Measurement of electromagnetic radiation force using a capacitance-bridge interferometer

This paper presents a tabletop interferometer using a mechanical cantilever and a capacitance-bridge geometry to measure nano-newton radiation forces induced by a high-power pulsed laser, utilizing standard undergraduate laboratory equipment.

The Gist

The Light-Pushing Experiment: Catching a Ghostly Breeze

Imagine you are standing on a beach on a perfectly still day. You can’t see the wind, but you see a tiny, delicate feather dancing in the air. Even though the air feels still, you know that if a tiny puff of wind hits that feather, it will move.

Now, imagine that instead of wind, you are using light.

Most people think of light as something that just illuminates a room. But in physics, light actually carries "momentum"—it’s like a stream of microscopic, invisible ping-pong balls hitting everything they touch. This "light pressure" is incredibly weak. To feel it with your hand would be like trying to feel the weight of a single grain of sand falling on a moving train. It is almost impossible to detect.

In this paper, researchers from IIT Bombay have built a "high-tech feather" to catch these invisible light-balls.

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The Setup: The World’s Most Sensitive Seesaw

To catch this tiny force, the scientists built a device that acts like a super-sensitive musical instrument. Here is how it works, broken down into three parts:

1. The "Feather" (The Cantilever)

Instead of a feather, they used a tiny, thin strip of brass (a metal). This strip is suspended over a circuit board, hanging there like a diving board. Because it is so thin and light, even the tiniest "nudge" from a laser beam will make it vibrate.

2. The "Microscope" (The Capacitance Bridge)

How do you measure a movement so small that even the best microscopes can't see it? You don't use eyes; you use electricity.

The metal strip and the circuit board act like two plates of a capacitor (a component that stores electricity). When the metal strip moves even a fraction of a hair's width closer to the board, the amount of electricity it can hold changes.

To detect this, they use a "Capacitance Bridge." Think of this like a perfectly balanced scale. On one side, you have the "moving" metal strip; on the other, a "still" reference strip. If the scale is perfectly balanced, the electricity flows smoothly. But the moment the laser hits the moving strip and nudges it, the scale tips. The scientists aren't looking for the movement itself; they are looking for the "electrical tilt" caused by that movement.

3. The "Drummer" (The Pulsed Laser)

If you hit a drum once, it makes a sound. If you hit it at just the right rhythm, it gets louder and louder. The scientists did the same thing with light. They used a laser that "pulses" (turns on and off) at a specific rhythm. By timing the pulses to match the natural "heartbeat" (the resonant frequency) of the metal strip, they made the tiny nudges add up, causing the strip to vibrate more noticeably.

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The Result: Measuring the Invisible

By watching how the electricity "tilted" on their electronic scale, the researchers were able to calculate exactly how much force the light was exerting.

They measured a force of about 1.47 nano-newtons. To give you an idea of how small that is: a nano-newton is one-billionth of a Newton. It is a force so small it is practically a ghost.

Why does this matter?

You might ask, "Why spend all this effort to measure a force that small?"

1. Education: The researchers designed this so it could be built in a standard college lab. It turns abstract, "invisible" math equations from textbooks into something students can actually see and measure on a screen.
2. Precision Technology: Understanding how light pushes on objects is vital for the future of technology—from the tiny sensors in your smartphone to the massive satellites in space that use "solar sails" to travel through the solar system using nothing but the pressure of sunlight.

In short: They built a way to "feel" the touch of light using a tiny metal diving board and an electrical scale.

Technical Summary

Technical Summary: Measurement of Electromagnetic Radiation Force Using a Capacitance-Bridge Interferometer

Authors: Devashish Shah, Pradum Kumar, and Pradeep Sarin (IIT Bombay)
Date: April 27, 2026

1. Problem Statement

While the momentum density and radiation force of electromagnetic waves are fundamental concepts in advanced electromagnetism, they are often treated as purely theoretical due to the extreme difficulty of quantifying these forces on macroscopic objects. Historically, measuring the nano-Newton (nN) scale forces exerted by light requires highly specialized, expensive equipment such as micromachined resonators or precision optical setups. There is a pedagogical and practical gap for a method that can demonstrate these phenomena using standard, accessible laboratory equipment.

2. Methodology

The researchers developed a tabletop, mechanical cantilever-based interferometer designed to detect minute capacitance changes induced by radiation pressure.

• Mechanical Setup: A thin brass cantilever (the Device Under Test, or DUT) is suspended parallel to a copper Printed Circuit Board (PCB) trace. This configuration forms an air capacitor ($C_{DUT}$).
• Sensing Mechanism (Capacitance Bridge): The experiment utilizes a "capacitance bridge" geometry. A reference capacitor ($C_{Ref}$) is placed in a bridge configuration with the $C_{DUT}$. By applying out-of-phase sinusoidal voltages from an Arbitrary Function Generator (AFG), the bridge is balanced ($V_{Bridge} \approx 0$). Any mechanical deflection of the cantilever changes the air gap ($d_0$), altering $C_{DUT}$ and creating a measurable voltage change ($\Delta V_{Bridge}$).
• Signal Processing: The tiny capacitance changes (on the order of femto-farads, fF) are amplified using an inverting amplifier circuit (based on the LF411C Op-Amp) integrated directly onto the PCB to minimize noise and parasitic capacitance.
• Excitation: A high-power (~1 W) blue laser ($\lambda \simeq 450$ nm) is pulsed at a frequency ($f_L$) tuned to the cantilever's resonant frequency. This resonant driving maximizes the mechanical oscillation amplitude, making the radiation force easier to detect.

3. Key Contributions

• Novel Low-Cost Interferometer: The authors present a design that achieves high sensitivity (detecting fF-scale changes) using equipment typically found in undergraduate physics laboratories (AFGs, oscilloscopes, and standard Op-Amps).
• Integrated Circuit Design: The development of a specialized PCB that integrates both the capacitance bridge and the amplification block, effectively managing parasitic capacitances ($C_{G1}, C_{G2}$) through LTSpice modeling and experimental calibration.
• Bridging Theory and Experiment: The paper provides a complete analytical framework—linking electromagnetic momentum density to mechanical cantilever dynamics and electrical bridge theory—to allow for the direct calculation of force from electrical signals.

4. Results

• Resonant Frequency: The cantilever's first fundamental frequency was experimentally determined to be 38.881 Hz, showing excellent agreement with the theoretical estimate of $40 \pm 4.9$ Hz.
• Force Measurement: By measuring the peak-to-peak change in the output voltage ($\Delta V_{Out}$) during resonant laser excitation, the authors calculated a radiation force of $1.47 \pm 0.53$ nN.
• Validation: The measured force corresponds to an average optical power of approximately 0.3 W, which aligns closely with the directly measured laser power of $0.4 \pm 0.02$ W (accounting for the uncertainty in the brass absorption/reflection coefficients).
• Harmonic Detection: The system successfully detected the second flexural harmonic of the cantilever at approximately 193 Hz.

5. Significance

This work is significant for both educational and scientific purposes. It provides a robust, "unambiguous" method for demonstrating Maxwell’s equations in a macroscopic, tangible way. By moving the measurement of radiation force from the realm of specialized micro-machined sensors to a tabletop setup, the authors have created an accessible platform for teaching electromagnetic wave theory, low-noise circuit design, and Fourier analysis. Furthermore, the setup's high sensitivity suggests potential applications as a general-purpose sensitive accelerometer or a tool for mechanical spectroscopy.