Optical differential low-noise receivers and related methods

Abstract

Low-noise optical differential receivers are described. Such differential receivers may include a differential amplifier having first and second inputs and first and second outputs, and four photodetectors. A first and a second of such photodetectors are coupled to the first input of the differential amplifier, and a third and a fourth of such photodetectors are coupled to the second input of the differential amplifier. The anode of the first photodetector and the cathode of the second photodetector are coupled to the first input of the differential amplifier. The cathode of the third photodetector and the anode of the fourth photodetector are coupled to the second input of the differential amplifier. The optical receiver may involve two stages of signal subtraction, which may significantly increase noise immunity.

Claims

1. An optical receiver comprising: a differential amplifier having first and second inputs and first and second outputs; a first photodetector configured to produce a first photocurrent and a second photodetector configured to produce a second photocurrent, the first and second photodetectors being coupled to the first input of the differential amplifier, wherein an anode of the first photodetector is coupled to a cathode of the second photodetector thereby producing a first differential current based on a difference between the first photocurrent and the second photocurrent; a third photodetector configured to produce a third photocurrent and a fourth photodetector configured to produce a fourth photocurrent, the third and fourth photodetectors being coupled to the second input of the differential amplifier, wherein a cathode of the third photodetector is coupled to an anode of the fourth photodetector thereby producing a second differential current based on a difference between the fourth photocurrent and the third photocurrent; and a photonic circuit configured to provide: a first optical signal to the first and third photodetectors, and a second optical signal to the second and fourth photodetectors.

2. The optical receiver of claim 1, wherein the anode of the first photodetector and the cathode of the second photodetector are coupled to the first input of the differential amplifier.

3. The optical receiver of claim 1, wherein the cathode of the third photodetector and the anode of the fourth photodetector are coupled to the second input of the differential amplifier.

4. The optical receiver of claim 1, wherein the first, second, third and fourth photodetectors are formed monolithically on a common substrate.

5. The optical receiver of claim 4, wherein the common substrate comprises a silicon substrate.

6. The optical receiver of claim 4, wherein the first, second, third and fourth photodetectors are disposed within an area of 0.1 mm.sup.2 on the substrate.

7. The optical receiver of claim 1, wherein the first, second, third and fourth photodetectors have equal responsivities.

8. The optical receiver of claim 1, wherein the first, second, third and fourth photodetectors are photodiodes.

9. The optical receiver of claim 1, further comprising an analog-to-digital converter coupled to the first and second outputs of the differential amplifier.

10. The optical receiver of claim 1, wherein the photonic circuit is configured to generate the first and second optical signals by combining a modulated optical signal with a reference optical signal.

11. A method for receiving an input signal, the method comprising: combining the input signal with a reference signal to obtain first and second optical signals; producing first and second photocurrents by detecting the first optical signal with a first photodetector and with a second photodetector, respectively, and producing third and fourth photocurrents by detecting the second optical signal with a third photodetector and with a fourth photodetector, respectively; and producing a pair of amplified differential voltages using first and second differential currents, the first differential current obtained based on a difference between the first photocurrent and the second photocurrent, and the second differential current obtained based on a difference between the fourth photocurrent and the third photocurrent.

12. The method of claim 11, wherein combining the input signal with the reference signal comprises combining the input signal with the reference signal with a directional coupler.

13. The method of claim 11, wherein producing the pair of amplified differential voltages comprises producing the pair of amplified differential voltages using the first and second differential currents with a differential operational amplifier.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) Various aspects and embodiments of the application will be described with reference to the following figures. It should be appreciated that the figures are not necessarily drawn to scale. Items appearing in multiple figures are indicated by the same reference number in all the figures in which they appear.

(2) FIG. 1 is a circuit diagram illustrating an example of a differential optical receiver, in accordance with some non-limiting embodiments.

(3) FIG. 2 is a schematic diagram illustrating a photonic circuit that may be coupled with the differential optical receiver of FIG. 1, in accordance with some non-limiting embodiments.

(4) FIG. 3A is a schematic diagram illustrating a substrate including a photonic circuit, photodetectors and a differential operational amplifier, in accordance with some non-limiting embodiments.

(5) FIG. 3B is a schematic diagram illustrating a first substrate including a photonic circuit and photodetectors, and a second substrate including a differential operational amplifier, where the first and second substrates are flip-chip bonded to each other, in accordance with some non-limiting embodiments.

