## Abstract

The optoelectronic process of light absorption and current formation in photodiodes is shown to be a significant source of optoelectronic chromatic dispersion (OED). Simple design rules are developed for fabricating a photodiode-based dispersion device that possesses large, small, zero, and either positive or negative OED. The OED parameter is proportional to a spectrally-dependent absorption term *α*^{−1}*dα*/*dλ* . Silicon-based devices are predicted to display significant OED throughout the near IR, while Ge and InGaAs have high OED in the C- and L-bands and 1650 nm region, respectively. The OED of a commercial Ge PN photodiode is measured to be 3460 ps/nm at 1560 nm wavelength with 500 kHz modulation, demonstrating 8 pm spectral resolution with the phase-shift technique. Temperature-tuning of the OED in the Ge photodiode is also demonstrated. The ubiquitous photodiode is a tunable OED device, with applications in high-resolution optical spectroscopy and optical sensing.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Since the advent of semiconductor-based photodiodes in the mid-20th century, this device has evolved from the simple PN-type detector into the various types that are in use today. This intense effort has been application-driven to reach the speed and efficiency requirements demanded by the various technologies that require light detection.

The dependence of the photodiode efficiency and RF bandwidth on the wavelength is well-established [1–6]. However, this wavelength dependency leads to a prominent feature of photodiodes that has been largely overlooked. The wavelength-dependent pathlength of the migrating charge-carriers is the source of an effective chromatic dispersion that can be extremely large, as well as controllable. Based on this inherent feature of photodiodes, we propose to regard this device as a source of optoelectronic chromatic dispersion (OED). We will present a model of OED which leads to simple design rules for fabricating OED devices having large, small, zero, positive or negative OED. The model predicts significant flexibility in designing a dispersion device based on a PN or PIN-type photodiode with desired OED characteristics. In particular, we predict and demonstrate large positive OED for a commercial Ge PN photodiode in the telecommunication C and L-bands, equivalent to over 200 km of standard optical fiber. Silicon-based photodiodes and PV cells can display positive, zero and negative OED in the visible and near IR. The OED can be tuned through various externally-applied mechanisms, such as temperature or strain, as well as internally through bias voltage control.

Material chromatic dispersion is a fundamental mechanism that affects light propagation in all types of materials [7]. As described by the Drude-Lorentz model, it is due to a wavelength-dependent time response of the bound and free charges to the illumination, also described as a dependence of the medium's real and imaginary refractive indices on the wavelength. It leads to an effective group index of refraction that is also wavelength dependent, and is the source of pulse spreading, posing a fundamental bandwidth limitation in fiber-optic communications [8]. However, besides its deleterious effects on propagating signals, chromatic dispersion has been widely utilized for optical sensing applications, and is the basis of many new ideas in fiber-optics. In particular, chromatic dispersion has been utilized for applications in signal processing, optical sensing and spectroscopy [9–17].

The physics of OED differs fundamentally from material chromatic dispersion. However, by regarding the photodiode as a new type of tunable dispersion device, there is untapped potential for utilizing this optoelectronic mechanism in a wide variety of optical sensing applications.

Figure 1 describes the fundamental source of OED in a PN-type photodiode, illuminated with sinusoidal-modulated light at frequency $\Omega = 2\pi f$. We consider an n-type light entrance region of width ${W_E}$, a p-type substrate region of width ${W_S}$, and a thin depletion region having width $d$ at the junction. Input light power ${P_{in}}$ is absorbed, generating electron-hole pairs that migrate due to diffusion to the PN junction. The figure depicts this process in the substrate side assuming illumination at ${\lambda _1}$ and ${\lambda _2}$ with ${\lambda _2} > {\lambda _1}$ . Due to the wavelength-dependent absorption coefficient $\alpha (\lambda )$, which for common semiconductors decreases for increasing $\lambda $ in the visible and IR, the power in each line is reduced by ${e^{ - 1}}$ at the penetration depths ${x_1}$ and ${x_2}$ respectively, so that the majority of the charge-pairs form in the region between the illumination entrance and the penetration depth. The figure depicts those charge-pairs that formed at the particular penetration depths of each of the two wavelengths. Due to the light modulation, the temporal dependence of the charge-pair formation follows this ac modulation, and after charge separation, the minority carriers (electrons in the p-type region) that ultimately contribute to the ac current have diffused on average to the p-side of the depletion region, where they are swept across to form this current. This RF modulated current accrues a modulation phase-delay ${\theta _{dif}} = \Omega {\tau _{dif}}$ where ${\tau _{dif}} = L_{{x_i}}^2/{D_e}$ is the electron diffusion time over the average pathlength ${L_{{x_i}}}$ to the depletion region edge, and ${D_e}$ is the electron diffusion coefficient. Since the ‘red’ electrons are formed along a larger distance from the junction as compared to the ‘blue’ electrons, they will accrue on average a larger phase-shift. This wavelength-dependent RF phase-shift is the fundamental source of OED. The figure depicts a vertical photodiode configuration, where the light propagation and charge motion are along the same axis. In a lateral configuration, in which the light path and charge motion are orthogonal, the OED would be negligible. In addition, we assume normal incidence illumination. Other types of coupling, such as butt or evanescent coupling, may affect the OED due to the angular spread of the illumination.

