In this document we'll describe a method for measuring the line width of single longitudinal mode lasers. Such lasers have very narrow (few MHz) spectral line widths, long coherence length, and very low phase noise.

A common figure of merit for an optical spectrometer quantifies its ability to distinguish between two adjacent wavelengths, \( \delta \lambda = \lambda_1 - \lambda_2\) or frequencies \( \delta \nu = \nu_1 - \nu_2\). Spectrometers encountered in a physics lab, based on prisms or diffraction gratings, typically have spectral resolving power, \(\lambda /\delta \lambda = \nu /\delta\nu\) , of a few hundred to a few tens of thousands. To achieve higher resolving power new methods, typically based on interferometers, are employed.

The heterodyne spectrometer described here achieves a spectral resolving power of almost \(10^9\). We use a Mach-Zehnder interferometer coupled with a heterodyne method to measure the spectral line shape and width of radiation emitted by a laser diode, as developed by T. Okoshi et al. (1980 "Novel method for high resolution measurement of laser output spectrum," Electron. Lett. 16, 630).

In the first section we review the hardware configuration. In the second section we review how to deduce the linewidth from the interferometer signal. A video tutorial on the Eikonal Optics web page shows the spectrometer hardware, presents experimental results, and illustrates real time fitting the laser line width using the theory developed below.

The light source (Fig. 1) examined here is a 1550 nm (193 THz) distributed feedback (DFB) laser diode. DFB diodes are one of the core technologies in dense wavelength division multiplexing (DWDM) fiber telecom systems and are characterized by very narrow linewidths (several MHz or less). The need for narrow linewidth comes from the requirement that DWDM systems multiplex up to 160 communication channels in a single SMF-28e fiber using 25 GHz channel spacings.

Figure 2 shows the experimental configuration where light from the diode is split into two paths with one injected into a Bragg cell also known as an acousto-optic modulator (AOM). In an AOM a piezoelectric actuator is energized by a radio frequency (80 MHz) signal, which launches an acoustic wave into a transparent optical medium. Photons interact with the acoustic phonons in a manner consistent with the Bragg condition for constructive interference (see Appendix A). To conserve energy and momentum the scattered photons are Doppler shifted in frequency by an amount equal to the phonon frequency. When light directly from the laser diode is recombined with that which passed through the AOM the two waves interfere, yielding beats at a frequency, \(\Omega\), equal to that of the RF signal driving the AOM. If the two waves are mixed on a fast photodiode then an electrical RF signal can be detected. This RF signal encodes information about the intrinsic line shape emitted by the laser diode.

In the absence of a delay between the two paths, the signals reaching the photodiode will be perfectly correlated and the diode will record a signal that is a delta-function at frequency \(\Omega\). However, if a sufficiently long time-delay is introduced, the two signals are temporally decorrelated. In this circumstance, any frequency fluctuations that are present in the original source will now be evident as a broadening of the beat frequency signal.

Figure 3 shows the physical set up. The experiment uses two fiber splitters-one to divide the light and one to recombine it. A splitter is the fiber equivalent of a free-space beam splitter or a partially reflecting mirror. A splitter is formed by stripping the cladding from a length of two fibers and wrapping the two resultant cores together. A bond is formed between the cores by tensioning the fibers and heating the junction until the two fibers are tapered and fused. Such a device is known as a fused biconical taper (FBT) splitter. This configuration can be used either as a splitter or as a combiner.

Unlike the sketch in Fig. 2, each splitter has two inputs and two outputs. Only one of the inputs is used for the input splitter; both outputs of the combiner are used to feed two photodiodes, although only one is needed for this application. These splitters/combiners are bidirectional, allowing any port to be used as an input.

A Faraday isolator is included between the laser diode and the interferometer. This prevents reflections from returning to the laser and modifying the conditions in the gain medium, which might cause fluctuations in intensity or frequency. An isolator consists of two linear polarizers separated by a Faraday rotator, which rotates the plane of polarization by 45\(^\circ\). Any reflections suffer a second pass through the Faraday rotator and an additional 45\(^\circ\) rotation so that the plane of polarization is now perpendicular to the original linear polarizer. Consequently, reflected light is suppressed, typically by 30 dB.

Figure 4 shows data for two different diodes characterized with this spectrometer. The next section explains why the shape of the noise power spectrum is Lorentzian with a width twice that of the laser line when the fiber delay is sufficiently long. For additional examples with different delays and variable line widths be sure to check out our video.The theory of the self-heterodyne method can be found in Richter et al. (1986 IEEE J Quant. Elect. 22, 2070). Here we add a few additional steps to help clarify the derivation and highlight the assumptions adopted.

First we develop a characterization of the spectral line shape \( S_E ( \omega ) \) produced by the laser. Suppose the resultant electric field generated by the laser is described by

\[ E(t) = E_0 \exp \{ \mathbb{i} \left[ \omega_o t + \phi(t) \right] \tag{1} \} , \]where departures from pure monochromatic behavior at laser angular frequency \( \omega_0 \) are characterized by time dependent random variations in \( \phi (t), \) known as phase jitter. Fluctuations exist in both amplitude and phase. In each round-trip in the laser cavity, some noise amplitude is added to the electric field. This changes both amplitude and phase of the field. However, because of gain saturation there is feedback which drives the power to the steady-state value and hence fluctuations in \( E_0 \) are damped. In contrast, there is no phase feedback, which therefore exhibits a random walk, leading to a finite emitted line width. Consequently, it is safe to ignore fluctuations in the amplitude of the electric field \( E_0 \).

We find the laser line shape by first evaluating the autocorrelation function associated with the fluctuations in \( \phi \) and then using the Wiener-Khinchin theorem to compute the associated spectrum, \( S_E (\omega ) \).

The autocorrelation function is defined as the expectation value \[ \Gamma(\tau) = \langle E^*(t) E(t + \tau ) \rangle, \tag{2} \] or in terms of the random phase change between times \(t \) and \( t + \tau \), \[ \Delta \phi(t,\tau ) = \phi(t+\tau) - \phi(t) . \] We assume that the fluctuations \(\Delta \phi(t,\tau) \) follow a Gaussian distribution with zero mean and variance given by \[ \sigma^2_\phi ( \tau ) = \langle \left[ \phi(t+\tau) - \phi(t) \right]^2\rangle . \] The phase error variance therefore follows a random walk that grows linearly with time, in which case the phase variance is conveniently written as \[ \sigma^2_\phi (\tau) = \tau/\tau_c , \] where the constant or proportionality \(\tau_c\) is called the coherence time.

Evaluating the autocorrelation function, Eq. (2) using Eq. (1) we find \[ \Gamma(\tau) =\langle \exp \left[ \mathbb{i} \Delta \phi(t,\tau ) \right] \rangle \exp(\mathbb{i} \omega_0 \tau) , \]

and the expectation value integral simplifies to \[ \Gamma(\tau) = \exp \left[ - \sigma^2_\phi (\tau )/2 \right] \exp(\mathbb{i} \omega_0 \tau) ,\]

using the Gaussian moment theorem \(\langle \exp (\mathbb{i} x ) \rangle = \exp ( -\langle x^2 \rangle/2 ) \) for a random variable \( x \) with zero mean.

Using the Wiener-Khinchin theorem, the Fourier transform of \( \Gamma (\tau) \) yields the line shape power spectral density \( S_E (\omega) \) \[ S_E (\omega) = \frac{E_0^2 }{\pi } \frac{ \gamma }{\gamma^2 + (\omega-\omega_0)^2} , \] where \( 2 \gamma = 1/\tau_c \) is the angular frequency full line width at half maximum of the resultant Lorentzian profile.

We now see that the constant \(2 \gamma \) is the laser line width given by the famous Schawlow-Townes (1958 Phys. Rev. 112 1940) formula in terms of the natural atomic line width and properties of the laser cavity.

We now describe the situation in this experimental configuration where the light from the laser is split and recombined.

The total electric field incident on the detector \[ E_T(t) = E_0 \exp\{\mathbb{i} [ \omega_0 t + \phi(t) ]\} + E_0 \exp\{\mathbb{i} [ (\omega_0 t + \Omega)(t-\tau_0) + \phi(t-\tau_0)] \}, \]

is the sum of the electric field from each arm of the interferometer. The quantity \( \tau_0\) is the differential time delay of the two interferometer paths,and \( \Omega \) is the frequency shift introduced by the acousto-optic modulator. We made the assumption that the electric field has unit amplitude in both arms of the interferometer.

In our experiment a photodiode is used to convert the phase fluctuations of \( E_T(t) \) into intensity noise and hence a fluctuating photocurrent which is recorded by the oscilloscope. Photocurrent is linearly proportional to the incident power which in turn is proportional to \( E_T^*(t) E_T(t)\). Therefore, the autocorrelation function for the photocurrent is \[ \Gamma_T (\tau) = \langle E_T^*(t) E_T (t) E_T^*(t+\tau) E_T (t+\tau) \rangle . \]

Substitution of the expression for \( E_T(t) \) and multiplying out yields 16 terms, ten of which average to zero because of the \( \exp (\mathbb{i} \Omega t ) \) oscillation. The remaining terms simplify to \[ \Gamma_T(\tau) = E_0^4 \left[ 4 + 2 \cos (\Omega \tau) \exp\left( -\sigma_\phi^2 (\tau_0) - \sigma_\phi^2 (\tau) + \sigma_\phi^2 (\tau+\tau_0)/2 + \sigma_\phi^2 (\tau-\tau_0)/2 \right) \right] , \]

again making use of the Gaussian moment theorem.

Ignoring the delta function at zero frequency associated with photon shot noise, the spectrum of photocurrent fluctuations is given by Fourier transformation to yield \[\begin{eqnarray} S_T(\omega) & = & \frac{1}{\pi} \frac{2\gamma }{(2\gamma)^2+(\omega-\Omega)^2} \left\{1-\left[\cos(\omega-\Omega)\tau_0+\frac{2\gamma \sin (\omega-\Omega)\tau_0}{(\omega-\Omega)}\right] \exp(- 2\gamma\tau_0 ) \right\}\ + \\ & & \\ & & \delta(\omega - \Omega) \exp(-2\gamma\tau_0) . \tag{3} \end{eqnarray}\]

The measured RF power is proportional \(E_0^4\). We have dropped the normalizing factor in Eq. (3) because we have not determined the coupling efficiency of the diode to the interferometer or calibrated the photodiode responsivity.

According to Eq. (3) as \( \tau_0 \) increases in relation to \(\tau_c\) the signal shifts from a delta function to a modified Lorentzian with interference ripples until the power spectrum becomes strictly Lorentzian when the delay is much longer than the coherence time.