(6) FIG. 3C is a schematic diagram illustrating a first substrate including a photonic circuit and photodetectors, and a second substrate including a differential operational amplifier, where the first and second substrates are wire bonded to each other, in accordance with some non-limiting embodiments.

(7) FIG. 4A is a flowchart illustrating an example of a method for fabricating an optical receiver, in accordance with some non-limiting embodiments.

(8) FIGS. 4B-4G illustrate an example of a fabrication sequence for an optical receiver, in accordance with some non-limiting embodiments.

(9) FIG. 5 is a flowchart illustrating an example of a method for receiving an optical signal, in accordance with some non-limiting embodiments.

DETAILED DESCRIPTION

(10) The inventors have recognized and appreciated that some conventional optical receivers are particularly susceptible to noise generated from voltage supplies, to noise arising from the fact that photodetectors inevitably produce dark currents, and to other forms of noise. The presence of noise reduces the signal-to-noise ratio, and therefore, the ability of these photodetectors to accurately sense incoming optical signals. This can negatively affect the performance of the system in which these photodetectors are deployed. For example, this can negatively affect the system's bit error rate and power budget.

(11) The inventors have developed optical receivers with reduced susceptibility to noise. Some embodiments of the present application are directed to optical receivers in which both the optical-to-electric conversion and the amplification are performed in a differential fashion. In the optical receivers described herein, two separate signal subtractions take place. First, the photocurrents are subtracted from one another to produce a pair of differential currents. Then, the resulting differential currents are further subtracted from each other to produce an amplified differential output. The inventors have recognized and appreciated that having an optical receiver involving multiple levels of signal subtraction results in multiple levels of noise cancellation, thus substantially reducing noise from the system. This can have several advantages over conventional optical receivers, including wider dynamic range, greater signal-to-noise ratio, larger output swing, and increased supply-noise immunity.

(12) Optical receivers of the types described herein can be used in a variety of settings, including for example in telecom and datacom (including local area networks, metropolitan area networks, wide area networks, data center networks, satellite networks, etc.), analog applications such as radio-over-fiber, all-optical switching, Lidar, phased arrays, coherent imaging, machine learning and other types of artificial intelligence applications, as well as other applications. In some embodiments, optical receivers of the types described herein may be used as part of a photonic processing system.

(13) FIG. 1 illustrates a non-limiting example of an optical receiver 100, in accordance with some non-limiting embodiments of the present application. As shown, optical receiver 100 includes photodetectors 102, 104, 106 and 108, though other implementations include more than four photodetectors. Photodetector 102 may be connected to photodetector 104, and photodetector 106 may be connected to photodetector 108. In some embodiments, the anode of photodetector 102 is connected to the cathode of photodetector 104 (at node 103), and the cathode of photodetector 106 is connected to the anode of photodetector 108 (at node 105). In the example of FIG. 1, the cathodes of photodetectors 102 and 108 are connected to voltage supply V.sub.DD and the anodes of photodetectors 104 and 106 are connected to the reference potential (e.g., to ground). The opposite arrangement is also possible in some embodiments. The reference potential may be at a potential equal to zero or having any suitable value, such as—V.sub.DD. V.sub.DD may have any suitable value.

(14) Photodetectors 102-108 may be implemented in any of numerous ways, including for example with pn-junction photodiodes, pin-junction photodiodes, avalanche photodiodes, phototransistors, photoresistors, etc. The photodetectors may include a material capable of absorbing light at the wavelength of interest. For example, at wavelengths in the O-band, C-band or L-band, the photodetectors may have an absorption region made at least in part of germanium, by way of a non-limiting example. For visible light, the photodetectors may have an absorption region made at least in part of silicon, by way of another non-limiting example.

(15) Photodetectors 102-108 may be integrated components formed monolithically as part of the same substrate. The substrate may be a silicon substrate in some embodiments, such as a bulk silicon substrate or a silicon-on-insulator. Other types of substrates can also be used, including for example indium phosphide or any suitable semiconductor material. To reduce variability in the characteristics of the photodetectors due to fabrication tolerances, in some embodiments, the photodetectors may be positioned in close proximity to one another. For example, the photodetectors may be positioned on a substrate within an area of 1 mm.sup.2 or less, 0.1 mm.sup.2 less or 0.01 mm.sup.2 or less.

