Why You Can't Use Lock-In Amplifiers for Fast Pulsed EPR Experiments

Lock-in amplifiers are among the most sensitive instruments available for detecting weak periodic signals. They are widely used in continuous-wave (CW) EPR, optical spectroscopy, scanning probe microscopy, and many other fields. Their ability to recover signals buried far below the noise floor often makes them appear to be a universal solution whenever signal levels become small.

However, lock-in amplifiers are fundamentally unsuited for observing fast transient phenomena such as free induction decays (FID), spin echoes, or resonator ringdowns in pulsed EPR experiments with nanosecond-scale relaxation times. This limitation exists regardless of whether the lock-in is implemented using analog electronics or modern digital signal processing.

The purpose of this article is not to criticize lock-in amplifiers. They remain indispensable tools for CW EPR. Instead, the goal is to explain (and essentially document the arguments - this emerged from a discussion with colleagues and is less of a blog article) why the measurement requirements of pulsed EPR are fundamentally different from those of CW experiments and why a transient acquisition system is required.

• What a Lock-In Amplifier Actually Measures
• Why This Works for CW EPR
• Why Pulsed EPR Is Fundamentally Different
• The Fundamental Bandwidth Problem
• Why a Digital Lock-In Does Not Solve the Problem
• When a Digital Lock-In Can Be Used
• Required Receiver Bandwidth
• Sampling Requirements
• Comparison with a Conventional Digitizer
• Resonator Ringdown as an Additional Limitation
• A Typical Pulsed EPR Receiver Architecture
• The Misconception of Repeated Filter Resets
  • What the Lock-In Sees
  • What Happens During a Filter Reset
  • Why Equivalent-Time Sampling Works but Filter Resets Do Not
  • A Thought Experiment
  • The Information-Theoretic View
  • The Correct Approach
• Conclusion
• References

What a Lock-In Amplifier Actually Measures

A lock-in amplifier multiplies an incoming signal by a phase-coherent reference signal and subsequently applies a low-pass filter:

[ I(t)=\mathfrak{LPF}\lbrack s(t)\cos(\omega_{ref}t) \rbrack ] [ Q(t)=\mathfrak{LPF}\lbrack s(t)\sin(\omega_{ref}t) \rbrack ]

The low-pass filter is not an implementation detail, it is the mechanism that suppresses noise and provides the sensitivity for which lock-in amplifiers are known.

The resulting outputs are therefore time-averaged estimates of the signal amplitude and phase at the reference frequency.

The longer the averaging time, the better the signal-to-noise ratio becomes. This is exactly why lock-in amplifiers are so effective in CW experiments.

Why This Works for CW EPR

In CW EPR experiments the measured signal evolves slowly compared to the lock-in time constant.

Typically, one modulates:

• The magnetic field
• The microwave amplitude
• The microwave frequency
• The excitation source

at frequencies ranging from a few hundred hertz to several tens of kilohertz. It is important to note that one has to modulate a quantity that is orthogonal to ones desired component or noise source.

The physical quantity of interest changes slowly enough that the lock-in output accurately follows the experiment while rejecting broadband noise. In this regime the lock-in performs exactly the task it was designed for.

Why Pulsed EPR Is Fundamentally Different

Pulsed EPR experiments do not attempt to measure a slowly varying amplitude.

Instead, they seek to observe the time evolution of a transient signal immediately after a microwave pulse.

Examples include:

• Free induction decays (FID)
• Hahn echoes
• Spin echoes
• Relaxation measurements
• Resonator ringdowns

For a spin system with

[ T_2 = 40,ns ]

the characteristic spectral width is approximately

[ \Delta f \approx \frac{1}{\pi T_2} ]

which yields

[ \Delta f \approx 8,MHz ]

This means that the signal contains information on time scales of only a few tens of nanoseconds. Such information must be recorded directly.

Once averaged away, it cannot be reconstructed.

The Fundamental Bandwidth Problem

The signal observed after a pulse can be represented as

[ s(t)=A_0 e^{-t/T_2} \cos(\omega_0 t+\phi) ]

where the envelope

[ A(t)=A_0 e^{-t/T_2} ]

contains the information of interest.

For a $40 ns$ decay, the envelope changes significantly within tens of nanoseconds. Any instrument that averages over microseconds, milliseconds, or seconds will inevitably smear this information. The output of the lock-in becomes the convolution of the actual transient and the lock-in filter response. Such a convolution can actually also be utilized to characterize short pulses, assuming their shape is known (which is exactly what I did during my diploma thesis when reading out semiconductor particle detectors).

Consequently, the measured signal is no longer the physical decay itself but rather a filtered representation dominated by the lock-in time constant. If one has the idea of cleaning the integration window on triggering one has to keep in mind that resetting the filter does not restore information that has already been removed by averaging.