Figure 5 plots some examples of the behavior predicted by Eq. (3). In this example the delta function at \( \omega = \Omega\) has been suppressed and the spectra are normalized to unity. The figure shows that in the limit of large delay times, \( \tau_0\), when the phase of the two optical fields at the photodiode have become decorrelated, the noise power spectrum becomes exactly Lorentzian with width \( 4 \gamma \), or twice the intrinsic laser linewidth. Figure 5 also shows the effect on the measured noise spectrum of finite \( \tau_0 \) (in units of the coherence time, \(\tau_c = 1/(2\gamma ) \)). For delay times which are comparable to or shorter than the coherence time, the quasi-Lorentzian part is broadened and scalloped.

The convolution in Eq. (3) originates from the multiplication of the time-varying local oscillator field with the signal field in the photodetector. Multiplication in the time-domain is equivalent to convolution in the frequency domain. Hence the measured power spectrum has a width twice that of the original line.

If you would like Eikonal to characterize a laser diode for you or design a spectrometer to measure your laser diode properties, please contact us.

where \( n \) is the index of refraction, \(\Lambda\) is the wavelength of the acoustic wave and \(\lambda \) is the optical wavelength. For typical parameters, angular deflections (2\(\theta\)) of a few degrees are practical given acoustic excitation at high frequencies (~ 10-100 MHz).

Since the acoustic wave is slow compared to the light wave ( \( v_s / c \simeq 10^{-5}\)), the acousto-optic perturbation can be thought of as a stationary volume grating. However, because the acoustic wave is moving, the light also experiences a Doppler shift in frequency according to \[ \Delta \nu /\nu = 2 n \nu \sin \theta / c ,\]

where \( v_s \sin \theta\) is the component of the acoustic wave speed in the direction of the incident beam. When the Bragg condition is satisfied the frequency shift is \(\Delta \nu = v_s /\Lambda \), or equal to the frequency of the acoustic wave.

Alternatively, suppose a photon with energy \( h \nu\) and momentum \( \hbar \vec{k} \) absorbs an acoustic phonon with energy \( h V \) and momentum \( \hbar \vec{K}\). Conservation of energy and momentum requires \( \nu' = \nu + V\) and \( \vec{k'} =\vec{k}+\vec{K}\). As optical frequencies (THz) are much greater then acoustic frequencies (MHz) \( k \gg K\), and this requires that \(2 k \sin \theta = K\). Furthermore, when \( k \gg K\) the small angle approximation yields the scattering angle \(2\theta \simeq K/k\). In the current example \( K/k\) = 80 MHz/193 THz = \( 4.2\times 10^{-7}\).

In summary the optical frequency of the scattered beam is increased or decreased by the frequency of the sound wave (depending on the propagation direction of the acoustic wave relative to the beam) and propagates in a slightly different direction.

© *Eikonal Optics*

A transimpedance amplifier is the configuration of choice when high-bandwidth and low noise operation is required. A transimpedance amplifier (TIA) converts an input current to a voltage. A common application is for use with a photodiode to convert a photocurrent into a readily measurable signal. Transimpedance amplifiers are useful because they can deliver an order of magnitude or more speed up relative to a simple photodiode plus load resistor configuration.

The characteristics of noise in transimpedance amplifiers is significant because photodiodes are frequently used to detect faint sources. Some noise contributions are fundamental in nature, e.g., thermal or Johnson noise associated with resistors and Poisson noise due to quantization of charge. Other noise sources depend on the specific op-amp device used in the design.

Design tools such as SPICE allow calculation of noise and commercial implementations, such as LTspice, include models for the manufacturer's op-amps. Characteristics of a specific device such as the input capacitance, current noise, and voltage noise factor into the achieved performance but it can be unclear which of these factors dominate the combined noise in a given design, which complicates op-amp selection.

Here we show how to compute the various noise terms present in a transimpedance amplifier. Existing resources on this topic are available in various texts and op-amp datasheets. However, the information is frequently fragmentary, and results are often quoted without derivation so that the assumptions and domain of applicability are unclear. The purpose of this post is to provide an explicit and accurate review of this topic.

The basic configuration of a transimpedance amplifier is shown in Figure 1. In contrast to a load resistor configuration, the TIA holds the input at the inverting input at a fixed voltage (ground in this example) which minimizes photodiode dark current and improves linearity.

Unlike op-amp circuits that are used as voltage amplifiers, the transimpedance amplifier has no input resistor. For a perfect op-amp configured as in Figure 1, the output is \(V_o = I_D R_2\) and the current gain (volts per amp) is measured in ohms.

The circuit shown in Figure 1 is not sufficiently detailed to analyze performance. Figure 2 shows a more realistic layout with a feedback network that includes both resistive and capacitive elements.

Real photodiodes have finite shunt resistance, \( R_1\), and capacitance, \(C_1\). Real op-amps also have common mode input capacitance, typically listed as \(C_{\rm {IN}} \) in datasheets, which will also contribute in parallel to \(C_1 .\) We also need to take account of limited bandwidth and finite DC gain of the op-amp.

The existence of significant photodiode capacitance explains why transimpedance amplifiers are necessary. If we used a biased photodiode with load resistor \(R_2\) then the -3 dB bandwidth is \(1/(2 \pi R_2 C_1)\).

It is a little more complicated to find the relationship between the photodiode input current, \(I_D\), and the output voltage, \(V_o\), for Figure 2.

First let's consider an op-amp with finite gain and bandwidth. If \(A(\omega\)) denotes the dependence of gain on frequency, \(f\), where \(\omega = 2 \pi f\), the output voltage, \(V_o\), is

\[ V_o = ( V_P - V_M)\; A(\omega), \tag{1} \]where \(V_P\) and \(V_M\) are the voltages at the non-inverting and inverting inputs, respectively. Op-amps are typically characterized by the gain bandwidth product, GBW, and the open-loop DC gain, \(A_0\). GBW is functionally equivalent to the frequency where the open loop-gain drops to unity. (See Appendix for our model of \(A(\omega)\)).

Next, to keep the expressions compact, write the parallel combination of \(R_1\) // \(C_1\) as \(Z_1\) and \(R_2\) // \(C_2\) as \(Z_2\). Referring to Figure 2, where \(I_1\) is the current flowing in \(Z_1\) and \( I_D\) is the photodiode current, the Kirchoff rules for voltage and current are

\[ \begin{eqnarray} I_2 & = & I_1 + I_D \\ V_M + I_2 Z_2 &=& V_o \\ I_1 Z_1 &=& V_M \\ V_P & =& 0. \end{eqnarray}\tag{2} \]Eliminating \(I_1\), \(I_2\), and \(V_M\) yields the TIA transfer function as a function of frequency,

\[ \boxed{ H(\omega) =\frac{V_o}{ I_D} = A(\omega)\; \frac{Z_1 Z_2}{Z_1 + A(\omega) Z_1 +Z_2} .} \tag{3}\]In the limit of large open-loop gain when \(A(\omega ) \gg 1\) we have \( V_o = I_D Z_2 \). To evaluate Eq. 3 we need an expression for the impedance of a resistor and capacitor in parallel, for which we use \[ Z_i = \left( R_i ^{-1} + \mathbb{i} \omega C_i \right)^{-1} , \tag{4}\] with \(i\) = 1 or 2.

Figure 3 provides examples of \( H(\omega) = V_o/ I_D\) as a function of frequency for a feedback resistor of \(R_2 \) = 10 k\(\Omega\) and two different values of input capacitance. Initially, assume that the photodiode shunt resistance, \(R_1\), is infinite and the feedback capacitance, \(C_2\), is zero. The op-amp has a GBW (500 MHz) and a DC gain (250,000) typical of a high performance device.

With input capacitance \(C_1 \) = 0 pF (red) the signal gain mirrors the performance of the op-amp and is flat up to the GBW frequency and then rolls off at 6 dB per octave. This result suggests that designing a high-bandwidth TIA is simply a matter for finding an op-amp with sufficient large GBW. When the input capacitance is increased to \(C_1 \) = 10 pF (blue), a value characteristic of the combined capacitance of a typical photodiode and op-amp. This shows a dramatically different outcome with pronounced signal gain peaking accompanied by a -180\(^\circ\) phase shift at about 28 MHz.

Figure 4 shows the step response associated with the blue transfer function plotted in Figure 3. The step response exhibits overshoot and ringing. The ringing appears at 28 MHz and persist for several hundred ns.

Figure 4 implies that the ideal TIA circuit in Fig. 1 is unusable. The combination of the feedback resistor and input capacitance has a characteristic frequency of \( f_{R_2 C_1} = 1/( 2 \pi R_2 C_1 )\). In this example, \( f_{R_2 C_1} \) = 1.59 MHz, which is much lower than the op-amp's gain-bandwidth product GBW of 500 MHz. The \( R_2 \)-\( C_1\) network in combination with the op-amp gain, \( A( \omega) \), introduces a quadratic term in frequency in the denominator of the transfer function (see below), which causes a prominent peak in the gain and large phase lag at a frequency of approximately \( \sqrt{ f_{R_2 C_1} {\rm GBW} }\).

We'll see below that adding a small amount of capacitance in parallel with the feedback resistor (i.e., non-zero \(C_2\)) can suppress these oscillations. Damping can also be achieved by adding some resistance in series with the photodiode. Typically the former option is chosen as the feedback resistor already has some parasitic capacitance; moreover, Johnson noise from a small series resistance can dominate the noise budget at high frequencies.

A convenient way to analyze overshoot and ringing is to rewrite the transfer function in a standard from that lets us analyze the system as a damped harmonic oscillator. This allows us to identify the two key parameters that define the system: the natural frequency and damping ratio. This in turn allows us to find the location of gain peaking, the degree of overshoot, and the -3 dB point. By adjusting the damping ratio we will be able to eliminate the ringing seen in Figure 4.

The transfer function of Eq. (3) can be written as \[ \frac{V_o}{I_D} = R_2 \left[ 1 + 2 \mathbb{i} \zeta \frac{ \omega}{\omega_{n}} - \frac{\omega^2}{\omega_{n}^2} \right]^{-1} , \tag{5} \]

where \(\omega_n \) is the natural frequency (where the gain peak occurs), and \(\zeta\) is the dimensionless damping ratio. When \(\zeta = 0\), oscillations are undamped and when \(\zeta = 1\), the system is critically damped with no overshoot. Figure 5 shows some examples of this behavior.