A well-known technique for measuring optical dispersion in materials is the modulation phase-shift method [18]. Sinusoidal-modulated light transverses the medium, and at the output the RF phase-shift of the light's modulation envelope $\Delta \theta = \Omega \Delta \tau $ is monitored as a function of wavelength shift $\Delta \lambda $, where $\Delta \tau = \widehat D\Delta \lambda $ due to chromatic dispersion $\widehat D$. In fiber-optics, it is customary to express $\widehat D = DL$ where *D* is the dispersion parameter and *L* is the fiber length. In OED, we lump this into a dispersion parameter ${\widehat D_{OED}}$, so that the modulation-phase-shift dependence on wavelength is $d\theta /d\lambda = 2\pi f{\widehat D_{OED}}$. We define the OED sensitivity as ${S_{OED}} \equiv d\theta /d\lambda $, and focus on developing the photodiode design rules for achieving a desired value for this performance parameter. Applications in highly sensitive wavelength monitoring, spectroscopy and other novel optical sensors would benefit from a high value of ${S_{OED}}$. Furthermore, this value can be made positive or negative. In other applications, it would be beneficial to design a zero-OED device.

## 2. Theory

In the early years of photodiode development, the theory of the RF frequency response of PN and PIN-type photodiodes was developed, based on the continuity equation for light-induced charge carrier generation and migration by diffusion and drift [1–4]. These widely accepted models form the basis for our description of OED in these devices. For modulated illumination, each of the two PN photodiode regions contribute linearly to the overall ac current, and therefore to the overall ${S_{OED}}$ as well. We will derive the contributions from each region, and then determine the total ${S_{OED}}$ from the PN photodiode.

#### 2.1 Entrance-region and substrate region contributions to OED

We begin with the Sawyer-Rediker model [1], which describes the dc and ac current components that form in the illumination entrance region of a PN photodiode with modulated light power of the form ${I_{in}} = {I_0}(1 + m{e^{i\Omega t}})$ where *m* is the modulation index. The subscript *E* will be used to denote parameters in the entrance region. We start with Eq. (14) of Ref. [1], which describes the ac current density component ${j_{ac,E}}$ as a solution of the diffusion equation for the entrance-region carrier density ${p_E}(x,t)$ with the specified boundary conditions: $d{p_E}/dx = 0$ at the entrance $x = 0$ and ${p_E}({W_E}) = 0$ at the PN-junction interface $x = {W_E}$, assuming negligible surface recombination (s=0):

Throughout this work, we will assume the entrance and substrate regions to be n-type and p-type respectively, however the main conclusions of this work are general in nature and not dependent upon this particular design choice. We recast the equation for ${j_{ac,E}}$ using three unitless parameters. The power absorption parameter ${P_E} \equiv \alpha {W_E}$ is the ratio of the entrance width ${W_E}$ to the wavelength-dependent penetration depth ${\alpha ^{ - 1}}$, so that smaller ${P_E}$ means less power absorption in this region. A second parameter is the modulation parameter ${M_E} \equiv \Omega {W_E}^2/{D_E}$, equal to the modulation phase accrued by a modulated migrating carrier density ‘wave’ that traverses the entire width ${W_E}$ (note that ${W_E}^2/{D_E}$ is the average diffusion time to traverse the entire entrance region). A third parameter is $\widehat {{W_E}} \equiv {W_E}/{L_E}$. With these substitutions, the ac current is

where*increasing*phase delay). They are dependent upon optical wavelength (through the absorption coefficient in ${P_E}$), modulation frequency (through ${M_E}$), and the other material parameters: ${W_E},\,{D_E}$ and ${L_E}$.