(16) As further illustrated in FIG. 1, photodetectors 102-108 are connected to a differential operational amplifier 110. For example, photodetectors 102 and 104 may be connected to the non-inverting input (“+”) of DOA 110 and photodetectors 106 and 108 may be connected to the inverting input (“−”) of DOA 110. DOA 110 has a pair of outputs. One output is inverting and one output is non-inverting.

(17) In some embodiments, as will be described in detail in connection with FIG. 2, photodetectors 102 and 106 may be arranged to receive the same optical signal “t” and photodetectors 104 and 108 may be arranged to receive the same optical signal “b.” In some embodiments, photodetectors 102-108 may be designed to be substantially equal to each other. For example, photodetectors 102-108 may be formed using the same process steps and using the same photomask patterns. In these embodiments, photodetectors 102-108 may exhibit substantially the same characteristics, such as substantially the same responsivity (the ratio between the photocurrent and the received optical power) and/or substantially the same dark current (the current generated when no optical power is received). In these embodiments, the photocurrents generated by photodetectors 102 and 106 responsive to reception of signal t may be substantially equal to each other. Such photocurrents are identified as “i.sub.t” in FIG. 1. It should be noted that, due to the orientations of photodetectors 102 and 106, the photocurrents generated by photodetectors 102 and 106 are oriented in opposite directions. That is, the photocurrent of photodetector 102 is directed towards node 103 and the photocurrent of photodetector 106 is oriented away from node 105. Furthermore, the photocurrents generated by photodetectors 104 and 108 responsive to reception of signal b may be substantially equal to each other. Such photocurrents are identified as “i.sub.b.” Due to the orientations of photodetectors 104 and 108 relative to each other, the photocurrents generated by photodetectors 104 and 108 are oriented in opposite directions. That is, the photocurrent of photodetector 108 is directed towards node 105 and the photocurrent of photodetector 104 is oriented away from node 103.

(18) In view of the orientations of the photodetectors, a current with amplitude i.sub.t−i.sub.b emerges from node 103 and a current with amplitude i.sub.b−i.sub.t emerges from node 105. Thus, the currents have substantially the same amplitudes, but with opposite signs.

(19) Photodetectors 102-108 may produce dark currents. Dark currents are typically due to leakage and arise from a photodetector regardless of whether the photodetector is exposed to light or not. Because dark currents arise even in the absence of incoming optical signals, dark currents effectively contribute to noise in the optical receiver. The inventors have appreciated that the negative effects of these dark currents can be significantly attenuated thanks to the current subtraction described above. Thus, in the example of FIG. 1, the dark current of photodetector 102 and the dark current of photodiode 104 substantially cancel out one another (or at least are mutually substantially reduced), and so do the dark currents of photodetector 106 and 108. Consequently, noise due to the presence of the dark currents is greatly attenuated.

(20) FIG. 2 illustrates a photonic circuit 200 arranged for providing two optical signals to photodetectors 102-108, in accordance with some non-limiting embodiments. Photonic circuit 200 may comprises optical waveguides for routing the optical signals to the photodetectors. The optical waveguides may be made of a material that is transparent or at least partially transparent to light at the wavelength of interest. For example, the optical waveguides be made of silicon, silicon oxide, silicon nitride, indium phosphide, gallium arsenide, or any other suitable material. In the example of FIG. 2, photonic circuit 200 includes input optical waveguides 202 and 204 and couplers 212, 214 and 216. As further illustrated, the output optical waveguides of photonic circuit 200 are coupled to photodetectors 102-108.

(21) In the example of FIG. 2, couplers 212, 214 and 216 comprise directional couplers, where evanescent coupling enables transfer of optical power between adjacent waveguides. However, other types of couplers may be used such as Y-junctions, X-junctions, optical crossovers, counter-direction couplers, etc. In other embodiments, photonic circuit 200 may be implemented with a multi-mode interferometer (MMI). Couplers 212, 214 and 216 may be 3 dB couplers (with a 50%-50% coupling ratio) in some embodiments, though other ratios are also possible, such as 51%-49%, 55%-45% or 60%-40%. It should be appreciated that, due to fabrication tolerances, the actual coupling ratio may deviate slightly from the intended coupling ratio.