Solving Eq. (5) for the half-power or -3 dB point, i.e., \( |H|^2 = 1/2\), yields

\[ \omega^2_{-3{\rm dB}} = \left[ 1 - 2 \zeta^2 + \sqrt{2 + 4\zeta^2 (\zeta^2-1)}\right] \; \omega^2_n . \tag{6} \]The -3 dB point according to Eq. (6) is shown on right panel of Fig. (5). Figure 6 illustrates how increasing \(\zeta\) decreases the bandwidth. When \(\zeta = 1/\sqrt 2\) then we have the simplification that \( \omega_{-3{\rm dB}} =\omega_n \). If \(\zeta = 1 \), which is the case when the system is critically damped with no overshoot, then \( \omega_{-3{\rm dB}} = (2^{1/2}-1)^{1/2} \omega_n \simeq 0.64 \omega_n\). ^{1}

Substituting for \(Z_1\) and \(Z_2\) from Eq. (4) in the transfer function (Eq. 3) and by grouping terms of powers of \(\omega\) we find that the coefficient of the quadratic term is

\[ \omega_n^2 = \frac{ 2\pi \; { \rm GBW}}{(C_1+C_2)R_2} \left[1+\frac{1}{A_0}\left(1+\frac{R_2}{ R_1}\right) \right] .\tag{7}\]Typically, the op-amp DC gain is very high, \(A_0 \gg 1\), and the photodiode shunt resistance exceeds the feedback resistor value by orders of magnitude, \(R_1 \gg R_2\). Hence, to a good approximation

\[ \boxed {\omega_n \simeq \sqrt{ \frac{2\pi \; \rm GBW}{ (C_1+C_2)R_2 } } .} \tag{8}\]In terms of frequency

\[ f_n \simeq \sqrt{ {\rm GBW}\; f_{\rm FB} , \tag{9} }\]with \(f_{\rm FB} = 1/\left[ 2\pi R_2 (C_1+C_2)\right]\), which can be interpreted as saying that the natural frequency is the harmonic mean of the op-amp GBW and the corner frequency of the feedback network. Figure 3 shows that \(f_n\) accurately locates the gain peak and the \(180^\circ\)phase shift.

The coefficient of the linear term in \(\omega\) gives the damping ratio ^{2}

where we have assumed that \(A_0 \gg 1\).

As \(\zeta\) approaches 1 ringing disappears, so Eq. (10) implies that for fixed feedback resistor and input capacitance, we can reduce ringing by increasing the op-amp GBW or adding a finite capacitance is parallel with the feedback resistor.

We can now choose \(\zeta\) by specifying an acceptable degree of overshoot and find the appropriate value of \(C_2 \) for fixed op-amp properties (GBW and \(A_0\)), feedback resistor, and input capacitance.

If \( V_{\rm eq}\) is the asymptotic equilibrium value of the step response, and \( V_{\rm max} \) is the peak overshoot, then \[ V_{\rm max}/V_{\rm eq} = \exp\left( - \frac{\pi \;\zeta}{\sqrt{1-\zeta^2} }\right) . \tag{11}\]

According to Eq, (11) the conventional choice of \(\zeta = 1/\sqrt{2}\) \(\simeq\) 0.7071, corresponds to an overshoot of \(e^{-\pi} \) or 4.3%. Using Eq. (10) and (11) we plot in Figure 7 the degree of overshoot corresponding to \(C_2\) for our current example. This plot also shows how increasing \(C_2\) and the corresponding increase in \(\zeta\) diminishes the bandwidth (see Fig. 6 and the dots marking the -3 dB point on right hand panel of Fig. 5).

Adding feedback capacitance increases \(\zeta\) and decreases the TIA bandwidth.

If we choose \(\zeta =1/\sqrt{2} \) then

\[ C_2 = \frac{(4 \pi \;{\rm GBW} \; C_1 R_1^2 R_2 - R_1^2 -2R_1 R_2 )^{1/2}-R_2 } {2 \pi\; {\rm GBW} R_1 R_2} \tag{12} \]so long as

\[ \frac{4 \pi\; {\rm GBW}\; C_1 R_1^2 R_2 }{(R_1+R_2)^2 } > 1. \]To a good approximation when \(R_1 \gg R_2\)

\[ \boxed{C_2 \simeq \sqrt{\frac{C_1}{\pi \; {\rm GBW}\; R_2 }}; \;\;\;\; \zeta = 1/\sqrt{2} .} \tag{13} \]Also, as \(\omega_{-3db} =\omega_n\) when \(\zeta =1/\sqrt 2\) the resultant -3 dB signal bandwidth for this choice of \(C_2\) is

\[ f_{-3dB } = \sqrt{\frac{\rm GBW}{2 \pi (C_1+C_2)R_2}} . \tag{14}\]As \(C_1\) and \(R_2\) may be fixed by choice of detector and maximum photocurrent, and \(C_2\) set by practical limits on parasitic capacitance, Eq. (14) can be rearranged to determine the op-amp GBW required to achieve a specific signal bandwidth.

A biased photodiode with capacitance \(C_1\) and load resistor \(R_2\) achieves a -3 dB bandwidth of \(f_{\rm LR } = 1/(2 \pi C_1 R_2 )\). Comparison with Eq. (14) shows that the TIA has a bandwidth advantage of approximately \( \sqrt{ {\rm GBW}/f_{\rm LR }}\). In the current example (\(R_2\) = 10 k\(\Omega\) and \(C_1\) = 10 pF), the speed advantage factor for a 500 MHz GBW op-amp is \(\times \)11.

By noise we mean noise generated in the op-amp and its associated components and not interference or unwanted external signals. Several noise sources contribute to the signal-to-noise achieved by a transimpedance amplifier. Specific to transimpedance amplifiers are internal current and voltage noise contributions that are present in non-ideal op-amps. There are also fundamental and ubiquitous noise contributions due to quantization of charge and the associated Poisson noise and the thermal or Johnson noise that arises in resistors (usually dominated by the feedback resistor).

These noise sources are normally uncorrelated. If they exhibit no temporal correlation, i.e. no covariance, each noise source can be evaluated independently and the total noise evaluated as root sum squares.

The contribution of the op-amp's voltage noise spectrum is denoted as \(e_n(\omega)\).^{3} Typically, \(e_n\) is listed on manufacturers' data sheets and is the * input referenced * voltage noise density with units of \( {\rm V}/\sqrt{\rm Hz}.\) Common values for high-performance op-amps are \(e_n\) = 1–10 nV\(/\sqrt{\rm Hz}\).

Figure 8 depicts an ideal, noise-free op-amp with an external voltage noise source labeled \(e_n\) connected to the inverting input. This configuration represents the conventional approach to modeling op-amp voltage noise.

Applying Kirchoff's circuit laws to Figure 8 and writing \(Z_i \) as the parallel combination \(R_i\) // \(C_i\) we have \[ \begin{eqnarray} V_1 & = & I Z_1 \\ V_M & = & V_1 + e_n (\omega) \\ V_o & = & I Z_2 + V_1 \\ V_o & = & A(\omega) (V_P-V_M) \end{eqnarray}. \tag{15} \]

Eliminating \(I\) and \(V_1\) we find the corresponding output voltage noise spectrum as

\[ e_{nV} (\omega) = - \frac{A(\omega)(Z_1 + Z_2) }{Z_1 + A(\omega)Z_1 + Z_2 } e_n(\omega) . \tag{16} \]When the op-amp gain is large, the feedback resistor is the dominant contribution to \(Z_2\), and the diode shunt resistance is effectively infinite, we have

\[ e_{nV} ( \omega) \simeq (1+ \mathbb{i} \omega C_1 R_2) e_n(\omega).\tag{17} \]Thus, at low frequencies the op-amp in Fig. 8 acts like a follower and the output voltage noise equals \(e_n\). As frequency increases the voltage noise source interacts with the input capacitance, \(C_1\), because feedback drives a current, \(e_n/Z_1\) so that the voltage at the inverting remains close to the ground potential of the non-inverting input. The associated noise current is \( \mathbb{i} \omega C_1 e_n\) corresponding to an output voltage of \( \mathbb{i} \omega C_1 R_2 e_n \). The noise induced by combination of \(e_n\) and input capacitance is designated \(e_n C\) noise, and increases with increasing frequency and input capacitance.

As an example, consider a range values of \(C_1\) and a flat input voltage noise spectrum with \(e_n \) = 5 nV/\(\sqrt{\rm Hz}\). The curves for the output noise voltage in Figure 9 show a prominent peak which increases in magnitude in proportion with the value of \(C_1\) and shifts to lower frequencies at \(f_n\) given by Eq. (9).

The dotted lines in Figure 9 show the impact of adopting the value of \( C_2\) given by Eq. (13) so that the damping ratio is \(1/\sqrt{2}\). Adding capacitance in parallel with the feedback resistor reduces but does not eliminate the \( e_N C \) noise.

Figure 10 shows the TIA configuration of an ideal op-amp combined with a current noise source \(i_n(\omega) \) that represents the op-amp's input referenced current noise. The input referenced current noise can range from fA/\(\sqrt{\rm Hz}\) to pA/\(\sqrt{\rm Hz}\), depending on op-amp architecture (e.g., BJT, JFET, or CMOS).

The current noise appears between the inverting input and ground and therefore in parallel to the photodiode current in Fig. 2. Consequently, the signal gain relation in Eq. (3) also describes the relation between \(i_n(\omega)\) and the output voltage noise,

\[ e_{nI}= \; \frac{ A(\omega) Z_1 Z_2}{Z_1 + A(\omega) Z_1 +Z_2} i_n(\omega) . \tag{18}\]As the voltage output associated with \(i_n\) is indistinguishable from input signal current and Fig. 5 (right) shows the shape of the corresponding output voltage noise. If \(i_n\) is flat, then the associated output voltage noise rolls over at \(f_n = \sqrt{ {\rm GBW} f_{\rm FB} }\) according to Eq. (9)..

Johnson noise is a fundamental noise source associated with thermal agitation of charge carriers in a resistor at any temperature above absolute zero. At radio frequencies (\( \nu \ll k_B T/h \simeq 6 \) THz at room temperature) the spectrum of Johnson noise is flat. The TIA further limits the output voltage noise at high frequencies.

Johnson noise in a resistor can be modeled as a noiseless resistor in series with a voltage noise power spectral density with \( e^2_{nJ} = 4k_B T R \), with units \( \rm{V} ^2 \ \rm{Hz}^{-1} \) and Boltzmann constant \(k_B\). For the Johnson noise associated with the TIA feedback resistor, \(R_2\), it is convenient to consider the complementary circuit of a noiseless resistor in parallel with a current source with noise power spectrum,

\[ i^2_{nJ} = 4 k_B T /R \ \ \ [ \rm{A} ^2 \ \rm{Hz}^{-1} ] . \tag{19}\]Figure 11 illustrates this circumstance where the Johnson noise current appears in parallel with \(Z_2\), where \(R_2\) and \(C_2\) in parallel comprise \(Z_2\).

Using the currents and voltages labeled on Figure 11, Kirchoff's circuit laws require \[ \begin{eqnarray} I_1 & =& I_2 + I_J\\ V_o & = & I_2 Z_2 + V_1 \\ V_1 & = & I_1 Z_1 . \tag{20} \end{eqnarray} \] Combining these relations yields the output voltage corresponding to \(I_J\) as \[ e_{nJ}(\omega) = -A(\omega) \frac{ Z_1 Z_2}{Z_1 + A(\omega) Z_1 +Z_2 } I_J . \tag{21} \]

The transfer function for photodiode current Eq. (3) and Johnson noise current are identical.

