Monotonic Decrease of Reflection Loss with Thickness When Material Characteristic Impedance Matches That of Free Space: A Mathematical Proof

Suppression of Mathematical Rigor: Analysis of Preprints.org’s Rejection of a Logically Rigorous Proof Challenging Mainstream Theory

Suppression of Mathematical Rigor: Analysis of Preprints.org’s Rejection of a Logically Rigorous Proof Challenging Mainstream Theory (2)

Preprint

Liu, Yue, Monotonic Decrease of Reflection Loss with Thickness When Material Characteristic Impedance Matches That of Free Space: A Mathematical Proof (November 21, 2025). Available at SSRN: https://ssrn.com/abstract=5780922 or http://dx.doi.org/10.2139/ssrn.5780922

Monotonic Decrease of Reflection Loss with Thickness When Material Characteristic Impedance Matches That of Free Space: A Mathematical Proof

Yue Liu

College of Chemistry and Chemical Engineering, Shenyang Normal University, Shenyang, China

yueliusd@163.com, ORCID: 0000-0001-5924-9730

Abstract

This paper rigorously proves that when a material’s characteristic impedance (Z_M) equals that of the free space impedance (Z₀), the reflection loss (RL/dB) monotonically decreases with increasing film thickness (d) at a fixed frequency, with no absorption peaks present. Through direct substitution of Z_M = Z₀ into the unmodified reflection loss formula and its mathematical simplification, we derive the exact relationship RL/dB = - 17.37ad, where a is the attenuation constant. Numerical verification confirms that both the unmodified and simplified formulas yield identical results, demonstrating strict monotonic decrease without fluctuations. Crucially, we clarify the fundamental distinction between two scenarios: (1) when Z_M = Z₀, the film’s microwave absorption behavior is identical to the material’s intrinsic absorption (governed solely by attenuation, with no wave interference effects); (2) when Z_M ！= Z₀, , the film’s absorption mechanism fundamentally differs from the material’s absorption due to wave interference effects. This work conclusively demonstrates that impedance matching theory contains logical flaws and should be abandoned, as it confuses Z_M = Z₀ with input impedance Z_in = Z₀.

Keywords: microwave absorption, impedance matching, reflection loss, characteristic impedance, wave interference, attenuation constant

1. Background

In microwave absorbing material research, traditional impedance matching theory posits that optimal absorption occurs when the film input impedance Z_in equals the free space characteristic impedance Z₀, resulting in a confusion between Z_in and the material characteristic impedance Z_M. This theory implies the existence of an “optimal matching condition” for the incident microwaves entering the film, leading researchers to focus extensively on achieving the condition. [1, 2]

This paper proves that when a material’s characteristic impedance Z_M (intrinsic impedance) matching with Z₀), at a fixed frequency, the reflection loss RL/dB strictly monotonically decreases with increasing film thickness d, with no absorption peaks present. This conclusion is verified through direct numerical substitution into the RL formula, fundamentally challenging the traditional theory’s assumption of an “optimal thickness.” [3 – 5]

2. Critical Concept Clarification: Film Absorption vs. Material Absorption

2.1 Essential Distinction Between Two Scenarios

In microwave absorption theory, it is crucial to distinguish between two fundamentally different scenarios for interface reflection in metal-backed film:

2.2 Key Explanation

• When Z_M = Z₀:

  • The material’s intrinsic properties already match free space, theoretically eliminating reflection at interface

  • All incident waves enter the material, with absorption determined solely by material attenuation

  • The metal-backed film’s microwave absorption behavior is identical to that of an infinite bulk material

  • Wave interference effects are absent (no reflected waves at the front interface of the metal-backed film)

  • RL/dB = - 17.37ad, strictly monotonically decreasing with thickness [3]

• When

  

  :

  • Material’s intrinsic properties do not match free space, resulting in reflection

  • Wave interference effects exist due to interactions between reflected waves

  • The film’s microwave absorption behavior fundamentally differs from the material’s intrinsic absorption [6]

  • RL/dB exhibits fluctuation with thickness d (multiple minima points)

  • Absorption performance is dominated by wave interference effects, not solely by material attenuation [7, 8]

Critical insight: When Z_M = Z₀, the film’s microwave absorption equals the material’s microwave absorption; when

, the film’s absorption mechanism fundamentally differs from the material’s absorption. These statements describe physically distinct phenomena under different conditions and are not contradictory.