Why a Digital Lock-In Does Not Solve the Problem

Modern lock-in amplifiers often advertise:

• Digital demodulation
• FPGA processing
• Programmable filters
• Digital signal processing

This sometimes leads to the misconception that they can automatically be used as transient recorders. However, a digital lock-in still performs the same fundamental operation:

• Demodulation
• Low-pass filtering
• Averaging

Replacing analog filters with digital filters does not eliminate the need for integration time. If the transient information is removed by filtering or decimation, it cannot later be reconstructed. A digital lock-in therefore remains a lock-in amplifier unless access to the high-speed data stream is provided.

When a Digital Lock-In Can Be Used

There is one important exception. Some modern instruments marketed as digital lock-in amplifiers contain:

• High-speed ADCs
• FPGA processing
• Large acquisition memories
• Programmable digital receiver chains

Certain instruments (but keep in mind that those are usually not the expensive ones sold as scientific equipment) provide access to:

• Raw ADC data
• Pre-decimation sample streams
• Dedicated transient acquisition modes

In such cases the instrument may indeed be used for pulsed EPR experiments. However, it is no longer functioning primarily as a lock-in amplifier. Instead, it is acting as:

• A digitizer
• A coherent digital receiver
• A digital signal-processing platform

The critical requirement is that the transient information remains accessible before any substantial filtering, averaging, or decimation occurs. If only the final lock-in outputs are available, the transient information has already been lost.

Required Receiver Bandwidth

As a real example. For

[ T_2 = 40 \mathrm{ns} ]

the theoretical linewidth is approximately

[ 8 \mathrm{MHz} ]

This value should be viewed as a lower limit rather than a practical design target. If receiver bandwidth is chosen too close to this limit, the receiver itself begins to distort the transient. In practice one generally desires a bandwidth several times larger than the linewidth.

For a $40 \mathrm{ns}$ decay, practical receiver bandwidths are more likely to lie between $20 \mathrm{MHz}$ to $100 \mathrm{MHz}$, depending on the experiment and desired fidelity. The exact value depends on the signal shape, desired timing resolution, pulse duration and receiver architecture.

The important point is that the receiver bandwidth must be comparable to the transient bandwidth rather than the lock-in time constant.

Sampling Requirements

After coherent downconversion, the transient must be digitized. For envelope bandwidths in the range of $20–100 \mathrm{MHz}$ the sampling rate typically falls into the range of:

[ 50-250,MS/s ]

or higher. Many practical systems sample significantly faster to improve timing resolution and simplify digital filtering.

Direct digitization at the microwave carrier frequency is generally avoided.

For a $30 \mathrm{GHz}$ EPR experiment, the signal is usually mixed down to baseband or low intermediate frequencies before digitization.

Comparison with a Conventional Digitizer

Once access to raw sample streams is required, the distinction between a digital lock-in and a digitizer becomes surprisingly small.

Both systems perform:

• Analog signal conditioning
• High-speed digitization
• Digital downconversion
• Digital filtering
• Numerical extraction of amplitude and phase

The primary difference is often instrument packaging and software rather than measurement physics. A digital lock-in that exposes raw high-speed data could be used for pulsed EPR.

A digital lock-in that exposes only low-bandwidth lock-in outputs cannot.

Resonator Ringdown as an Additional Limitation

In many pulsed EPR systems the resonator itself becomes a limiting factor. The resonator decay time is approximately

[ \frac{Q}{\pi f_0} ]

For an operating frequency of

[ f_0 = 30,GHz ]

one obtains:

Q	Resonator Ringdown
1000	10.6 ns
5000	53 ns
10000	106 ns

For a spin system with $T_2 = 40ns$ a resonator with $Q \approx 5000$ already rings down on essentially the same time scale as the spin coherence itself.

In such situations the receiver may be unable to distinguish between:

• Resonator recovery
• Receiver recovery
• Actual spin dynamics

This problem exists regardless of the lock-in amplifier. Even an ideal lock-in amplifier cannot recover information hidden beneath resonator ringdown. Low-Q resonators and fast receiver recovery are often more important than lock-in performance in short $T_2$ pulsed EPR experiments.

A Typical Pulsed EPR Receiver Architecture

A typical pulsed EPR receiver consists of:

• Microwave Resonator
• Receiver Protection / Blanking
• Low Noise Amplifier
• I/Q Mixer
• Fast ADCs
• Digital Signal Processing
• Averaging

The measured complex signal is

[ S(t) = I(t) + \mathrm{i} Q(t) ]

From this signal one extracts the signal amplitude, phase relation, frequency offset, relaxation times and echo amplitudes.