In terms of the output voltage power spectral density, the Johnson noise contribution is \[ e_{nJ}^2(\omega) = \left[ A(\omega) \frac{ Z_1 Z_2}{Z_1 +A(\omega) Z_1 +Z_2 }\right]^2 \ i^2_{nJ} . \tag{22} \]

For an ideal op-amp with \(A\gg 1\), the output voltage Johnson noise is \( e_{nJ} = - i_{nJ} Z_2 \). As \( Z_2 \) is made up of the feedback resistor \(R_2\) and the parallel feedback capacitance \(C_2\), the output voltage noise is \(e_{nJ} = -i_{nJ} /(1/R_2 + \mathbb{i} \omega C_2 )\) and the Johnson noise rolls over at a 3dB point with frequency \(1/(2\pi R_2 C_2) \).

We adopt the LTC6268 op-amp as an example for noise analysis. This device has GBW = 500 MHz, a high open loop gain (\(A_0 = 250,000\)), a FET-input with low input bias current (3 fA) and low input capacitance (0.45 pF). It also has low input referenced voltage and current noise noise making it suitable for fast transimpedance amplifier designs. Figure 12 shows the measured input referenced voltage noise for an LTC6268.

Figure 13 shows \(i_n(\omega)\) for the LTC6268 op-amp.

The contributions of \(e_n\), \(i_n\), and \(i_{nJ}\) to the output voltage noise are plotted in Figure 14 for a TIA using an LTC6268 with a 10 k\(\Omega\) feedback resistor and a total input capacitance of 10 pF. A compensating capacitor of \(0.8\ {\rm pF}\), is placed in parallel with the feedback resistor to achieve a damping ratio of \(\zeta = 1/\sqrt{2}\). The Johnson noise from the feedback resistor is also plotted.

Equations (16), (18), (19), and (22) are used to find the individual noise contributions. The combined noise is found assuming that these sources are uncorrelated so that the root-sum-square yields the total. Figure 14 shows that at low frequencies ( \(\lt \) 4 MHz) Johnson noise from the feedback resistor (13 nV/ \( \sqrt{\rm Hz}\) ) dominates. At high frequencies the \(e_n C\) noise peak is dominant because of the relatively large value of input capacitance (10 pF). Input current noise makes a relatively small contribution.

Figure 14 also shows the noise prediction using the manufacturers model for the LTC6268 op-amp built in to LTspice (dot-dashed lines). The LTspice model generates the total noise and the Johnson noise contribution but not the individual contributions of \(e_n\) and \(i_n\).

In the LTspice TIA circuit model we assume that the value of the op-amp common mode input capacitance quoted in the datasheet of \(C_{in} \) = 0.45 pF and a photodiode capacitance of 9.55 pF, for a total \(C_1 \) = 10 pF to match the analytic calculation. The agreement of Eqs. (16), (18), (19), and (22) and LTspice is excellent over the full range of frequencies. Above 20 MHz a discrepancy is noticeable where LTspice predicts a total noise that is about 5% lower than our results. Comparisons of LTspice calculations for lower values of \(C_1\) where \(e_n\) is less dominant suggests that the difference arises because the model for the LTC6268 in LTspice adopts a lower value for \(i_n\) than that plotted in the datasheet and used here.

If the signal is photocurrent \( I_D(\omega) \), then the signal-to-noise ratio is output voltage given by Eq. (3) divided by the total noise combined as root-sum-square, i.e.,

\[ SNR = \frac{ I_D H(\omega) }{ \sqrt{ e_{nP}^2 + e_{nV}^2 + e_{nI}^2 +e_{nJ}^2 }} . \tag{23}\]The first term in the denominator of Eq. (23) accounts for the Poisson or shot noise associated with the photodiode current, which includes not only the photocurrent but also any dark current, \(I_{\rm DC}\), contribution. Photon counting statistics sets the fundamental quantum limit on SNR when all other noise sources have been eliminated. The current shot noise power spectral density is \( 2 q_e (I_D + I_{\rm DC}) \) with units A\(^2/\rm{Hz}\) and \(q_e\) is the elementary charge. The corresponding output voltage noise, \(e_{nP}\), is found using Eq. (3) or Eq. (18). The other three noise terms are the op-amp voltage noise (Eq. 16), op-amp current noise Eq. (18), and Johnson noise Eq. (22), respectively.

The noise-equivalent power (NEP) is a measure of the sensitivity of a photodiode and TIA combination defined as the signal power that gives unity signal-to-noise ratio in a one hertz bandwidth. To compute the NEP we need the system responsivity, \(\mathcal{R} = I_D/P \), which gives the photocurrent \(I_D\) in terms of the incident radiant power \(P\). As the photon energy is \( h \nu\) then the photocurrent is \[ I_D = \frac{\eta P q_e}{h\nu } \] and \[ \mathcal{R} = \frac{\eta\; q_e}{h\nu },\]

where we have included a photon detection efficiency factor, \(\eta\), known as the quantum efficiency. The NEP is then computed as the ratio of the total noise spectrum divided by the responsivity.

If you would like Eikonal to design a transimpedance amplifier for your application or evaluate the noise performance of an existing design, please contact us. Examples of our transimpedance amplifiers include: Large Area Fast Amplified Si Photodiode and LiPo-Powered Amplified Si Photodiode.

A convenient parameterization describes the gain as a function of frequency, \(\omega = 2 \pi f\), as a single pole, low-pass filter with open-loop DC gain \(A_0\) \[ A(\omega) = \frac{A_0}{1+ \mathbb{i}( \omega /\omega_p)} \] and the location of the pole is given by \( \omega_p\). The op-amp's bandwidth is usually specified in manufacturers' datasheets as the gain-bandwidth product, GBW, or equivalently where the magnitude of the gain falls to unity. Hence, \[ |A(\rm{GBW})|^2 = 1 \] or \[ \frac{A_0^2}{1+ ( 2\pi \; {\rm GBW} /\omega_p)^2} = 1 . \] Hence, \[ \omega_p = 2 \pi \frac{ {\rm GBW}}{\sqrt{A_0^2-1}} .\] As the open loop, DC gain is very large \[ \boxed{ \omega_p \simeq 2\pi \frac{ {\rm GBW}}{A_0} , } \tag{A1} \] is an excellent approximation.

^{ 1. } Horowitz and Hill in The Art of Electronics: The x-Chapters introduce a quantity \(\zeta\) in § 4x.3.2, which they call the damping ratio. The caption to their Fig. 4x.20 states \(\zeta =1 \) corresponds to critically damped, in agreement with common usage. A critically damped system will just fail to overshoot; however, the step response for \(\zeta=1\) (Fig. 4x.21, Case b), shows pronounced overshoot. Inspection of the numerically computed transfer functions in Fig 4x.20 and step responses in Fig. 4x.21 suggests that Case b is not \(\zeta = 1\) but \(\zeta = 1/\sqrt{2} \simeq 0.707\) and Case c is the critically damped case with \(\zeta = 1\). Case-a (identified as 1.25 dB gain peaking in Fig. 4x.20) is for a damping ratio of \(\zeta = 1/2\). ↩

^{ 2. } An expression for \(\zeta\) appears on p.14 of the LTC6268 datasheet . This expression as presented is dimensionally inconsistent and must contains typographical errors. ↩

^{ 3. } The squared quantity, \(e^2_n (f) \), is the voltage power spectrum with units of \({\rm V^2\, Hz^{-1}}\). For example, the Johnson noise associated with a resistor, \(R\), at absolute temperature, \(T\), is \(e^2_n = 4 k_B T R \). The quantity \(e_n \) is convenient because if the spectrum is flat (a constant) like Johnson noise, the RMS voltage measured in bandwidth \(\Delta \nu\) is \( e_n \sqrt{\Delta\nu}\). ↩

© *Eikonal Optics*

An ideal imaging system delivers a perfect, converging, spherical wavefront where the optical path length (OPL) along all rays from a given object point to the corresponding point on the image are identical. The Strehl ratio is a common and easily computed figure of merit for describing image quality in near diffraction-limited systems. Here we derive an approximate expression for the Strehl ratio and compare with exact results for a few cases.

The peak brightness in a perfect image is designated \(I_0\). When aberrations are present in an optical system the wavefront converging towards the image is no longer spherical but distorted. Aberrations distort the image and reduce the peak brightness, \(I\), compared to the ideal case. The ratio of achieved image brightness relative to \( I_0\) is known as the Strehl ratio, \[ SR \triangleq \frac{I}{I_0} . \] The Strehl ratio is easy to compute and therefore a convenient measure of image quality.

In "Principles of Optics," Born & Wolf^{1} derive a Taylor series approximation to the on-axis intensity in the image plane for the case of small wavefront aberrations. The first-order term in this expansion for the Strehl ratio yields,

is the physical wavefront variance; \(OPL \) is measured in units of length.

The approximation for the Strehl ratio can be understood and extended by considering image formation as the interference from \(N\) sub-regions into which the converging wavefront has been divided. The electric field amplitude, \(E\), at the image plane is the vector sum of the corresponding \( N\) equal amplitude phasors (see Figure 1),

\[ E= \sum_{j=1}^N \exp(\mathbb{ i}\phi_j) ,\]where \(\phi_j=(2\pi/\lambda ) OPL_j\), and \( \mathbb{i} = \sqrt{-1}\).

If all \(N\) optical path lengths are identical, i.e., the wavefront is spherical, all the phases are the same and the RMS phase error is zero. Consequently, all the phasors line up coherently to give intensity, \( I_0 \propto E_0E_0^* = N^2\) at the location of the geometric image.

If the optical paths vary and the associated phase errors are zero mean, uncorrelated, and normally distributed then we can compute the electric field in terms of the expectation value of the complex phasor. From the definition of the expectation value, \( \langle \cdot \rangle\), \[ E=N \langle \exp(\mathbb{i} \phi) \rangle, \] and using the result ^{3} for a normally distributed quantity with zero mean \[ \langle \exp(\mathbb{i} \phi ) \rangle = \exp(-\sigma_\phi^2 /2), \] we have \[ E= N \exp(-\sigma_\phi^2/2) ,\] and \[ SR = \frac{I}{I_0} = \frac{EE^*}{N^2} = \exp(-\sigma_\phi^2) . \] The Strehl ratio quantifies the peak intensity of an image formed by a distorted or aberrated wavefront relative to the peak intensity of an unaberrated wave; this last equation is known as the extended Marechal approximation. In the radio astronomy literature the Strehl ratio is analogous to antenna gain and this result is known as the Ruze formula ^{4}.