We now turn to the ac current that forms in the p-type substrate region. We denote all substrate parameters with the subscript *S*. The photocurrent in this region is due to minority carrier density ${p_S}(x,t)$ that forms in this region. Following the same approach as for the entrance region, with the boundary conditions $d{p_S}/dx = 0$ at $x = {W_E} + {W_S}$ (neglecting surface recombination as well the thin depletion region) and ${p_S}(x = {W_E}) = 0$ at the PN junction, the substrate ac current density is

${P_S} \equiv \alpha {W_S},\,{M_S} \equiv \Omega {W_S}^2/{D_S},\,\widehat {{W_S}} \equiv {W_S}/{L_S},\,{L_S} = {({D_S}{\tau _S})^{1/2}}$ with ${D_S}$ the substrate-region diffusion coefficient, ${\tau _S}$ is the minority carrier lifetime in the substrate region, and ${u_S} \equiv {({{{\widehat {{W_S}}}^2} + i{M_S}} )^{1/2}}\,.\,{F_S}({P_S},{M_S},\widehat {{W_S}}) = |{{F_S}} |{e^{ - i{\theta _S}}}$ describes the ac amplitude and phase response of the substrate region.

#### 2.2 PN photodiodes – total OED

The total ac current ${j_{ac,tot}}$ is the sum of the contributions from the two regions:

*q*, the OED will be influenced by both regions. These important features will be further elucidated in the following section. With these definitions, ${F_{tot}} = |{{F_S}} |{e^{ - i{\theta _S}}} + |{{F_E}} |{e^{ - i{\theta _E}}} = |{{F_{tot}}} |{e^{ - i{\theta _{tot}}}}$ describes the total ac amplitude and phase-shift.

With a focus on the OED sensitivity, and realizing that the wavelength dependence in Eq. (3) is only through the *P* parameters, we can express the OED sensitivity as follows:

*q*and ${W_{tot}}$), diffusion coefficients, and the modulation frequency (which together with the diffusion coefficients ${D_E}$ and ${D_S}$ determines ${M_E}$ and ${M_S}$). This evaluation process is described below for germanium and silicon photodiodes.

#### 2.3 Predictions of the model

We first discuss the implications of the second term. Figure 2 plots ${\alpha ^{ - 1}} \cdot d\alpha /d\lambda $ vs. $\lambda $ for some common semiconductors [19] showing prominent peak (negative) values in the respective band-edge regions of the material. Propitiously, for germanium, large OED is possible in the C and L-bands. Silicon stands out as having potential broadband OED sensitivity in the near IR. This is due to the fact that in this region, silicon possesses a significantly lower absorption coefficient than other common semiconductors. At higher wavelengths, InGaAs will show significant OED in the 1650 nm regime. Finally, both Si and GaAs show potentially large OED in various portions of the 350-700 nm range. However, this absorption term is only one part of the picture. In order to realize the full potential for large OED, the first term describes how to design the photodiode accordingly.

We now turn to an evaluation of the first term, ${P_{tot}} \cdot d{\theta _{tot}}/d{P_{tot}}$. Its dependence on ${P_{tot}}$ is shown in Fig. 3(a), left y-axis (note the inverted sign on this axis), for various photodiode designs as quantified by *q*. We assumed the following nominal values: ${W_{tot}} = 100\mathrm{\mu}\textrm{m}$, ${D_E} = 25\textrm{c}{\textrm{m}^\textrm{2}}\textrm{/s}$, ${D_S} = 50\textrm{c}{\textrm{m}^\textrm{2}}\textrm{/s,}\,\widehat {{W_{tot}}} = 1$ and $f = 0.5\textrm{MHz}$. We assumed diffusion coefficient values that are characteristic of doped Ge. However, as we explain in pt. 5 below, it is the values of the modulation parameters ${M_E}$ and ${M_S}$ which determine the OED characteristics, as they lump together all of the other critical parameters. This universality in the OED characteristics simplifies the analysis. The right y-axis displays the OED sensitivity vs. ${P_{tot}}$ for a specific case of a germanium photodiode at $\lambda = 1560\,\textrm{nm}$. As described in Eq. (4), these values are obtained from the product $[{{P_{tot}} \cdot d{\theta_{tot}}/d{P_{tot}}} ][{{\alpha^{ - 1}} \cdot d\alpha /d\lambda } ]$. Figure 3(b) displays the ac amplitude vs. ${P_{tot}}$ for the same parameter set. The following 7 points summarize the salient features of these graphs.

