity variations of ±0.08 units caused solely by trace-to-weave alignment, whereas free-space optical systems smooth these variations into a volumetric baseline.

Designing sub-terahertz microstrip circuits using unclad bulk dielectric permittivity figures without copper roughness corrections causes target 50-ohm transmission lines to yield measured characteristic impedances below 44 ohms.

Table 2 provides measured permittivity discrepancy data across laminate material classes, copper profile treatments, and test methods at 140 GHz.

The extraction failure modes below illustrate physical mechanisms that degrade high-frequency performance when tensor shifts are left unmodeled.

• Profile Reactance Overestimation ~ Treating copper roughness delay as dielectric permittivity distorts physical trace lengths in phase-critical matching networks.
• Substrate Thinning Error ~ Applying thick-substrate profile models to cores under 75 micrometres introduces substantial impedance errors.
• Weave-Alignment Blindness ~ Routing sub-terahertz paths without tracking glass bundle pitch creates unmodeled phase skew across parallel differential channels.
• Tensor Component Substitution ~ Using in-plane split-cylinder permittivity values in z-axis microstrip calculations causes systematic phase velocity errors.

Failing to resolve these extraction discrepancies leads to multi-layer boards that miss return loss specifications, forcing expensive redesigns and pushing out schedules.

Stack

Layer allocation and dielectric thickness turn material properties into physical board dimensions. Designers must translate tensor permittivity values into executable artwork geometries and laminate specs. When building multi-layer stackups for 140 GHz radar or communications modules, layer thickness sets the tolerance window for controlled impedance lines.

Typical core thicknesses at these frequencies include 50-micrometre, 75-micrometre, and 100-micrometre ultra-low-loss dielectrics combined with thin prepreg bonding sheets.

Pressed dielectric thickness after lamination diverges from nominal unpressed catalog specs. As prepreg flows during lamination, liquid resin fills the spaces between etched inner-layer traces. This movement thins prepreg over solid copper planes while shifting local glass-to-resin ratios: a prepreg sheet specified at 60 micrometres nominal often presses down to 48 micrometres over copper planes and 54 micrometres over clearance areas.

Stackup calculations must rely on pressed thickness values rather than nominal sheet specs.

Trace Width Adjustments for Anisotropic Permittivity

Calculating trace widths for microstrips and striplines requires dual-tensor inputs to hit target impedances. Conventional field solvers operating on single scalar permittivity generate widths that miss physical circuit performance. On a 75-micrometre core with an in-plane permittivity (ε_xx) of 3.30 and an out-of-plane permittivity (ε_zz) of 3.02, a scalar solver using ε_zz yields a 178-micrometre trace for 50 ohms.

Accounting for the full tensor matrix shows that lateral fringing fields increase effective capacitance, requiring a 162-micrometre trace to hit 50 ohms.

Fabrication tolerances compound this discrepancy on the shop floor. Etch factor distortions turn rectangular trace profiles into trapezoids. A trace specified with a 162-micrometre top width can end up with a 178-micrometre base because of chemical undercut during etching.

That trapezoidal profile increases ground capacitance and pulls characteristic impedance lower, requiring fabricators to apply etch compensation while accounting for dual-tensor values.

IPC-6012 Class 3 performance specifications mandate tight dielectric thickness controls that require fabricators to measure pressed stackup cross-sections using calibrated optical microsections.

Table 3 shows a stackup impedance sensitivity matrix comparing single-scalar design assumptions against full-tensor design models for a 140 GHz microstrip channel.

Fabrication Drawing Notes for High Frequency Boards

Procuring sub-terahertz boards accurately requires drawing notes that explicitly hold suppliers to tensor-aware material controls. Relying on generic IPC callouts leaves room for fabricators to substitute lookalike materials from different slash sheets with very different tensor ratios.

The steps below outline the procedure for releasing high-frequency panel stackup drawings to production fabricators.

1. Specify the baseline substrate by brand, core construction style, resin content, and copper foil profile code on the master drawing.
2. Define dual-tensor permittivity acceptance limits (ε_xx and ε_zz) at the operating frequency, specifying the test method required for lot verification.
3. Include etch-compensated trace geometry tables listing target nominal widths alongside acceptable trapezoidal limits for each impedance layer.
4. Require TDR impedance coupon trace data generated from IPC-2631 style coupons placed on all four panel margins.
5. Mandate microsection coupons on every production panel to verify pressed dielectric thickness and copper tooth penetration prior to shipment.

Standard IPC-6012 Clause 3.6.2 sets core dielectric thickness tolerances within ±10 percent of drawing nominal unless overridden by master drawing notes.

Ledger

Panel procurement costs depend heavily on material choice, copper foil treatment, and dielectric tolerances. Higher performance at sub-terahertz frequencies comes at a steep price: standard high-Tg FR-4 runs roughly 15 to 22 dollars per square metre, whereas premium hydrocarbon ceramic laminates run between 180 and 260 dollars. Fluoropolymer PTFE materials with woven micro-glass reinforcement cost 350 to 520 dollars per square metre, and liquid crystal polymer core sheets exceed 800 dollars per square metre in production quantities.

Panel utilization dictates unit cost for bare boards. Standard production panels in advanced PCB plants measure 457 millimetres by 610 millimetres (18 inches by 24 inches). A sub-terahertz module measuring 40 millimetres by 40 millimetres fits 108 units onto a panel once routing channels, margins, and test coupons are subtracted.

Wasted panel area directly increases the unit cost allocated to each usable board.

Panel Yield Mechanics under Strict Tolerance Windows

Permittivity tolerances restrict panel yields quickly. Standard commercial laminate specs guarantee dielectric constants within ±0.05 units of nominal, but sub-terahertz designs often demand ±0.02 units to maintain phase alignment across antenna arrays. Meeting that tighter window requires laminate suppliers to sort batches and reject up to 40 percent of standard production runs ~ a scrap expense passed directly to the buyer as a surcharge.

Yields drop sharply as layer count and feature density rise. A four-layer sub-terahertz board with 75-micrometre traces and 100-micrometre microvias achieves around 88 percent panel yield in high-volume plants. Tightening impedance tolerances from ±10 percent to ±5 percent drops yield to 62 percent, and holding ±3 percent reduces yield below 35 percent.

Purchase orders absorb the cost of scrapped panels during tight-tolerance fabrication runs.

Commercial Sourcing Strategies for Sub-Terahertz Laminates

Sourcing sub-terahertz circuit boards requires balancing technical specs against supplier availability. Specifying single-source laminate grades leaves builds vulnerable to supply disruptions and long lead times, which routinely stretch from 6 to 18 weeks during material bottlenecks. Qualifying alternative laminates from secondary vendors takes thorough dual-tensor testing to confirm matching performance on the bench.

Procurement dockets for sub-terahertz bare boards require complete documentation to verify material traceability and performance.

• Certificate of Conformance ~ Lot traceability documentation linking panel batch numbers to raw material lamination runs.
• Permittivity Test Report ~ Coupon extraction data reporting dielectric values measured at operating frequencies.
• Copper Roughness Profilometry ~ Conductor profile certification confirming peak-to-valley roughness meets drawing limits.
• Microsection Dossier ~ Cross-sectional optical micrographs proving pressed dielectric thickness and line dimensions meet IPC standards.

Panel yield improves when stackup specs align with standard fab capabilities instead of pushing tight process limits.