Figure 2 shows some numerical diffraction calculations for an unobscured circular pupil (left hand column). The far field diffraction pattern (center and right hand columns) is computed using the Fraunhofer approximation implemented using fast Fourier transforms. The figure shows three rows with increasingly large wavefront errors. These errors are normally distributed with a mean of zero and an RMS that is listed at the bottom of each surface plot in the left column. The central column shows a false-color image of the diffraction pattern or point spread function (PSF) displayed using a logarithmic scale. The right hand column shows a 1-d plot of a horizonal line cut through the center of the image.

The first row of Figure 2 shows the aberration free result, where the PSF is an Airy function. The first and second Airy rings (4.7% and 1.6% of the peak) are easily identifiable in the false color image. In the second row normally distributed wavefront errors are included with an RMS of 1/8 of a wave. In the corresponding image the central intensity has decreasd by about a factor of two relative to the Airy function (the numerically computed Strehl ratio is 0.54). Only the first Airy ring is clearly identifable and numerous speckles at 1% of peak brightness are scattered across the image. In the third row the RMS is increased to 1/4 of a wave. The core of the Airy function persists, but it is now only a tenth of its original brightness (\(SR\) = 0.1) and speckles are pervasive.

Figure 2 considers pure Gaussian random aberrations, with no spatial correlation. Comparison of the numerically computed Strehl ratios shows that the extended Marechal approximation is exact under these circumstances.

Because surface polishing defects leads to spatially correlated errors and and optical misalignment tends to yield smoothly varying wavefront shapes, polynomial desciptions are commonly adopted to describe actual wavefront shapes. Figure 3 shows an example of a low order aberration. In this case we have chosen Zernike spherical aberration described by \[ W(\rho ) = C_{40} ( 6 \rho^4 - 6\rho^2 +1 ), \] where \(C_{40}\) is the Zernike polynomial coefficient determining the strength of the aberration and \(\rho \) is the radial pupil coordinate. Zernike spherical aberration is a balanced aberration where spherical aberration, \(\rho^4\), is balanced with defocus, \(\rho^2\), to minimize the resultant RMS wavefront error. The peak-to-valley amplitude of this aberration is \( 3C_{40}/2\) and the corresponding RMS wavefront error for this aberration is \( C_{40}/ \sqrt{5} \simeq 0.447 C_{40}\).

The wavefront surface plots in Figure 3 are labeled by the value of \(C_{40}\) and the exact Strehl ratio is listed on the image of the PSF in the central column. The PSF images and line plots show the charactertics of spherical aberration which decreases the peak of the PSF fills in the dark zones between the Airy rings thereby reducing overall contrast.

Figure 4 compares the exact results for the Strehl ratio of Zernike spherical aberration with the extended Marechal approximation. Unlike the case of Gaussian errors, the analytic formula is not exact. However, for values of \(C_{40} < 0.4 \), the approximation for the Strehl ratio is better than 10%.

The well-known Rayleigh \(\lambda/4\) criterion for "diffraction limited" perfomance is based on the observation that a quarter wave of spherical aberration reduces the Strehl ratio to \(0.8\). One quarter of a wave corresponds to \(C_{40} = 1/6\), which consultation of Fig. 4 shows corresponds to this Strehl ratio.

**Footnotes:**

^{1. Born, M., & Wolf, E., 1980, "Principles of Optics", Sixth Ed., \(\S\)9.1.3, p. 464, Pergammon Press ↩}

^{2. Marechal (1947, Rev. d'Opt., 26, 257) showed that \( S\simeq (1 - \sigma_\phi^2/2)^2\). The expression cited in Born and Wolf is the result achieved when terms in \(\sigma^4\) and higher are neglected. ↩}

^{3. The expectation value of \(\exp({\mathbb{i}\phi})\) for zero mean, normally distributed errors, \(\phi\), is ↩} \[ \langle \exp({\mathbb{i}\phi})\rangle = \int_{-\infty}^{\infty} \frac{ \exp{(\mathbb{i}\phi}) }{\sqrt{2\pi}\sigma_\phi} \exp{\left( -\frac{\phi^2}{2\sigma_\phi^2} \right) }\; d\phi = \exp{(-\sigma_\phi^2/2)}. \]

^{4. Ruze, J. 1966 Proc. IEEE, 54, 633 ↩}

© *Eikonal Optics*

Multimode fibers are conveniently used to couple low spectral resolving power \( ( \lambda /\delta \lambda \simeq 1000 ) \) spectrometers to light sources. However, as spectral resolution increases modal noise in fibers introduces systematic errors in fiber-fed spectrographs. Here we summarize the origin of this noise and implement a remedy in the form of a ball lens based fiber coupler for a high spectral resolving power spectrometer \( (\lambda /\delta \lambda \simeq 10,000 ) \).

Figure 1 shows the far-field diffraction pattern from a 50 \(\mu\)m multimode fiber. When this fiber is illuminated using a quasi-monochromatic 650-nm diode laser, the resultant speckle pattern due to interference between the various modes propagating in the fiber is evident. Since the optical path length and hence the resultant interference pattern is wavelength dependent, the transmission of the fiber also varies with wavelength. This wavelength dependent transmission is superimposed on any spectra measured using this fiber, causing systematic errors. Small perturbations to the fiber changes the optical path length, e.g., by bending the fiber, hence the speckle pattern is not static. Consequently, the transmission is variable and cannot easily be measured and removed.

One solution to modal noise is to agitate the fiber mechanically, which changes the speckle pattern. If the pattern can be refreshed many times during an exposure, then a form of averaging occurs and the effect is mitigated. The alternative is to use a fiber that supports more modes and therefore more speckles. This approach relies on spatial averaging rather than temporal averaging. The simplest way to increase the number of modes is to use a larger diameter fiber. However, as the spectral resolving power of a dispersive spectrometer is typically determined by the size of the entrance aperture (the fiber diameter) a simple replacement with a larger diameter fiber will degrade resolution.

Originally, in our application a high-resolution spectrometer was fed using inexpensive telecom OM2 fiber that is used for 10 Gigabit Ethernet. This graded index, multimode fiber has a 50 \(\mu\)m-diameter core, \( 2a\), a 125 \(\mu \)m cladding, and a numerical aperture, \( NA = 0.2\). The normalized frequency or dimensionless \(V\) number,

\[ V = \frac{2\pi a }{\lambda}\; NA , \]determines the number of propagating modes in a fiber. For a circular cross section, graded-index fiber the number of supported modes is approximately \( M =V^2/4 \) for a fiber with power law profile that minimizes modal dispersion. At 650 nm an OM2 fiber (\(NA =0.2 \)) has \(V =110 \) and \(M = 580 \). This is a relatively small number of modes, as the signal-to-noise ratio is set by Poisson statistics and is approximately equal to \(M^{1/2} =24 \).

Alternatives to telecom fibers are larger 100, 200, or 400 \(\mu\)m diameter step index, \(NA =0.22\) fibers. The smallest diameter is preferred given that the spectral resolving power decreases inversely with fiber diameter. The number of modes supported by the preferred choice of 100 \( \mu\)m step index fiber is

\[ M = \left( \frac{2V}{\pi} \right)^2\]or \(M\) = 5050, which represents a substantial reduction in speckle noise.

The simplest way to couple a larger diameter fiber to the spectrograph without degrading the spectral resolving power is to re-image the end of the fiber onto a entrance slit with width equal to the diameter of the original fiber. This approach loses light which spills over the edges of the slit, but because light up and down the slit can be imaged onto the sensor the light loss for a 100 \(\mu\)m fiber and 50 \(\mu\)m slit is only approximately 40%.

The adopted design to couple a fiber to a slit using ball lenses is shown in Figure 2.

Ball lenses are compact and inexpensive and although they suffer spherical aberration. Figure 3 shows the geometric spot size (20 \(\mu\)m diameter), which is sufficient to illuminate the spectrograph entrance slit without additional losses.

The fiber coupler is compact; the total track from fiber input to slit is about 13 mm. The ball lenses are glued into compact lens adaptors (Figure 4 & 5) that are 2 mm thick. The small size of the assembly means that no major changes are needed to the mechanical structure of the spectrometer and the 1:1 magnification permits reuse of the collimator optics.

An image of the 50 \(\mu\)m spectrograph entrance slit is shown in Fig. 6.

The final assembly of the fiber coupler, slit, and 100-mm focal length collimator is shown in Fig. 7. A cage system was adopted for the mechanical structure. Since the working distance from the ball lenses is only 1.1 mm, care has to be taken to allow the fiber tip and the slit to be close enough to the lens holder so that they can be in focus. Accurate element spacing was established using precision gauge blocks. The slit holder (Fig. 6) was glued into a 1-inch threaded lens tube and a retaining ring was used to allow this holder to be oriented so that the image of the slit is vertical at 590 nm. The slit is held in a stage with adjustable X/Y position so that the image of the fiber is accurately centered on the slit.

Figure 8 shows a high resolution spectrum of the Sun obtained with a 100 \(\mu\)m fiber and the new fiber coupler. In this very high signal-to-noise data, multiple weak atomic absorptions lines are visible as well as molecular bands due to oxygen and water vapor in the Earth's atmosphere.

© *Eikonal Optics*

The wavelength solution for a given instrument is not static in time. Environmental changes in temperature or humidity, mechanical changes such as shock or vibration, and optical configuration changes such as slit replacement or fiber illumination all impact the wavelength scale. Depending on the precision and accuracy required for a specific application, wavelength calibration may be needed at a minimum for each new set-up or, for critical applications, for each new measurement.

If we observe a source of atomic emission lines of known wavelength from a Hg/Ar or Ne lamp, we can identify the pattern of lines and associate each line with its measured pixel position. The mapping between line wavelength and pixel position depends on the grating geometry, groove density, camera focal length, and pixel size. However, rather than fitting a physical model, it is convenient and conventional to capture this relationship empirically as a polynomial.

The code below provides an example of how to perform your own wavelength calibration, avoiding the costly and time-consuming process of returning your spectrometer to the manufacturer for calibration.

The below code is an embedded trinket written in Python, which can be run in your browser without downloading data or software. You can customize the code for your own specific application or change it to explore how it works. The four included files are the code itself (main.py), the calibration Hg/Ar spectrum used for establishing the relationship between pixel position and wavelength (hgar.csv), a dark spectrum taken immediately following the Hg/Ar spectrum and with the same exposure time (dark.csv), and a line list for Hg I and Ar I developed for this particular spectrum that associates observed line positions with known NIST wavelengths (atlas.csv).

If you have not used trinket before, to run the code, choose the "play" button. To edit the code, select the appropriate file tab and click in the window. You can toggle to full-screen mode from the hamburger menu in the upper left. You can download the code from this menu as well.