3. Rigorous Mathematical Proof

3.1 Definitions and Notation

Key condition: Z_M = Z₀ (material’s intrinsic impedance matches Z₀)

3.2 Derivation of Simplified RL/dB Expression

Substitute Z_M = Z₀ into the input impedance formula:

Substitute into the reflection coefficient formula for metal-backed film:

Critical simplification (hyperbolic function identity):

Therefore:

Compute the magnitude with

Final expression for RL/dB:

Conclusion: When Z_M = Z₀, RL/dB strictly monotonically decreases with thickness d for metal-backed film, with no extremum points.

4. Fundamental Conflict with Impedance Matching Theory

4.1 Core Assumption of Traditional Theory

• Optimal absorption occurs when Z_in = Z₀ where all incident microwaves enter the film

• An “optimal matching thickness” d exists that minimizes RL

4.2 Actual Situation When Z_M = Z₀ with Full Penetration

1. Z_in = Z₀ is never achievable with finite thickness:

Only as

does

2. RL/dB has no minimum point:

   • RL/dB monotonically decreases with d, contradicting traditional theory’s prediction of absorption peaks

   • While increasing thickness improves absorption, the rate of improvement diminishes with greater thickness

   • No “optimal thickness” exists in the traditional sense

3. Physical mechanism:

   • When material’s intrinsic impedance matches free space (Z_M = Z₀), incident waves enter without reflection at the front interface

   • Absorption of the metal-backed film is determined solely by material attenuation: RL/dB = - 17.37ad

   • Increasing thickness d directly enhances attenuation, without needing to adjust to a “matching point”

5. Conclusions and Implications

1. Mathematical rigor:

   • Direct substitution of Z_M = Z₀ into the RL formula proves RL/dB = - 17.37ad

   • Unmodified and simplified formulas yield identical results, confirming monotonic decrease

2. Fundamental rejection of impedance matching theory:

   • When Z_M = Z₀, no absorption peaks exist, completely invalidating the concept of “optimal matching thickness”

   • Impedance matching theory contains logical flaws in its foundational assumptions and should be abandoned

   • The condition Z_M = Z₀ is neither necessary nor sufficient for optimal absorption:

     • Necessity: When Z_in = Z₀, the optimal absorption doesn’t require Z_M = Z₀

     • Sufficiency: Even when Z_M = Z₀, RL/dB may not be optimal

3. Correct design principles:

   • When Z_M = Z₀: Film absorption equals material absorption, with RL/dB monotonically decreasing with thickness

   • When

Film absorption differs fundamentally from material absorption due to wave interference [6]

• Absorption performance must be evaluated considering both material properties and the effects at the interfaces of film

• Optimization requires balancing multiple factors including attenuation and phase effects

6. Appendix: Universality of Wave Mechanics Theory

This work is fully grounded in wave mechanics theory (Maxwell’s equations + boundary conditions), demonstrating [9]:

• When Z_M = Z₀, wave interference effects vanish, and absorption is determined solely by material attenuation a

• When

wave interference effects dominate absorption behavior

• Wave mechanics theory applies universally to all scenarios (Z_M = Z₀ or

)

The correct theoretical framework for microwave absorption must distinguish between two scenarios:

• Z_M = Z₀: Film absorption equals material absorption, with RL/dB monotonically decreasing with thickness

Film absorption is affected by wave interference, with RL/dB fluctuating with thickness

References:

1. Ying Liu, Yi Ding, Yue Liu, Michael G. B. Drew. Unexpected Results in Microwave Absorption – Part 1: Different absorption mechanisms for metal-backed film and for material, Surfaces and Interfaces, 2023, 40, 103022

2. Liu Y, Drew MGB, Li H, Liu Y. An experimental and theoretical investigation into methods concerned with “reflection loss” for microwave absorbing materials. Materials Chemistry and Physics 2020, 243 : 122624.

3. Yue Liu, Michael G.B. Drew, Ying Liu, Theoretical Insights Manifested by Wave Mechanics Theory of Microwave Absorption—Part 1: A Theoretical Perspective, Preprints.org, Preprint, 2025, DOI:10.20944/preprints202503.0314.v5, supplementary.docx

4. Liu Y, Liu Y, Drew MGB. A theoretical investigation of the quarter-wavelength model — part 2: verification and extension. Physica Scripta 2022, 97(1) : 015806.