- 1. For $q \ge 0.5$, most of the device consists of an entrance region, so that it is essentially entrance-dominant throughout most of the usable spectrum. In this region, ${P_{tot}} \cdot d{\theta _{tot}}/d{P_{tot}}$ is positive and ${S_{OED}}$ is negative for all values of ${P_{tot}}$. The reason is as follows: for a given device width ${W_{tot}}$, decreasing ${P_{tot}}$ means decreasing $\alpha $, i.e. increasing $\lambda $. As $\lambda $ increases, the light penetration depth increases and approaches the PN junction from the left. Therefore, the average minority carrier migration distance decreases, and so does the phase delay, rendering $d\theta /d\lambda $ negative. Therefore, for an entrance-region-dominant operation, ${S_{OED}} = d{\theta _{tot}}/d\lambda $ is always negative, and $d{\theta _{tot}}/d{P_{tot}}$ is always positive. In addition, the model predicts that the magnitude of the OED sensitivity will peak within the region $2 < {P_{tot}} < 5$, and will increase for increasing
*q*. - 2. As
*q*decreases below 0.5, most of the device consists of the substrate layer. Therefore, for decreasing*q*, there is an increasing tendency for the light to penetrate into the substrate region, becoming the dominant region for $q < < 1$. As a result, a prominent feature appears: a sign flip from positive to negative values for ${P_{tot}} \cdot d{\theta _{tot}}/d{P_{tot}}$ and negative to positive values for ${S_{OED}}$. This is due to a basic asymmetry of the optoelectronic interaction in the two regions. As opposed to the previous case, here the migration distance*increases*with increasing $\lambda $, so that $d\theta /d\lambda > 0$ . This effect first appears for low values of ${P_{tot}}$, i.e. low values of absorption, and broadens to a wider band of ${P_{tot}}$ values as*q*continues to decrease. As is the case for the entrance-dominant device, the peak values of $|{{S_{OED}}} |$ for the substrate-dominant device also appear in the range of $2 < {P_{tot}} < 5$, but are relatively lower than those of the entrance-dominated device. - 3. The OED can be zeroed-out at the particular value of ${P_{tot}}$ where the ${S_{OED}}$ crosses over from positive to negative values. In Fig. 3(a), with the parameters taken in this simulation, and with $q = 0.3$, the OED is predicted to be zero for ${P_{tot}} \simeq 4$.
- 4. The ${S_{OED}}$ values for the specific case of germanium at 1560 nm, shown on the right y-axis, are a product of ${P_{tot}} \cdot d{\theta _{tot}}/d{P_{tot}}$ on the left y-axis, and the value of ${\alpha ^{ - 1}}{({d\alpha /d\lambda } )_{Ge,1560nm}} ={-} 0.066\,\textrm{n}{\textrm{m}^{ - 1}}$, taken from Fig. 2. The model predicts, for example, that for a photodiode with $q = 0.01$, the maximum OED sensitivity ${S_{OED,Ge}} = [{{P_{tot}} \cdot d{\theta_{tot}}/d{P_{tot}}} ][{{\alpha^{ - 1}} \cdot d\alpha /d\lambda } ]= ({ - 12.34\deg } )( - 0.066\textrm{n}{\textrm{m}^{ - 1}}) \simeq 0.8\textrm{deg/nm}$ at ${P_{tot}} \simeq 3.2$. Therefore, with ${\alpha _{Ge}}({1560\textrm{nm}} )= 1286\textrm{c}{\textrm{m}^{\textrm{ - 1}}}$, the optimum device width to achieve maximum OED would be ${W_{tot}} = {P_{tot}}/\alpha \simeq 249\mathrm{\mu}\textrm{m}$, and with $q = 0.01$ the entrance width would be about $2.5\mathrm{\mu}\textrm{m}$. However, we also note that the maximum OED is not strongly dependent on ${P_{tot}}$; the graphs show a wide tolerance for variations of at least 50% around the optimum ${P_{tot}}$.
- 5. The choice of modulation frequency, or more generally the values of the modulation parameters