To see the uncalibrated spectrum and a list of algorithmically-identified peak positions, set `identify_peaks` on line 32 to `True`. This first figure marks each prospective line with a red tick. Type "c" in the results frame continue. These line positions are used with the NIST atomic lines database to construct a spectral atlas (atlas.csv) for wavelength calibration of the spectrum. On line 81, setting `determine_poly_degree = True` allows testing the results of different polynomial orders for the wavelength solution. With good centroid measurements and the appropriate polynomial, the rms for the wavelength solution is 0.28 pix or 0.07 nm. Again type "c" in the results frame to continue. The final figure applies the polynomial wavelength solution to the reference spectrum, with "Calibrated wavelength" shown on the x-axis.

Here is a textual summary of the procedure to collect and analyze the measurements you need to update your spectrometer wavelength calibration.

- Illuminate your spectrometer with a spectrometer calibration lamp.
- Collect a spectrum with high signal-to-noise and high dynamic range. If bright lines are saturating, use a shorter exposure time and average an ensemble of frames.
- Collect a dark spectrum, if necessary.
- Export your data to CSV and dark subtract.
- Identify pixel positions for peaks manually or algorithmically.
- Compare your spectrum with the NIST database for the atomic species appropriate for your lamp (e.g, Hg I, Ne I, Ar I). Discard closely-spaced, blended lines.
- Construct a list mapping peak pixel positions to NIST wavelengths (atlas.csv in this example).
- Use this list to calculate the line centroids (line 64 in the above code).
- Determine the appropriate order of polynomial for the wavelength solution and confirm the residuals are satisfactory.
- Print the coefficients for the polynomial that returns wavelength for a given pixel position. Inspect the current coefficients uploaded in your spectrometer's control software. Before modifying the coefficients in your firmware, check your newly derived coefficients for large departures from the default. Optionally, upload the new coefficients to your spectrometer.

© *Eikonal Optics*

We explore the mapping from pixel position to wavelength and how it depends on grating geometry, groove density, camera focal length, and pixel size. The polynomial approximation can be justified based on this examination. A practical example is explored and a method to fix the order of the polynomial provided.

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The wavelength solution for a spectrometer is established by observing a source with known monochromatic wavelengths, e.g., cold cathode Ne and Ar discharge lamps, and measuring the pixel position of the resultant lines on the sensor array. In this example we use both neon and argon lines. For a grating-based spectrometer we derive this relationship and show why the commonly adopted polynomial approximation provides a suitable wavelength solution. The degree of the polynomial adopted is not known * a priori*. Here we propose a method to make this choice. As a concrete example we will work through the wavelength calibration of an Ocean Optics USB2000 compact spectrometer.

The USB2000 spectrometer is based on a diffraction grating and a one-dimensional CCD detector array of 2048 pixels. Light enters via a slit located at the end of a threaded receptacle, which can be used to connect to a multimode optical fiber that is terminated with a SMA 905 connector. This instrument achieves a spectral resolution of about 0.6 nm between wavelengths of 350 to 660 nm. The spectrograph is based on a Czerny-Turner optical design and has no moving parts.

Property |
Value |

Model | USB 2000 |

Grating | 1200 mm\(^{-1}\) holographic grating |

Bandwidth | 350-660 nm |

Options | L2 lens, 25 mm slit, WG305 filter |

CCD | Sony ILX411; 2048 pixels; 14 \(\mu\)m\(\times\)200 \(\mu\)m |

Focal lengths | 42 mm collimator; 68 mm camera |

The optical layout of the USB2000 is shown schematically below.

Light enters via the SMA fiber receptacle. The light diverges and fills a cone defined by the numerical aperture of the fiber. The light is collected by the collimator mirror. The collimator illuminates the grating with parallel light, where it is diffracted, with longer wavelength light emerging at a larger angle than shorter wavelengths. These rays are imaged by the camera mirror, which relays rays at different angles of diffraction onto different locations on the CCD sensor.

The function of the collimator, grating, and camera are clearer in a simplified ray diagram, where we have replaced the mirrors with lenses and the reflection grating by a transmission grating.

The basic function of the optical layout shown above is to make an image of the entrance slit on the CCD array. The slit and the CCD are placed one focal length from the collimator and camera, respectively, so that light between the two is parallel or collimated. With no grating in the optical path, all wavelengths would come to a focus at a single point on the optical axis. However, the grating introduces a angular deflection so that the position of the image depends on the wavelength. The spectrometer makes monochromatic images of the slit with position that depends on wavelength. Thus, the spectrometer encodes wavelength as position. As the slit has finite width, the ability of a slit spectrometer to separate two wavelength (spectral resolving power) depends on the width of the slit.

To understand the basis of wavelength calibration we need to understand the relationship between the location of the image of the entrance slit on the CCD and wavelength. The grating equation (Eq. 1) determines the angle of diffraction, \( \beta \), relative to the grating normal. If \( \lambda \) is the wavelength under consideration and \( \sigma\) is the grating groove spacing, then the condition for constructive interference requires that \[ m \lambda /\sigma = \sin \alpha + \sin \beta , \tag{1} \] where we have assumed a 2-d geometry in which the light that is incident at an angle \( \alpha \) relative to the grating normal lies in the plane perpendicular to the grooves on the grating.

By considering the camera as an ideal thin lens we can see from the sketch above that \( \beta\) fixes the position of the image on the CCD array at wavelength \( \lambda\).

In the general case where the grating normal and the optical axis of the camera are not parallel, a ray passing through the center of the camera lens, focal length, \(f_{cam}\), at angle \( \beta \) intersecs the image plane at a point \[ p = p_0 + f_{cam} \tan (\beta-\beta_0) , \tag{2} \] where \( \beta_0\) is the angle between the grating normal and the camera optical axis and \(p_0 \) is a reference pixel where the wavelength is \( \lambda_0\). Using the grating equation for \( \beta \), the full relation between \(p\) and \( \lambda \) is

\[ p = p_0 + f_{cam} \tan \left[ \arcsin \left( m\lambda/\sigma -\sin \alpha \right)-\beta_0 \right] \tag{3} .\]For our USB spectometer we know the pixel size of the Sony ILX511 CCD (14 \(\mu\)m), the groove density of the grating (\(1/\sigma = \) 1200 mm\(^{-1}\)), and the order of diffraction (\(m\) = 1) in Eq. (3). We should be able to use the measurements of line positions to make a least squares estimate of the remaining four parameters and therefore the wavelength of any pixel in the array.

At first sight, Eq. (3) does a good job describing the measurement of line positions. The non-linearity evident in the data (blue points) seems to be nicely matched by upward curvature of the model (red line). However, it is always good practice to plot the * residuals * or the difference between the observed and predicted values. This allows us to focus on the discrepancy between the data and the model and highlight any systematic trends.

This figure shows a systemic pattern with errors as large as 2.6 pixels or 0.4 nm. The fidelity of Eq. (3) could be improved to achieve smaller residuals and a more precise and accurate wavelength calibration. For example, the camera suffers from optical distortion because \(f_{cam}\) is not constant, but depends on field angle. A more expedient solution, an empirical polynomial solution: \[ p = p_0 + p_1 \lambda + p_2 \lambda^2 + p_3 \lambda^3 + ... , \tag{4}\] is typically adopted by manufacturers.

A polynomial wavelength solution is justified by making a Taylor expansion of Eq. (3) about some wavelength, \( \lambda_0\) , which yields

\[\begin{eqnarray} p & = & p_0 + f_{cam} \frac{m}{\sigma \cos \beta_0 } (\lambda - \lambda_0) \\ & + & f_{cam} \frac{m^2 \tan \beta_0}{2\sigma^2 \cos^2\beta_0} (\lambda - \lambda_0)^2 \\ & + & f_{cam} \frac{m^3 }{2\sigma^3 \cos^5\beta_0} (\lambda - \lambda_0)^3 \\ &+& O\left( (\lambda - \lambda_0)^4 \right). \tag{5} \end{eqnarray}\]A polynomial wavelength solution (Eq. 4) and a Taylor series expansion (Eq. 5) are functionally equivalent, justifying common practice. Both odd and even powers are present in the expansion, so no terms can be dropped. The coefficients are not independent, but this information is discarded when a polynomial solution is adopted. In principle a series in powers of \( (\lambda - \lambda_0)\) is preferable because of the potential to exceed the largest floating point values if 32-bit representation is used. This approach provides no guidance on where to terminate the series.

Examining the residuals or difference between the observed line position, \(p_i\), and the predicted one, \( p(\lambda_i)\), based on a polynomical fit provides a practical way of establishing the order of fit to use. It is necessary to use the lowest order consistent with the uncertainties in the position measurements because a polynomial of sufficently high order will pass through every point and amplify the noise in the wavelength solution.

If the uncertainty in each line position measurment \( \sigma_i\) is known then the \(\chi_n ^2\) statistic,

\[ \chi_n^2 = \frac{1}{n}\sum_{i=1}^n \frac{ \left(p_i - p(\lambda_i)\right)^2 } {\sigma_i^2} \]can be used to assess the quality of the fit. The rule of thumb is when \( \chi_n^2 =1 \), the fit is adequate.

Often the measurement errors are not well known. But the quality of the fit can be examined by plotting the residuals and computing the RMS, \[ RMS = \sqrt{\frac{1}{n-m-1}\sum_{i=1}^n \left(p_i - p(\lambda_i)\right)^2 } , \] where \( m\) is the order of the polynomial.

The linear fit shows a residual with a predominantly parabolic shape. The quadratic fit accounts for this departure, but leaves an S-shaped cubic residual, which in turn is removed by the next higher order fit. At each step the RMS drops: from 23 to 2.3 to 0.5 and then to 0.1 pixels for the quartic polynomial. The final panel shows that no improvement is achieved going from a quartic to a quintic polynomial. Adding higher order terms would just fit the noise in the data, so the quartic fit is preferred.

© *Eikonal Optics*

A silicon photomultiplier (SiPM) is solid state detector comprised of an array of reverse biased photodiodes. The bias voltage is large so that the photoelectron/hole pair created by photon absorption is accelerated by the resultant electic field and acquires enough energy to liberate secondary charge carriers. This process repeates, causing an avalanche, hence the name avalanche photodiode (APD). The resultant charge multiplication or gain is large (\(10^5 - 10^7 \)) and the avalanche is fast (ns) so that individual photons generate current pulses of sufficient amplitude to be detectable.

This summary suggests that a SiPM can be used to count photons and is therefore an ideal linear photodetector. The linear regime cannot extend without limit because an APD is "blind" for some period after detecting a photon and above some intensity it will fail to detect a fraction of subsequent events. As we will see, by assembling an array of APDs, the SiPM design affords greater dynamic range than a single APD but there remain limits to the linearity of the device.

One of the unique features of a SiPM is that it is assembled from hundreds to thousands of individual APDs.

