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• > Vector Analysis
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Book contents

• Frontmatter
• Introduction
• 1 Summary of vector algebra
• Exercises A
• 2 The geometrical background to vector analysis
• 3 Metric properties of Euclidean space
• Exercises B
• 4 Scalar and vector fields
• Exercises C
• 5 Spatial integrals of fields
• 6 Further spatial integrals
• Exercises D
• 7 Differentiation of fields. Part 1: the gradient
• 8 Differentiation of fields. Part 2: the curl
• 9 Differentiation of fields. Part 3: the divergence
• 10 Generalisation of the three principal theorems and some remarks on notation
• Exercises E
• 11 Boundary behaviour of fields
• Exercises F
• 12 Differentiation and integration of products of fields
• 13 Second derivatives of vector fields; elements of potential theory
• Exercises G
• 14 Orthogonal curvilinear coordinates
• Exercises H
• 15 Time-dependent fields
• Exercises I
• Answers and comments

Published online by Cambridge University Press:  23 December 2009

If one wishes to express in precise and general terms any statement in physics in which positions, directions and motions in space are involved the most appropriate language to use is the language of vectors. In the mechanics of particles and rigid bodies vectors are used extensively and it is assumed in this monograph that the reader has some prior knowledge of vector algebra , which is the part of vector theory required in mechanics. Nevertheless chapter 1 provides a summary of vector algebra. There the notation to be used is made explicit and a brief survey of the whole field is given with stress laid on a number of particular results that become especially important later. The reader is also given the opportunity to test his understanding of vector algebra and his facility in applying it to detailed problems: a fairly extensive set of examples (exercises A) follows the chapter, with some comments and answers provided at the end of the book.

A considerably widened theory of vectors becomes necessary when one turns to such parts of physics as fluid dynamics and electromagnetic theory where one deals not just with things at certain particular points in space but with the physical objects as distributed continuously in space. Quantities that are continuous functions of the coordinates of a general point in space are called fields and some of the fields of greatest interest in physics are vector fields .

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• Book: Vector Analysis
• Online publication: 23 December 2009
• Chapter DOI: https://doi.org/10.1017/CBO9780511569524.003

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Keyboard Shortcuts

Lesson 0: matrices and vectors, overview: why matrix algebra section  .

Univariate statistics is concerned with a random scalar variable \(Y\) .

In multivariate analysis, we are concerned with the joint analysis of multiple dependent variables. These variables can be represented using matrices and vectors. This provides simplification of notation and a format for expressing important formulas.

Example 0-1: Section  

Suppose that we measure the variables \(x_1\) = height (cm), \(x_2\) = left forearm length (cm) and \(x_3\) = left foot length for participants in a study of the physical characteristics of adult humans.  These three variables can be represented in the following column vector :

\[\mathbf{x}= \left(\begin{array}{l}x_1\\ x_2\\x_3 \end{array}\right)\]

The observed data for a specific individual, say the i th individual, might also be represented in an analogous vector. Suppose that the \(i^{th}\) person in the sample has height = 175 cm, forearm length = 25.5 cm and foot length = 27 cm. In vector notation these observed data could be written as:

\[\mathbf{x_i} = \left(\begin{array}{l}x_{i1}\\x_{i2}\\x_{i3}\end{array}\right)=\left(\begin{array}{l}175\\25.5\\27.0\end{array}\right)\]

Notice the use and placement of the subscript i to represent the \(i^{th}\) individual.

Definitions of Matrix and Vector Section  

The data matrix in multivariate problems.

Usually, the observed data are represented by a matrix in which the rows are observations and the columns are variables. This is exactly the way the data are normally prepared for statistical software such as SAS or Minitab.

The usual notation is n = the number of observed units (people, animals, companies, etc.) and p = the number of variables measured on each unit. Thus the data matrix will be an n × p matrix.

Example 0-2: Section  

Suppose that we have scores for n = 6 college students who have taken the verbal and the science subtests of the College Qualification Test (CQT). We have p =2 variables: (1) the verbal score and (2) the science score for each student.  The data matrix is the following 6 × 2 matrix:

\[\mathbf{X}=\left(\begin{array}{ll}41&26\\39&26\\53&21\\67&33\\61&27\\67&29\end{array}\right)\]

In the matrix just given, the first column gives the data for \(x_1\) = verbal score whereas the second column gives data for \(x_2\) = science score. Each row gives data for a student in the sample. To repeat – the rows are observations, the columns are variables.

Notation notes: Section  

Note that we have used a small \(\textbf{x}\) to denote the vector of variables in Example 1 and a large \(\textbf{X}\) to represent the data matrix in Example 2. It should also be noted that, in matrix terms, the i th row in the data matrix \(\textbf{X}\) is the transpose of the data vector

\(\mathbf{x_i}=\left(\begin{array}{l}x_{i1}\\x_{i2}\end{array}\right)\), as we defined data vectors in Example 1.

5.2 Vector Addition and Subtraction: Analytical Methods

Learning objectives.

By the end of this section, you will be able to do the following:

• Define components of vectors
• Describe the analytical method of vector addition and subtraction
• Use the analytical method of vector addition and subtraction to solve problems

Components of Vectors

For the analytical method of vector addition and subtraction, we use some simple geometry and trigonometry, instead of using a ruler and protractor as we did for graphical methods. However, the graphical method will still come in handy to visualize the problem by drawing vectors using the head-to-tail method. The analytical method is more accurate than the graphical method, which is limited by the precision of the drawing. For a refresher on the definitions of the sine, cosine, and tangent of an angle, see Figure 5.18 .

Since, by definition, cos θ = x / h cos θ = x / h , we can find the length x if we know h and θ θ by using x = h cos θ x = h cos θ . Similarly, we can find the length of y by using y = h sin θ y = h sin θ . These trigonometric relationships are useful for adding vectors.