*M*, will dictate the characteristics of the OED. In general, the model predicts that the best overall performance in terms of a relatively strong amplitude response, shown in Fig. 3(b), together with a maximum OED (assuming, of course, that it is desired to maximize the OED) would be in the region of $\widehat M \sim \pi $,where we define $\widehat M \equiv \Omega W_{tot}^2/{D_{avg}}$ with ${D_{avg}} \equiv ({D_E} + {D_S})/2$. The frequency at which $\widehat M = \pi $ can be considered to be a cut-off frequency. For $\widehat M < \pi $, the frequency is below cut-off, the amplitude response is maximum, ${\widehat D_{OED}}$ is constant and ${S_{OED}}$ increases proportionally with increasing $\widehat M$. For $\widehat M > \pi $, the ac amplitude decreases with increasing $\widehat M$, and ${S_{OED}}$ may continue to increase (for entrance dominant operation) or reach a plateau (for substrate-dominant operation), depending on*q*and ${P_{tot}}$. The operating parameters used in generating the graphs in Figs. 3(a) and 3(b) reflect the response for the parameter set that gave $\widehat M = 2.1$. A universality emerges: for any given set of values for $\widehat M$ and*q*, it can be stated that all PN photodiodes, regardless of the type of semiconductor material, will have very similar (albeit not identical) graphs of ${P_{tot}} \cdot d{\theta _{tot}}/d{P_{tot}}$ and amplitude response vs. ${P_{tot}}$. - 6. Silicon PN detectors possess significant OED in the near IR. For example, at $\lambda = 900\textrm{nm,}\,{\alpha _{Si}}(900\textrm{nm}) \simeq 295\textrm{c}{\textrm{m}^{\textrm{ - 1}}}$, and from Fig. 2, ${\alpha ^{ - 1}}{({d\alpha /d\lambda } )_{Si,980\textrm{nm}}} \simeq{-} 0.012\textrm{n}{\textrm{m}^{\textrm{ - 1}}}$. Now in order to compute the OED value, we refer back to Fig. 3(a). These graphs would be relevant for silicon as well, assuming the same operating condition $\widehat M = 2.1$ (see previous point). To achieve maximum OED, the device would be designed to have ${P_{tot}} \simeq 3$ so that ${W_{tot}} \simeq 3/295\textrm{c}{\textrm{m}^{\textrm{ - 1}}}\textrm{ = 102}\mathrm{\mu}\textrm{m}$. Assuming $q = 0.01$, this leads to ${S_{OED,Si}} = P(d\theta /dP){\alpha ^{ - 1}}({d\alpha /d\lambda } )= ({ - 12.34\deg } )( - 0.012\textrm{n}{\textrm{m}^{\textrm{ - 1}}}) \simeq 0.15\textrm{deg/nm}$. Due to the lower values of the diffusion coefficients for silicon, this OED would be achieved with proportionally lower modulation frequencies, on the order of $100\,\textrm{kHz}$. If the geometry is flipped to form an entrance-dominated device with $q \sim 1$, the sensitivity will be ${S_{OED,Si}} = P(d\theta /dP){\alpha ^{ - 1}}({d\alpha /d\lambda } )\simeq \,({48.5\deg } )( - 0.012\textrm{n}{\textrm{m}^{\textrm{ - 1}}}) \simeq{-} 0.6\textrm{deg/nm}$ i.e. opposite in sign and a factor of 4 larger in magnitude. These predictions also point to silicon-based photovoltaic cells as an inexpensive and sensitive device for OED.
- 7. In order to achieve a combination of high OED and strong amplitude response, an entrance-dominant device would be preferable - the model predicts an OED that is over a factor of 3 larger than the substrate-dominant device (see points 2, 5 and 6 above). However, many commercial PN detectors are fabricated with a thin entrance region, i.e. $q < < 1$, so that their OED characteristics tend to be substrate-dominant over most of the spectral range of interest. Furthermore, in order to achieve zero OED, the device should be in the $q \sim 0.3$ region. Therefore, some intriguing features of OED may require photodiodes that are presently not commercially available, although straightforward to manufacture.