The image shows the structure of an * onsemi* MICRO-FC-30035 SiPM with 4774 APDs or microcells. The device is 3mm \( \times \) 3 mm and each microcell is 35 \(\mu\)m \( \times \) 35 \(\mu\)m and appears as a tiny black square. Unlike a CCD or a CMOS sensor the photodiodes which comprise the SiPM are not separately readout or addressed\(-\)they are all connected in parallel and the output signal cannot be traced back to an individual diode.

To understand SiPM non-linearity we must first understand how the avalanche triggered by the absorption of a photon terminates. Consider the equivalent circuit for an APD adopted by Hamamatsu.

In this model, an APD is idealized as a perfect switch (*APD*) with diode junction capacitance, *CJ*. Each APD in the SiPM array has its own quench resistor, *RQ*. The resistor, *RS*, represents the resistance of the APD during discharge, which is initiated by the * TRIGGER * signal. The breakdown voltage of the APD is represented by voltage source, *VBR*, and the externally applied bias voltage is *VBIAS*.

In the absence of light (and neglecting thermal excitations which cause dark current) the photodiode junction capacitance charges up to *VBIAS* through *RQ*. When a photon generates an electron/hole pair the switch turns on. Two things now happen: 1) a current pulse (red) begins to flow in the circuit and; 2) the voltage across the APD (blue) starts to fall. The current pulse, with a sub-ns rise time set by the \( R_S C_J\) time constant, signifies the detection of a photon. Once current starts to flow, the diode junction capacitance discharges faster than it can be replenished through * RQ * and the voltage drops to the breakdown voltage (horizontal blue dotted line), terminating or * quenching * the avalanche.

The diode switches off and the voltage recovers as * CJ * recharges through * RQ * with a time constant \( R_Q C_J \). This interval represents a deadtime in which an individual APD is unresponsive. Here, the rise and fall times are representative of the * onsemi * MICRO-FC-30035 in the Eikonal SIPM-01 module (0.6 ns and 82 ns respectively). This disparity in these times reflects the fact that the quench resistor is much larger than the resistance through which the diode discharges.

If two microcells are triggered simultaneously the output is a linear superposition of two current pulses. Thus, the brighter the incident intensity, the stronger the response. However, if the same microcell absorbs two photons on a timescale comparable to the recharge time \( R_Q C_J \), only a single current pulse ensues. For high intensity nanosecond duration pulses we can expect to experience non-linear response.

The following image helps visualize the circumstances when a brief pulse is incident on a SiPM. The image depicts an array of 69 \( \times \) 69 = 4761 microcells. Each microcell has 100% detection efficiency and when a microcell receives one or more photon it is displayed as white. The upper right image shows the array when the pulse contains 10 photons. In this example 10 microcells were triggered. In the next panel 100 photons triggered 97 microcells because 3 photons landed in microcells that had already recorded a detection. In the third panel with 1000 photons only 909 events were recorded and in the last panel with 10,000 photons only 4160 microcells produced events. As the number of photons in a pulse increases, the chances that an incident photon lands on a microcell that has already triggered increases. Therefore, the responsivity of the SiPM decreases with light pulse intensity and the response is non-linear.

The procedure illustrated in the simulated images showing triggering of microcells by a group of photons is a statistical experiment known as a Bernoulli trial. We are given \( n \) trials (the number of photons) for which the probability of success is \( p\) (the probability a photon triggers one of \( M \) microcells), and we want to know the probability of achieving \( k \) successes. If the detection efficiency for a single photon is 100% the probability that one photon lands in a given microcell is \( p = 1/M \). If we want to include the quantum efficiency, \( \eta < 1 \), then \( p = \eta /M \).

Rather than calculating the probabilty of a microcell receiving one or two or three or more photons, it is easier to calculate the probability that a microcell receives *none*. The probability of a given microcell failing to receive a photon is just \( 1- \eta/M\), so in \( n \) trials the probability of a a microcell not being triggered is \[ P(k=0) = \left(1 - \eta/M \right)^{n} . \] Probabilities must sum to unity, therefore \( P(k=0) + P(k\ge 1) = 1\) and \[ \begin{eqnarray} P(k\ge 1 ) &=& 1- \left(1 - \eta/M \right)^{n} \\ & \simeq & 1- \exp \left( - \eta \frac{ n}{M}\right) ,\end{eqnarray}\] using the Bernoulli approximation for \( e \). The exponential approximation is sufficient for all practical purposes unless there are only a handful of microcells and \(M \sim 1\) .

On average, when \( n \) photons are incident on \(M \) microcells, the number of microcells with one or more photons is the product \( M P(k\ge 1) \). As we are assuming that the light pulse is shorter than the microcell recharge time and a single microcell can only produce a single current pulse, the number of photon events recorded by the SiPM is \[ N = M \left[ 1- \exp \left( - \eta \frac{n}{M}\right) \right] .\] For small numbers of photons (\(\eta n \ll M\)), \[ N = \eta n ,\] the SiPM responds linearly to the number of incident photons. When \(\eta n \ll M\) the number of recorded events per detected photon, \( \partial N / \partial (\eta n ) \) , is unity. If we define the non-linearity as \(\rm{N.L.} = 1 - \partial N / \partial (\eta n ) \) then \[ \rm{N.L.} = 1- \exp\left(- \eta n/M \right). \] As the number of photons in the pulse approaches the number of microcells, the SiPM becomes increasingly non-linear as shown in the plot below. As a rule of thumb, better than 10% non-linearity is achieved when there are more than 10 microcells per detected photon in the light pulse being measured.

We measured the linearity of the Eikonal plug-and-play SiPM detector module (SIPM-01) in response to nanosecond laser pulses. The Eikonal SiPM detector module is equipped with an * onsemi * MICRO-FC-30035 detector, which has 4774 microcells and a recovery time of 82 ns. The Eikonal nanosecond pulsed laser diode (NLD-01) is ideal for this exploration because it delivers up to 0.4 nJ or \(1.3 \times 10^6\) photons in a few nanoseconds. We used a set of neutral density filters with optical densities up to OD 4.5 to explore the transition from linear to non-linear response. Measurements of the SiPM signal were collected at 2 GS/s using a Tektronics TDS380 scope and integrated numerically to record the strength of the laser pulse. The plot shows the measured SiPM response together with the theoretical expectation (red curve) derived above.

Below \( 10^3 \) photons per pulse the results show good agreement with linear response. At \( 5\times 10^3 \) photons per pulse (\( n \simeq M \)) both the model and data begin to reveal a marked deviation from linear behavior. At the last data point (\( 2\times 10^4 \) photons per pulse), the measured signal is well into the non-linear regime. The gain of the RF amplifier in the SIPM-01 module is chosen such that saturation of the amplifier and significant SiPM non-linearity occur at approximately the same photon flux.

The microcell architecture of SiPMs means that they can be used to measure nanosecond light pulse intensity over many orders of magnitude. However, it is important to be aware that non-linear effects must be taken into account when the number of incident photons on the SiPM within the microcell recharge time approaches the number of microcells. SiPMs are available with a large range of number of microcells; please contact us to customize your integrated SiPM detector module to fit your needs.

© *Eikonal Optics*

Generally, noise is any unwanted signal such as interference from a switching power supply or a WiFi router. These types of external noise can often be reduced or even eliminated by moving the source or adding appropriate shielding. However, some types of noise are intrinsic and unavoidable, even in a circuit made with ideal components. Quantum mechanical shot noise is a well known example. Shot noise is evident at low light levels when individual photons (and corresponding photoelectrons) comprising the signal can be discerned.

Johnson noise is perhaps less familiar but also ubiquitous because it is a type of noise that is associated with resistors. Historically, the discovery of voltage and current fluctuations in a resistor was made at Bell Labs in the 1920s. Measurements by J. B. Johnson were subsequently explained by H. Nyquist.

Johnson noise sets a threshold against which signals must be discriminated. Johnson noise is present at any temperature above absolute zero. It is therefore helpful to understand its origin and nature so its effects can be minimized.

A resistor is an ohmic device that introduces a voltage drop when current flows and necessarily dissipates electrical energy as heat. On a microscopic level, the action of a resistor is a result of collisions between charge carriers and the conducting material of the resistor. These collisions impede current flow and transfer electrical energy to vibrational energy of the atomic lattice (heat). Like all physical processes, the transfer of energy from a charge carrier to the lattice must have an inverse. Thermal motions can therefore excite fluctuations in the energy of charge carriers, which appear as current and voltage fluctuations unless the resistor is at a temperature of absolute zero (0 K). Notice the importance of dissipation: ideal capacitors and ideal inductors do not exhibit Johnson noise.

In principle, the nature of Johnson noise can be inferred from a detailed physical understanding of the transport of charge carriers and their interaction with a given material. However, Nyquist showed this is not necessary, as the properties of Johnson noise can be understood in terms of the statistical mechanics of a transmission line.

Consider a circuit consisting of a capacitor, \(C\), and resistor, \(R\), connected in parallel. In this example the resistor is enclosed in a heat bath at absolute temperature, \(T\). The circuit experiences thermal fluctuations as energy is exchanged between its components.

According to classical statistical mechanics, the principle of equipartition says that the average energy of each degree of freedom is \( k_B T/2\) , where \(k_B\) is Boltzmann's constant. Our \( RC \) circuit has a single degree of freedom associated with the energy stored in the capacitor carrying charge, \( q\). This energy is: \[ E = \frac{1}{2}\frac{q^2}{C} =\frac{1}{2} C V^2 .\]

Using angle brackets to denote time averages, \[ \langle E \rangle =\frac{1}{2} C\langle V^2 \rangle , \] and from equipartition \[ \frac{1}{2} C\langle V^2 \rangle = \frac{1}{2}k_B T.\] or \[ \langle V^2 \rangle = \frac{k_B T}{C} . \] Equivalently, the fluctuating charge on the capacitor is \[ \langle q^2 \rangle = kT_B C , \] which is why the Johnson noise appearing on a capacitor is sometimes called "kTC" noise.

To find the variance (standard deviation squared) we use \( \sigma_x^2 = \langle x^2\rangle - \langle x\rangle^2 \). In this example, the average charge and average voltage are both zero because both positive and negative charge fluctuations are equally likely. Therefore, \( \sigma_q^2 = \langle q^2\rangle \) and \( \sigma_V^2 = \langle V^2\rangle \).

Counterintuitively, the value of \( \langle V^2 \rangle \) for Johnson noise is independent of \( R \). The Fourier spectrum of voltage fluctuations on the capacitor, \( S_V^C(f) \), occurs over a range of frequencies. If \( R \) is large, the bandwidth of the \( RC\) circuit is small. For the total rms to be independent of \( R\), as we have just found, the amplitude of the spectrum of voltage fluctuations must be proportional to \( R\). Thus the fact that the expression for \( \langle V^2 \rangle \) only references \( C\) does not mean that the capacitor is the source of Johnson noise. Alternatively, we could have arrived at the same conclusion about the spectrum by considering current fluctuations in a resistor paired with an ideal inductor, implying that the resistor and its associated dissipation is always the noise source.