(22) Signal s.sub.1 may be provided at input optical waveguide 202 and signal s.sub.2 may be provided at input optical waveguide 204. Signals s.sub.1 and s.sub.2 may be provided to the respective input optical waveguides using for example optical fibers. In some embodiments, s.sub.1 represents a reference local oscillator signal, such as the signal generated by a reference laser, and s.sub.2 represents the signal to be detected. As such, the optical receiver may be viewed as a homodyne optical receiver. In some such embodiments, s.sub.1 may be a continuous wave (CW) optical signal while s.sub.2 may be modulated. In other embodiments, both signals are modulated or both signals are CW optical signals, as the application is not limited to any particular type of signal.

(23) In the example of FIG. 2, signal s.sub.1 has amplitude A.sub.LO and phase ϑ, and signal s.sub.2 has amplitude A.sub.s and phase φ. Coupler 212 combines signals s.sub.1 and s.sub.2 such that signals t and b emerge at respective outputs of coupler 212. In the embodiments in which coupler 212 is a 3 dB coupler, t and b may be given by the following expression:

(24) $\left(\frac{t}{b}\right)=\frac{1}{\sqrt{2}}\left(1ii1\right)\left(\frac{A_{\mathrm{LO}}{e}^{i\vartheta }}{A_s{e}^{i\phi }}\right)$
and the powers T and B (of t and b, respectively) may be given by the following expressions:
T=[A.sub.LO.sup.2+A.sub.S.sup.2+2A.sub.LOA.sub.S sin(ϑ−φ)]
B=[A.sub.LO.sup.2+A.sub.S.sup.2−2A.sub.LOA.sub.S sin(ϑ−φ)]

(25) Thus, in the embodiments in which couplers 214 and 216 are 3 dB couplers, photodetectors 102 and 106 may each receive a power given by T/2 and photodetectors 104 and 108 may each receive a power given by B/2.

(26) Referring back to FIG. 1, and assuming that the responsivities of photodetectors 102-108 are all equal to each other (though not all embodiments are limited in this respect), the currents emerging from node 103 and 105, respectively, may be given by the following expressions:
i.sub.t−i.sub.b=2A.sub.LOA.sub.S sin(ϑ−φ)
i.sub.b−i.sub.t=−2A.sub.LOA.sub.S sin(ϑ−φ)

(27) DOA 110 is arranged to amplify the differential signal received at the “+” and “−” inputs, and to produce an amplified differential output, represented in FIG. 1 by voltages V.sub.out,n and V.sub.out,p. In some embodiments, DOA 110, in combination with impedances z, may be viewed as a differential transimpedance amplifier, in that it produces a differential pair of voltage (V.sub.out,n, V.sub.out,p) based on a differential pair of current (i.sub.b−i.sub.t, i.sub.t−i.sub.b). In some embodiments, each of V.sub.out,n, V.sub.out,p may be proportional to the difference between current i.sub.t−i.sub.b and current i.sub.b−i.sub.t, thus giving rise to the following expressions:
V.sub.out,p=2z(i.sub.t−i.sub.b)
V.sub.out,n=2z(i.sub.b−i.sub.t)

(28) This differential pair of voltages may be provided as input to any suitable electronic circuit, including but not limited to an analog-to-digital converter (not shown in FIG. 1). It should be noted that optical receiver 100 provides two levels of noise rejection. The first level of noise rejection occurs thanks to the subtraction of the photocurrents, the second level of noise rejection occurs thanks to the subtraction taking place in the differential amplification stage. This results in a significant increase in noise rejection.

(29) In the example of FIG. 1, impedances z are shown as being equal to each other, however different impedances may be used in other embodiments. These impedances may include passive electric components, such as resistors, capacitors and inductors, and/or active electronic components, such as diode and transistors. The components constituting these impedances may be chosen to provide a desired gain and bandwidth, among other possible characteristics.

(30) As discussed above, optical receiver 100 may be integrated monolithically on a substrate. One such substrate is illustrated in FIG. 3A, in accordance with some non-limiting embodiments. In this example, photodetectors 102-108, photonic circuit 200 and DOA 110 are monolithically integrated as part of substrate 301. In other embodiments, photodetectors 102-108 and photonic circuit 200 may be integrated on substrate 301 and DOA 110 may be integrated on a separate substrate 302. In the example of FIG. 3B, substrates 301 and 302 are flip-chip bonded to one another. In the example of FIG. 3C, substrates 301 and 302 are wire bonded to one another. In yet another example (not illustrated), photodetectors 102-108 and photonic circuit 200 may be fabricated on separate substrates.