Now we turn to Fig. 4 which displays graphs of the predicted ${S_{OED}}$ vs. wavelength for germanium and silicon photodiodes, for the same values of *q* used in Fig. 3. For the germanium photodiode, we assumed ${W_{tot}} = 100\mathrm{\mu}\textrm{m,}\,{D_E} = 25\textrm{c}{\textrm{m}^\textrm{2}}\textrm{/s,}\,{D_S} = 50\textrm{c}{\textrm{m}^\textrm{2}}\textrm{/s,}\,\textrm{and}\,f = 0.5\textrm{MHz}$. For the silicon photodiode, the values were ${W_{tot}} = 100\mathrm{\mu}\textrm{m,}\,{D_E} = 5\textrm{c}{\textrm{m}^\textrm{2}}\textrm{/s,}\,{D_S} = 10\textrm{c}{\textrm{m}^\textrm{2}}\textrm{/s,}\,\textrm{and}\,f = 100\textrm{kHz}$. We summarize the main features of these graphs as follows:

- 1. Silicon shows significant OED in the $0.8\mathrm{\mu}\textrm{m} - 1\mathrm{\mu}\textrm{m}$ region. For large
*q*, i.e. entrance-dominant operation, the OED peaks at approx. $- 0.56\textrm{deg/nm}$ at $\lambda \simeq 0.89\mathrm{\mu}\textrm{m}$. For small*q*, the device is substrate-dominant, and the OED peaks at $\lambda \simeq 0.93\mathrm{\mu}\textrm{m}$ with a value of approx. $0.14\textrm{deg/nm}$. For $q = 0.3$, the OED switches from negative to positive values around the zero-dispersion wavelength ${\lambda _{ZD}} \simeq 0.87\mathrm{\mu}\textrm{m}$. - 2. In the case of germanium, there is significant OED in the C and L-bands. For large
*q*, the entrance-dominant OED peaks at about $- 2.8\textrm{deg/nm}$ at $\lambda \simeq 1.58\mathrm{\mu}\textrm{m}$. At*q*=0.01, the substrate-dominant OED is $0.75\textrm{deg/nm}$ at the same wavelength. A zero-dispersion wavelength ${\lambda _{ZD}} \simeq 1.577\mathrm{\mu}\textrm{m}$ exists for $q = 0.3$.

## 3. Experimental results

We measured the OED in Ge PN-type and InGaAs PIN-type commercial photodiodes with the set-up of Fig. 5. Light from a tunable laser was modulated at $500\,\textrm{kHz}$ and was directed through an optical fiber before reaching the detector. Using the phase-shift technique, we measured the RF phase-shift as the wavelength was varied in the C and L-bands between 1530-1610 nm. Since each of the elements in the system can in principle be the source of a wavelength-dependent phase-shift due to either chromatic dispersion (in the fiber components) or OED (in the detector), the experiments were designed to calibrate the system and eliminate ambiguity. In the first experiment, the optical fiber was a 1meter SMF-28 fiber jumper and the detector was an unbiased PIN-type InGaAs detector (Thorlabs DET08CFC/M). In the second experiment, the fiber was replaced with a dispersion compensating fiber module (Cisco ONS 15216, DCU 1150) with dispersion $DL ={-} 1150\textrm{ps/nm}$, with the same detector in place. Finally, in the third experiment, we reverted back to the short fiber jumper, and the detector was replaced with an unbiased Ge PN-type photodiode (GPD Optoelectronics, GM3). At 500 kHz*,* both the short fiber as well as the Mach-Zehnder modulator can be regarded as giving essentially zero wavelength-dependent phase shift. This was verified experimentally.