The variance of the voltage on the capacitor is related to the power spectral density of voltage fluctuations by \[ \langle V^2 \rangle = \int_0^{\infty } |S_V^C (f)|^2 \; df , \] where \( |S_V^C(f)|^2 = S_V^C(f) S_V^C(f)^*\) and the symbol \( * \) denotes the complex conjugate.

We now have to relate the voltage fluctuations that appear on the capacitor with the voltage fluctuations that arise in the resistor. As shown in the figure below, imagine replacing the resistor and its heat bath with its Thevenin equivalent of a perfect noise-free resistor in series with a voltage noise source described by the Fourier spectrum, \(S_V^N(f) \).

The transfer function, \( H(f) \), relates the the voltage across the capacitor to the applied voltage from the noise source, \( H(f) = S_V^C(f) /S_V^N(f) \). The resistor and capacitor form a voltage divider for which the transfer function is \[\begin{eqnarray} H(f) & = & \frac{1/(i2\pi f C) }{1/(i 2 \pi f C) + R} \\ \\ & = & \frac{1}{ 1 + i 2 \pi f RC}, \end{eqnarray}\] with \( i = \sqrt{-1} \) .

Using this result we have \[ \frac{k_BT}{C} = \int_0^\infty |S_V^N H(f)|^2 \; df . \]

The voltage fluctuations in the resistor are uncorrelated in time, so the associated power spectral density of fluctuations is constant with frequency and can be taken outside of the integral, \[ \frac{k_BT}{C} = |S_V^N|^2 \int_0^\infty \left| \frac{1}{ 1 + i 2 \pi f RC} \right| ^2 \; df , \] where the definite integral evaluates as \[ \int_0^\infty \left| \frac{1}{ 1 + i 2 \pi f RC} \right| ^2 \; df = \frac{1}{4RC}. \] This integral yields a quantity known as the equivalent power bandwith \( \Delta f = 1/(4 R C)\) for a single pole \(RC\) filter. Note, this is distinct from the 3 dB point which is defined by \( |H(f_{3dB})|^2 = 1/2\) or \( f_{3dB} = 1/(2 \pi RC)\).

From these results we conclude that the power spectral density of Johnson noise is \[ \boxed{|S_V^N|^2 = 4 k_BT R .}\] The SI units of \( |S_V^N|^2\) are \(\rm{V}^2\; {Hz}^{-1}\).

Referring back to the original equipartition argument we can see that the result for the voltage fluctuations on the capacitor can be written with explicit reference to the resistor value as \[\begin{eqnarray}\langle V^2 \rangle & = & 4k_BT R \Delta f \\ & & \\ & = & \frac{ k_BT R}{ \tau } . \end{eqnarray}\] Thus, as stated before with smaller bandwidths or longer time constants, \(\tau = RC \), the precision improves. This is related to our assumption that the Johnson noise fluctuations are uncorrelated in time and that averaging independent measurements reduces the standard error as the square root of the number of measurements or equivalently as the square root of the total integration time.

Often the results for a noise spectrum are quoted in as the square root of the power spectral density since the units are volts per root hertz not volts squared per hertz and perhaps easier to compare with a signal measured in volts. In this convention the numerical value of the Johnson noise spectrum is \[ | S_V^N| = 41 \left( \frac{ R}{100 \; \rm k\Omega }\right)^{1/2} \left( \frac{T}{ 300 \; {\rm K}} \right) ^{1/2} {\rm nV}\ {\rm Hz^{-1/2}} ,\] where the scaling is from adopted convenient values of \(R\) = 100 k\(\Omega\) and \(T \) = 300 K.

Note that the although the Johnson noise is proportional to \( R^{1/2} \) the signal is proportional to \(R\), hence the signal-to-noise ratio increases as \( R^{1/2}\). Thus, \(R\) should be made as large as possible, consistent with the desired bandwidth. In this example, the TIA-F-01 is limited only by Johnson noise at frequencies below 100 kHz.

In a photon-counting or integrating detector, which uses a capacitor to accumulate charge, e.g., a CCD, the rms error in units of the electron charge, \(q_e\), is \[ \begin{eqnarray}\sigma_e & = & \frac{\sqrt{k_BTC}}{q_e} \\ &=& 401 \left(\frac{C}{pF}\right)^{1/2} \left(\frac{T}{300~ K}\right)^{1/2}\; e^- \rm{rms} . \end{eqnarray}\] A 1 pF capacitor at room temperature has rms charge fluctuations corresponding to about 400 electrons. This seems to set a severe detection limit if we want to detect individual photons. However, if the integrating capacitor is isolated from the rest of the circuit while integrating (\(R\) is very large), the bandwidth becomes very small and the any fluctuation on the capacitor becomes fixed on very long timescales. Thus, the expedient of double correlated sampling, where the voltage on the integrating capacitor is measured before and after the integration defeats Johnson noise.

The derivation here is based on J. R. Pierce, "Physical Sources of Noise," in Proc. IRE, vol. 44, no. 5, pp. 601-608, 1956.

A history of electrical noise measurements and a rigorous derivation are in D. Abbott, B. R. Davis, N. J. Phillips & K. Eshraghian, "Simple derivation of the thermal noise formula using window-limited Fourier transforms and other conundrums," in IEEE Transactions on Education, vol. 39, no. 1, pp. 1-13, 1996.

Nyquist's approach to deriving Johnson noise is outlined in §17.5 of B. E. A. Saleh & M. C. Teich, Fundamentals of Photonics, Wiley

The spectrum of Johnson noise is found using the classical equipartition law similar to that used in the derivation of the Rayleigh-Jean's law for black-body radiation. The current result is valid only a "low" frequencies, i.e., below THz at room temperatures. At high frequencies or low temperatures, the classical value is modified by a correction factor of \(x/(e^x-1)\), with \(x=hf/k_BT\). This topic is discussed in our discussion of photon statistics.

© *Eikonal Optics*

Photons are not classical objects. Their behavior is governed by quantum statistics. At short wavelengths, Poisson counting is a good approximation so that when \(n \) photons from a thermal source are detected, the fluctuations are given by \( \sqrt{n} \). At infrared and radio wavelengths, the Poisson approximation breaks down. At these long wavelengths, photons do not arrive independently and the fluctuations are of order \(n \), rather than \( \sqrt{n} \).

Consider the number of photons, \( n \) of thermal radiation, in a cavity at temperature, \( T \), and frequency, \( \nu \). As photons are emitted and absorbed by the walls, the number of photons in the cavity will change. Boltzmann’s law gives the probability that a state with energy, \( E = n h \nu \), occurs, where \( n \) may take on any value \( 0, 1, 2, 3, \dots \): \[P(n) = \frac{1}{U} \exp \left(-n h \nu / k T \right), \] where \( h \) is Planck’s constant, and \( k \) is Boltzmann’s constant. The normalization, \( U \), is found from the condition that the probability must sum to unity: \[ \begin{eqnarray} U & = & \sum_{n=0}^{\infty} \exp \left( -n h \nu / k T \right) \\ &=& \frac{ \exp \left( h \nu / k T \right) }{ \exp \left( h \nu / k T\right) -1 }.\end{eqnarray}\] The quantity \( U \) is sometimes called the partition function.

The average energy at frequency, \( \nu \), is \[\begin{eqnarray} \langle E \rangle &=& \sum_{n=0}^{\infty} E (n) P(n) \\ &=& \frac{1}{U} \sum_{n=0}^{\infty} n h \nu \exp \left( -n h \nu / k T \right) \\ &=& \frac{h \nu}{\exp(h \nu / kT) -1}. \end{eqnarray}\] We can use this result to compute the energy density of photons. Photons are spin zero particles, and therefore, unlike half-integer spin particles like electrons, can congregate in a cell of phase space. The number of phase cells per unit volume for photons with momenta \( p \) to \( p + dp \) is \[ dn = 2 \frac{4 \pi p^2}{h^3} dp = \frac{8 \pi \nu^2}{c^3} d \nu, \] where the factor of two accounts for the two polarization states. Thus, the energy density of radiation at frequency \(\nu \) to \( \nu + d \nu \) is \[\begin{eqnarray} \rho_\nu d \nu & = & \langle E \rangle dn \\ & = & \frac{8 \pi \nu^2}{c^3} \frac{h \nu}{\exp(h \nu / kT) -1} d\nu, \end{eqnarray}\] which is the famous Planck formula for blackbody radiation.

The mean number of photons is \[ \langle n \rangle = \sum_{n=0}^{\infty} n P(n) = \frac{1}{\exp(h \nu / kT) -1}, \] and the average value of \( n^2 \) is \[ \langle n^2 \rangle = \sum_{n=0}^{\infty} n^2 P(n) = \frac{\exp(h \nu / kT) +1}{\left(\exp(h \nu / kT) -1\right)^2}. \] Hence, the variance, which is \( \sigma^2 = \langle n^2 \rangle - \langle n \rangle^2 \), is \[ \sigma^2 = \frac{\exp(h \nu / kT)}{\left(\exp(h \nu / kT) -1\right)^2}, \] or by substituting for \( \langle n \rangle \), we see that the variance is proportional to \( \langle n \rangle \) times a correction factor: \[\begin{eqnarray} \sigma^2 &=& \langle n \rangle \frac{1}{1 - \exp(h \nu / kT)} \\ &=& \langle n \rangle (1 + \langle n \rangle ).\end{eqnarray}\] Often the correction factor is very close to unity. The figure below shows the correction factor for an incandescent lamp (3000 K) and room temperature (300 K) at wavelengths between 100 nm and 100 \( \mu\)m.

In the ultraviolet and visible, we have to a very good approximation: \[ \sigma^2 = \langle n \rangle, \] i.e., the fluctuations are Poisson. This feature of photon statistics is used in our measurement of the gain of a CCD.

However, at infrared and radio wavelengths the Poisson approximation for thermal radiation becomes increasingly less accurate. When \(h \nu / kT \gg 1 \), the expression for \( \langle n \rangle \) shows that \( \langle n \rangle \ll 1\), and therefore the expression for the variance becomes \( \sigma^2 = \langle n \rangle \). For the detection of such light, individual photon events are uncorrelated. However, when \(h \nu / kT \ll 1 \), \( \langle n \rangle \gg 1\) and thus \( \sigma^2 = \langle n \rangle ^2 \). The increased fluctuations in this limit are a consequence of "photon bunching." Photons are correlated in this case in such a way as to increase the noise in the detected signal above that given by Poisson noise.

**Further Reading**

Lewis, W. B. (1947). *Fluctuations in streams of thermal radiation. Proceedings of the Physical Society, 59(1), 34–40.* doi:10.1088/0959-5309/59/1/307

* The image at the top of the post shows a two-dimensional image of Poisson counts with a mean value of unity, displayed using a false color map.

© *Eikonal Optics*