Importantly, averaging is performed after the transient has already been captured. The transient information is therefore preserved.

The Misconception of Repeated Filter Resets

A common argument is that a digital lock-in amplifier could be synchronized to the pulse sequence and its internal filter could be reset after each pulse. By repeating the experiment many times and varying the reset timing, it may appear possible to reconstruct the transient signal. At first glance this idea seems reasonable because pulsed EPR experiments are inherently repetitive. Similar techniques are routinely used in sampling oscilloscopes and equivalent-time acquisition systems. However, the crucial difference lies in what quantity is actually being measured.

What the Lock-In Sees

The lock-in does not observe the transient directly. Instead, the transient first passes through the lock-in’s demodulation and low-pass filtering stages. Mathematically, the measured output is

[ y(t)=h(t) * s(t) ]

where

• $s(t)$ is the actual transient,
• $h(t)$ is the lock-in filter response,
• $*$ denotes convolution.

The lock-in output therefore represents a filtered version of the transient rather than the transient itself.

What Happens During a Filter Reset

Resetting the lock-in filter merely changes the initial condition of the filter. It does not change the fact that the signal is still being processed through the same filter. Consequently, after every reset the instrument still produces

[ y(t)=h(t)*s(t) ]

with the same filter bandwidth and the same temporal limitations.

The measured signal remains bandwidth-limited.

Why Equivalent-Time Sampling Works but Filter Resets Do Not

A sampling oscilloscope can reconstruct extremely fast signals from repetitive events by acquiring a small number of actual waveform samples during each repetition. Each repetition contributes new information about the original waveform. After many repetitions, the complete waveform can be reconstructed. The same is possible with a digitizer.

In contrast, a lock-in amplifier does not store samples of the original waveform. Instead, it stores the output of an averaging filter.

Repeating the experiment therefore does not reveal new information about the transient. It merely produces multiple measurements of the same filtered quantity.

A Thought Experiment

Consider a spin system with

[ T_2 = 40 \mathrm{ns} ]

and a lock-in time constant of

[ \tau = 1 \mu s ]

The lock-in response is approximately 25 times slower than the physical decay. The spin coherence disappears long before the lock-in output has responded significantly. Resetting the filter every pulse does not alter this fact. The transient has already vanished before the lock-in output can accurately follow it.

The Information-Theoretic View

The essential problem is not synchronization but information loss. The lock-in filter intentionally removes high-frequency temporal information in order to improve signal-to-noise (SNR) ratio. Once this information has been removed, it cannot be recovered by:

• resetting the filter
• repeating the experiment
• changing the trigger timing
• post-processing the lock-in output

To reconstruct a fast transient, the receiver must preserve the transient bandwidth before averaging occurs. Averaging can always be performed later, lost bandwidth cannot be recovered later.

The Correct Approach

If the goal is to measure nanosecond-scale EPR transients, the receiver must first capture the transient itself using sufficient analog bandwidth and high-speed digitization. Only after the transient has been recorded should averaging, digital filtering, phase-sensitive detection or lock-in-style processing be applied.

In other words: Capture first. Average later.

A lock-in amplifier performs these operations in the opposite order, which is precisely why it is very well suited for continuous wave EPR and poorly suited for fast pulsed EPR. A digitizer or a sampling oscilloscope is the right tool for the job.

Conclusion

Lock-in amplifiers are extremely powerful instruments for CW EPR and other weak-signal measurements. However, fast pulsed EPR experiments have fundamentally different requirements. A transient with $T_2 = 40 \mathrm{ns}$ contains information on nanosecond time scales and requires receiver bandwidths of tens of megahertz together with high-speed digitization.

Because lock-in amplifiers intentionally improve sensitivity by averaging away high-bandwidth temporal information, they cannot normally serve as transient recorders.

This limitation applies equally to analog and digital lock-in amplifiers.

Only instruments that expose the raw high-speed data stream before filtering and decimation can be used for such measurements, and in that mode they are effectively operating as digitizers and coherent digital receivers rather than classical lock-in amplifiers.

For short-coherence pulsed EPR experiments, receiver bandwidth, digitizer performance, resonator recovery, and coherent transient acquisition are the critical design parameters - not lock-in integration time.

In addition, digitizers are way more suited for this kind of experiment than lock-in amplifiers.

References

• Heterodyne, Superheterodyne, Homodyne and Lock-In Amplifier
• Measurements from small DIY NMR spectrometer
• Nuclear magnetic resonance and electron paramagnetic resonance: Basics
• Electric dipole interaction of atoms with radiated electromagnetic waves

This article is tagged: Physics, Basics, How stuff works, Measurement, Measurements, Experiment