Figure 6 shows the experimental results and theoretical predictions. The InGaAs PIN detector exhibits a relatively small ${S_{OED,InGaAs}} \simeq 0.027\deg /\textrm{nm}$, as expected for a substrate-dominant device with illumination wavelengths that do not match the band-gap region ($\ge 1.62\mathrm{\mu}\textrm{m}$ for InGaAs). Although this is not a PN device, we can get a rough estimate of the predicted ${S_{OED}}$ value based on the PN model. From Fig. 2, ${\alpha ^{ - 1}}{({d\alpha /d\lambda } )_{InGaAs,1560\textrm{nm}}} \simeq{-} 0.004\textrm{n}{\textrm{m}^{\textrm{ - 1}}}$, which is a factor of 17 less than that for Ge at the same wavelength. Therefore, we expect this device to exhibit an OED reduced by at least this amount as compared to Ge. This will be verified below. After replacing the fiber jumper with the DCF module (while the detector is unchanged), we measured a slope of $- 0.21\textrm{deg/nm}$, in excellent agreement with the manufacturer’s specifications of $DL ={-} 1150\textrm{ps/nm}$ which leads to the same value $d\theta /d\lambda = 2\pi fDL = 0.21\textrm{deg/nm}$ at 500 kHz. After establishing the validity of the measurement system, we reinserted the short fiber and measured the OED sensitivity of the PN Ge photodiode. The experimental data displays significant OED in the C and L-bands as predicted, with an ${S_{OED}}$ value of approx. $0.64\textrm{deg/nm}$ at $\lambda = 1556\textrm{nm}$, or a factor of 3 greater in magnitude than the DCF module and with opposite sign (the DCF module compensates for about 70 km of SMF28 fiber). This corresponds to ${\widehat D_{OED}} = $ 3460 ps/nm, equivalent to the dispersion in 204 km of SMF28 fiber at 500 kHz. The theoretical prediction is shown as well, using published data for the Ge absorption spectrum [19] and assuming the following parameters: ${W_{tot}} = 70\mathrm{\mu}\textrm{m,}\,\,q = 0.01,\,{D_S} = 40\textrm{c}{\textrm{m}^\textrm{2}}\textrm{/s},\,\,{D_E} = 20\textrm{c}{\textrm{m}^\textrm{2}}\textrm{/s}$ (we measured the total width of the photodiode using a SEM). The overall shape and maximum slope of the predicted response (${\simeq} 0.6\textrm{deg/nm}$ at 1570nm) agrees well with the experiment, however there is a spectral shift of 5-10 nm between the predictions and experiment. We attribute this offset to the following possibilities: 1) a discrepancy between the tabulated absorption spectra for intrinsic Ge and actual absorption values for this photodiode, due to doping and/or temperature dependence, and 2) inaccurate values for the diffusion coefficients (which are also dependent upon the doping). Finally, as we discussed above, a comparison of the measured OED for Ge and InGaAs gives a factor of approx. 18 between them in the C-band, in excellent agreement with theory. Although the InGaAs OED is significantly smaller than that of Ge in this spectral region, we point out that it is still sizable – equivalent to approx. 12 km of optical fiber. Furthermore, the data in Fig. 2 indicates that in its band-edge region $\ge 1.62\mathrm{\mu}\textrm{m}$, a PN InGaAs device will display a maximum ${S_{OED}}$ that is about 20% of the value for Ge in the C-band, i.e. approx. 0.12deg/nm.

We measured the 3-sigma phase noise that accompanied the signal. With an electronic bandwidth $B = 500Hz$, the phase noise was $5x{10^{ - 3}}\deg $ for all the experiments and was proportional to ${B^{1/2}}$. Several possible noise sources include: the statistical behavior of the light absorption profile and charge transport mechanisms (especially diffusion), dark current, shot and thermal noise, and instrument noise. With the measured ${S_{OED}} \approx 0.64\,\,\deg /\textrm{nm}$ for Ge, this noise floor points to a spectral resolution (at SNR=1) of approx. $8\,\textrm{pm}$ for applications in optical spectroscopy.

In a further experiment, we demonstrated temperature-tuning of the OED in the Ge photodiode, shown in Fig. 7. For this experiment, the photodiode was placed in an oven to control the ambient temperature. The high OED region, which follows the absorption-edge region, shifts to higher wavelengths with increasing temperature as expected. The amount of the measured shift $\Delta \lambda /\Delta T$, on the order of $1\,\textrm{nm}/{\,^0}C$, agrees with published data for Ge [20–22]. This also indicates that for high-resolution spectroscopy applications, it would be necessary to temperature-stabilize the photodiode.