(31) Some embodiments of the present application are directed to methods for fabricating optical receivers. One such method is depicted in FIG. 4A, in accordance with some non-limiting embodiments. Method 400 begins at act 402, in which a plurality of photodetectors are fabricated on a first substrate.

(32) Once fabricated, the photodetectors may be connected together, for example in the arrangement shown in FIG. 1. In some embodiments, the photodetectors may be positioned on the first substrate within an area of 1 mm.sup.2 or less, 0.1 mm.sup.2 less or 0.01 mm.sup.2 or less. At act 404, a photonic circuit is fabricated on the first substrate. The photonic circuit may be arranged to provide a pair of optical signals to the photodetectors, for example in the manner shown in FIG. 2. At act 406, a differential operational amplifier may be fabricated on a second substrate. An example of a differential operational amplifier is DOA 110 of FIG. 1. At act 408, the first substrate may be bonded to the second substrate, for example via flip-chip bonding (as shown in FIG. 3A), wire bonding (as shown in FIG. 3B), or using any other suitable bonding technique. Once the substrates are bonded, the photodetectors of the first substrate may be electrically connected to the differential operational amplifier of the second substrate, for example in the manner shown in FIG. 1.

(33) Examples of fabrication processes are depicted schematically at FIGS. 4B through 4G, in accordance with some embodiments. FIG. 4B depicts a substrate 301 having a lower cladding 412 (e.g., an oxide layer such as a buried oxide layer or other types of dielectric materials) and a semiconductor layer 413 (e.g., a silicon layer or a silicon nitride layer, or other types of material layers). At FIG. 4C, semiconductor layer 413 is patterned, for example using a photolithographic exposure, to form regions 414. Regions 414 may be arranged to form optical waveguides in some embodiments. In some embodiments, the resulting pattern resembles photonic circuit 200 (FIG. 2), where waveguides 202 and 204, and couplers 212, 214 and 216 are embedded into one or more regions 414. At FIG. 4D, photodetectors 102, 104, 106 and 108 (and optionally, other photodetectors) are formed. In this example, an optical absorbing material 416 is deposited to be adjacent a region 414. The optical absorbing material 416 may be patterned to form the photodetectors. The material used for the optical absorbing material may depend on the wavelength to be detected. For example, germanium may be used for wavelengths in the L-Band, C-Band or O-Band. Silicon may be used for visible wavelengths. Of course, other materials are also possible. The optical absorbing material 416 may be positioned to be optically coupled to regions 414 in any suitable way, including but not limited to butt coupling, taper coupling and evanescent coupling.

(34) At FIG. 4E, DOA 110 is formed. In some embodiments, DOA 110 includes several transistors formed via ion implantation. FIG. 4E depicts implanted regions 418, which may form part of one or more transistors of DOA 110. While only one ion implantation is illustrated in FIG. 4E, in some embodiments, formation of DOA 110 may involve more than one ion implantations. Additionally, DOA 110 may be electrically connected to the photodetectors, for example via one or more conductive traces formed on substrate 301.

(35) The arrangement of FIG. 4E is such that photonic circuit 200, photodetectors 102-108 and DOA 110 are formed on a common substrate (as shown in FIG. 3A). Arrangements in which DOA 110 is formed on a separate substrate (as shown in FIG. 3B or FIG. 3C) are also possible. In one such example, DOA 110 is formed on a separate substrate 302, as shown in FIG. 4F, where implanted regions 428 are formed via one or more ion implantations.

(36) Subsequently, substrate 301 is bonded to substrate 302, and photodetectors 102-108 are connected to DOA 110. At FIG. 4G, conductive pads 431 are formed and placed in electrical communication with optical absorbing material 416, and conductive pads 432 are formed and placed in electrical communication with implanted regions 428. The conductive pads are electrically connected via wire bonding (as shown in FIG. 4G) or via flip-chip bonding.

(37) Some embodiments are directed to methods for receiving input optical signals. Some such embodiments may involve homodyne detection, though the application is not limited in this respect. Other embodiments may involve heterodyne detection. Yet other embodiments may involve direct detection. In some embodiments, reception of optical signals may involve optical receiver 100 (FIG. 1), though other types of receivers may be used.