5. Ying Liu, Michael G. B. Drew, Yue Liu, A physics investigation on impedance matching theory in microwave absorption film—Part 2: Problem Analyses, Journal of Applied Physics, 2023, 134(4), 045304, DOI: 10.1063/5.0153612

6. Yue Liu，Ying Liu，Michael G. B Drew，Wave Mechanics of Microwave Absorption in Films - Distinguishing Film from Material，Journal of Magnetism and Magnetic Materials，2024，593, 171850

7. Ying Liu, Yue Liu, Drew M.G.B, A re-evaluation of the mechanism of microwave absorption in film – Part 2: The Real mechanism, Mater. Chem. Phys,. 2022, 291, 126601

8. Yue Liu，Ying Liu，Michael G. B Drew. Review：Wave mechanics of microwave absorption in films: a short review, Optics and Laser Technology, 2024, 178, 111211

9. Blog, https://blog.sciencenet.cn/blog-3589443-1510827.html

10. Yue Liu, The True Significance of Wave Mechanics Theory in Microwave Absorption Research: Redirecting Scientific Inquiry Toward Meaningful Theoretical Advancement, Preprints.org, Preprint, 2025, DOI:10.20944/preprints202511.1214.v1

11. Yue Liu, Ying Liu, Michael G. B. Drew. (2025) Objective Evaluation of Impedance Matching Theory versus Wave Mechanics Theory for Microwave Absorption: A Theoretical Analysis Using Transmission Line Principles Subtitle: A new development of microwave theory in the field of material and device. Journal of Electromagnetics, 8, 15-20; Objective Evaluation of Impedance Matching Theory versus Wave Mechanics Theory for Microwave Absorption: A Theoretical Analysis Using Transmission Line Principles

[Preprints.org] Preprints ID: preprints-186039 - Your Preprint Is Declined for Announcement

• 发件人：

  jade.zhou<jade.zhou@preprints.org>+

• 收件人：

  我<yueliusd@163.com>

• 抄送人：

  jade.zhou<jade.zhou@preprints.org>+

• 时 间：

  2025年11月21日 09:44 (星期五)

Dear authors,

It is with regret that we must inform you that your following submission to Preprints has been declined for announcement:

Preprints ID: preprints-186039
Type: Article
Title: Monotonic Decrease of Reflection Loss with Thickness When Material Characteristic Impedance Matches That of Free Space: A Mathematical Proof
Authors: Yue Liu
Emails: yueliusd@163.com
Submission received: 2025-11-20

You may revise your manuscript according to our instructions here: https://www.preprints.org/instructions-for-authors.

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Re:[Preprints.org] Preprints ID: preprints-186039 - Your Preprint Is Declined for Announcement

• 发件人：

  yueliusd<yueliusd@163.com>

• 收件人：

  jade.zhou<jade.zhou@preprints.org>

• 时 间：

  2025年11月21日 18:12 (星期五)

Suppression of Mathematical Rigor: Analysis of Preprints.org’s Rejection of a Logically Rigorous Proof Challenging Mainstream Theory

Monotonic Decrease of Reflection Loss with Thickness When Material Characteristic Impedance Matches That of Free Space: A Mathematical Proof

Dear Preprints.org Editorial Team,

I received notification (November 21, 2025) that my manuscript (ID: preprints-186039) was declined for announcement. The notification references “our instructions” but provides no specific explanation of what guidelines my manuscript violates.

I have reviewed Preprints.org’s instructions for authors and find that my manuscript:

1. Presents rigorous mathematical analysis within appropriate academic category

2. Contains no ethical violations or plagiarism

3. Falls within stated scope of Preprints.org

I respectfully request:

1. Specific identification of which guideline(s) my manuscript violates

2. Explanation of how rigorous mathematical proof challenging established theory violates Preprints.org policies

3. Identification of specific mathematical or logical errors, if such exist

4. Pathway for remediation if legitimate concerns exist

If Preprints.org’s policy is to suppress mathematical proofs challenging mainstream theories, this policy should be:

• Made explicit in guidelines

• Justified on scientific grounds

• Applied consistently to all such work

Publishing decisions that cannot be justified on the basis of mathematical rigor or logical coherence undermine Preprints.org’s credibility and contradict its stated mission of providing unrestricted publication.

I await substantive response.

Respectfully,

Yue Liu