## 4. Discussion and conclusions

The main focus of this work was on developing the model and experimental verification of OED in PN-type photodiodes. We have also developed the theory of OED in PIN-type photodiodes, which is a straightforward extension of the PN model to include an additional intrinsic layer as well as the drift current in the depletion region under reverse-bias [23,24]. Due to the relatively high drift velocity, the additional OED in the drift-dominant region may be negligible as compared to the diffusion-dominant regions. Furthermore, if the intrinsic region width is relatively wide, the overall OED of the device may be smaller than that of the PN device, depending on the wavelength, width of the entrance region and amount of bias applied. However, the PIN-type device allows for the important feature of electronically tuning the OED through bias control. It is well-known that the depletion width as well as the drift velocity increase with bias voltage. The theory of OED in the PIN structure predicts that the OED value as well as its sign is controllable through the applied bias. This is easy to understand based on our model of the PN-type OED. For example, increasing the reverse-bias voltage leads to an increased and non-symmetric encroachment of the depletion region on the p and n regions, reducing the effective widths of the regions in which the diffusion mechanism dominates. Depending on the relative change in these widths, it is clear that the OED value will change, and can even be switched from negative to positive due to a transformation between two states: from entrance-dominant to substrate-dominant. We have experimentally verified these predictions of voltage bias-controlled OED, using a silicon PIN detector in the 800 nm region [25].

Regarding the OED in other types of structures, such as the avalanche photodiode (APD) or separated-absorption-charge-multiplication APD, our model predicts that the significant source of OED is the relatively slow diffusion current. Therefore, other types of structures can also be a source of high OED, as long as the diffusion mechanism is not negligible.

This work has not addressed the potential use of OED for dispersion compensation in high bit-rate fiber-optic links. Our analysis and experiments described the photodiode OED as a response to sinusoidal modulation, and will be constant only up to the modulation cut-off frequency, as dictated by the modulation parameters M. Further work will elucidate the relevance of OED for this spectrally-broadband RF application. Obviously, a negative OED will be needed to compensate for the positive dispersion in standard optical fiber, which will require an entrance-dominant device in the wavelength range of interest.

For applications in optical spectroscopy, we have demonstrated picometer-level sensitivity for wavelength monitoring in the C-band, due to the extremely large OED in germanium. This would be of interest, for example, in FBG-based sensors [13] and other techniques in optical spectroscopy [12]. Finally, we mention the availability of two-band photodiodes based on two semiconductor materials as another intriguing OED device [26].

Finally, we wish to point out that photodiode OED may be of fundamental as well as practical importance in the field of optoelectronic oscillators (OEO) [17,27]. The OEO is used to generate high spectral purity microwave photonic signals, based on a high Q-factor optical storage element or resonator and a hybrid positive optoelectronic feedback loop. Traditionally, the photodiode has only played the role of optical-to-electronic signal transformation in this loop. However, by regarding the photodiode as a potential source of large tunable dispersion, this can be utilized for tunable RF phase-delay. Obviously, this will require the optimum design of photodiodes for this application, which usually operates in the GHz frequency range.

In conclusion, we have proposed to regard the photodiode as a device for optoelectronic chromatic dispersion. A model for OED in PN-type photodiodes predicts that due to the wavelength-dependent diffusion migration times, these simple devices can exhibit a wide range of positive or negative OED values, depending on the specific material, structure, wavelength and modulation frequency. We show how to design desired OED features through the evaluation of the product of the spectral absorption term and the design term. Silicon and germanium photodiodes stand out as possessing high OED in the near IR.

Surprisingly, the OED can be extremely large. Among the common semiconductor materials, it appears that germanium possesses the highest values of OED, due to a very high ${\alpha ^{ - 1}}d\alpha /d\lambda $ in the C and L-band regions. At 500 kHz modulation, a commercial detector showed an OED sensitivity of 0.64 deg/nm with the phase-shift technique, or an OED parameter of approx. 3500 ps/nm, equivalent to about 204 km of SMF28 optical fiber. After accounting for noise, this resulted in a wavelength-shift resolution of 8 pm. In an entrance-dominant device, the OED can be further increased by a factor of at least 3. It is also possible to zero-out the OED at certain wavelengths.

The OED can be tuned through control of the photodiode bandgap. We demonstrated this using temperature control. A further possibility would be the application of mechanical strain. This also suggests applications in temperature and strain sensing. Another important feature is the ability to tune the OED electronically through voltage-control.

Chromatic dispersion has been widely utilized in the past for optical sensing applications. By viewing the optoelectronic process in a photodiode as a new tunable source of chromatic dispersion, new applications emerge in RF photonics, spectroscopy, optical sensing, as well as a new tool for investigating the physics of the optoelectronic process.

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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