(38) An example of a method for receiving an input optical signal is depicted in FIG. 5, in accordance with some embodiments. Method 500 begins at act 502, in which the input signal is combined with a reference signal to obtain first and second optical signals. The input signal may be encoded with data, for example in the form of amplitude modulation, pulse width modulation, phase or frequency modulation, among other types of modulation. In some of the embodiments involving homodyne detection, the reference signal may be a signal generated by a local oscillator (e.g., a laser). In other embodiments, the reference signal may also be encoded with data. In some embodiments, the input signal and the reference signal are combined using a photonic circuit 200 (FIG. 2), though other types of optical combiners may be used, including but not limited to MMIs, Y-junctions, X-junctions, optical crossovers, and counter-direction couplers. In the embodiments in which photonic circuit 200 is used, t and b may represent the signals obtained from the combination of the input signal with the reference signal.

(39) At act 504, the first optical signal is detected with a first photodetector and with a second photodetector and the second optical signal is detected with a third photodetector and with a fourth photodetector to produce a pair of differential currents. In some embodiments, act 504 may be performed using optical receiver 100 (FIG. 1). In some such embodiments, the first optical signal is detected with photodetectors 102 and 106, and the second optical signal is detected with photodetectors 104 and 108. The produced pair of differential currents is represented, collectively, by currents i.sub.b−i.sub.t and i.sub.t−i.sub.b. Being differential, in some embodiments, the currents of the pair may have substantially equal amplitudes, but with substantially opposite phases (e.g., with a π-phase difference).

(40) At act 506, a differential operational amplifier (e.g., DOA 110 of FIG. 1) produces a pair of amplified differential voltages using the pair of differential currents produced at act 504. In the embodiments that use DOA 110, the produced pair of differential voltages is represented by voltages V.sub.out,n and V.sub.out,p. Being differential, in some embodiments, the voltages of the pair may have substantially equal amplitudes, but with substantially opposite phases (e.g., with a π-phase difference).

(41) Method 500 may have one or more advantages over conventional methods for receiving optical signals, including for example wider dynamic range, greater signal-to-noise ratio, larger output swing, and increased supply-noise immunity.

(42) Having thus described several aspects and embodiments of the technology of this application, it is to be appreciated that various alterations, modifications, and improvements will readily occur to those of ordinary skill in the art. Such alterations, modifications, and improvements are intended to be within the spirit and scope of the technology described in the application. It is, therefore, to be understood that the foregoing embodiments are presented by way of example only and that, within the scope of the appended claims and equivalents thereto, inventive embodiments may be practiced otherwise than as specifically described. In addition, any combination of two or more features, systems, articles, materials, and/or methods described herein, if such features, systems, articles, materials, and/or methods are not mutually inconsistent, is included within the scope of the present disclosure.

(43) Also, as described, some aspects may be embodied as one or more methods. The acts performed as part of the method may be ordered in any suitable way. Accordingly, embodiments may be constructed in which acts are performed in an order different than illustrated, which may include performing some acts simultaneously, even though shown as sequential acts in illustrative embodiments.

(44) All definitions, as defined and used herein, should be understood to control over dictionary definitions, definitions in documents incorporated by reference, and/or ordinary meanings of the defined terms.

(45) The indefinite articles “a” and “an,” as used herein in the specification and in the claims, unless clearly indicated to the contrary, should be understood to mean “at least one.”

(46) The phrase “and/or,” as used herein in the specification and in the claims, should be understood to mean “either or both” of the elements so conjoined, i.e., elements that are conjunctively present in some cases and disjunctively present in other cases.

(47) As used herein in the specification and in the claims, the phrase “at least one,” in reference to a list of one or more elements, should be understood to mean at least one element selected from any one or more of the elements in the list of elements, but not necessarily including at least one of each and every element specifically listed within the list of elements and not excluding any combinations of elements in the list of elements. This definition also allows that elements may optionally be present other than the elements specifically identified within the list of elements to which the phrase “at least one” refers, whether related or unrelated to those elements specifically identified.

(48) The terms “approximately” and “about” may be used to mean within ±20% of a target value in some embodiments, within ±10% of a target value in some embodiments, within ±5% of a target value in some embodiments, and yet within ±2% of a target value in some embodiments. The terms “approximately” and “about” may include the target value.