When a vector acts in more than one dimension, it is useful to break it down into its x and y components. For a two-dimensional vector, a component is a piece of a vector that points in either the x- or y-direction. Every 2-d vector can be expressed as a sum of its x and y components.

For example, given a vector like   A     A   in Figure 5.19 , we may want to find what two perpendicular vectors,   A x     A x   and   A y     A y   , add to produce it. In this example,   A x     A x   and   A y     A y   form a right triangle, meaning that the angle between them is 90 degrees. This is a common situation in physics and happens to be the least complicated situation trigonometrically.

A x   A x   and   A y     A y   are defined to be the components of   A     A   along the x - and y -axes. The three vectors,   A   A ,   A x   A x , and   A y   A y , form a right triangle.

If the vector   A     A   is known, then its magnitude   A     A   (its length) and its angle   θ     θ   (its direction) are known. To find   A x     A x   and   A y   A y , its x - and y -components, we use the following relationships for a right triangle:

where   A x     A x   is the magnitude of A in the x -direction,   A y     A y   is the magnitude of A in the y -direction, and   θ     θ   is the angle of the resultant with respect to the x -axis, as shown in Figure 5.20 .

Suppose, for example, that   A     A   is the vector representing the total displacement of the person walking in a city, as illustrated in Figure 5.21 .

Then A = 10.3 blocks and θ = 29.1 ∘ θ = 29.1 ∘ , so that

This magnitude indicates that the walker has traveled 9 blocks to the east—in other words, a 9-block eastward displacement. Similarly,

indicating that the walker has traveled 5 blocks to the north—a 5-block northward displacement.

Analytical Method of Vector Addition and Subtraction

Calculating a resultant vector (or vector addition) is the reverse of breaking the resultant down into its components. If the perpendicular components A x A x and A y A y of a vector A A are known, then we can find A A analytically. How do we do this? Since, by definition,

we solve for θ θ to find the direction of the resultant.

Since this is a right triangle, the Pythagorean theorem (x 2 + y 2 = h 2 ) for finding the hypotenuse applies. In this case, it becomes

Solving for A gives

In summary, to find the magnitude A A and direction θ θ of a vector from its perpendicular components A x A x and A y A y , as illustrated in Figure 5.22 , we use the following relationships:

Sometimes, the vectors added are not perfectly perpendicular to one another. An example of this is the case below, where the vectors A A and B B are added to produce the resultant R , R , as illustrated in Figure 5.23 .

If   A     A   and   B     B   represent two legs of a walk (two displacements), then   R     R   is the total displacement. The person taking the walk ends up at the tip of   R   R . There are many ways to arrive at the same point. The person could have walked straight ahead first in the x -direction and then in the y -direction. Those paths are the x - and y -components of the resultant,   R x     R x   and   R y .     R y .   If we know   R x     R x   and   R y   R y , we can find   R     R   and   θ     θ   using the equations   R = R x 2 + R y 2     R = R x 2 + R y 2   and   θ = t a n – 1 ( R y / R x )   θ = t a n – 1 ( R y / R x ) .

and find the y component of the resultant (as illustrated in Figure 5.25 ) by adding the y component of the vectors.

Now that we know the components of R , R , we can find its magnitude and direction.

• To get the magnitude of the resultant R, use the Pythagorean theorem. R = R x 2 + R y 2 R = R x 2 + R y 2
• To get the direction of the resultant θ = tan − 1 ( R y / R x )   . θ = tan − 1 ( R y / R x )   .

Watch Physics

Classifying vectors and quantities example.

This video contrasts and compares three vectors in terms of their magnitudes, positions, and directions.

• 0 units . All of them will cancel each other out.
• 5 units . Two of them will cancel each other out.
• 10 units . Two of them will add together to give the resultant.
• 15 units. All of them will add together to give the resultant.

Tips For Success

In the video, the vectors were represented with an arrow above them rather than in bold. This is a common notation in math classes.

Using the Analytical Method of Vector Addition and Subtraction to Solve Problems

Figure 5.26 uses the analytical method to add vectors.

Worked Example

Add the vector   A     A   to the vector   B     B   shown in Figure 5.26 , using the steps above. The x -axis is along the east–west direction, and the y -axis is along the north–south directions. A person first walks   53 .0 m     53 .0 m   in a direction   20 .0°     20 .0°   north of east, represented by vector   A .     A .   The person then walks   34 .0 m     34 .0 m   in a direction   63 .0 °     63 .0 °   north of east, represented by vector   B .   B .

The components of   A     A   and   B     B   along the x - and y -axes represent walking due east and due north to get to the same ending point. We will solve for these components and then add them in the x-direction and y-direction to find the resultant.

First, we find the components of   A     A   and   B     B   along the x - and y -axes. From the problem, we know that   A = 53.0  m   A = 53.0  m ,   θ A = 20.0 ∘   θ A = 20.0 ∘ ,   B     B   =   34 .0 m   34 .0 m , and   θ B = 63.0 ∘   θ B = 63.0 ∘ . We find the x -components by using   A x = A cos θ   A x = A cos θ , which gives

Similarly, the y -components are found using   A y = A sin θ A   A y = A sin θ A

The x - and y -components of the resultant are

Now we can find the magnitude of the resultant by using the Pythagorean theorem

Finally, we find the direction of the resultant

This example shows vector addition using the analytical method. Vector subtraction using the analytical method is very similar. It is just the addition of a negative vector. That is,   A − B ≡ A + ( − B )   A − B ≡ A + ( − B ) . The components of – B   B   are the negatives of the components of   B   B . Therefore, the x - and y -components of the resultant   A − B = R     A − B = R   are

and the rest of the method outlined above is identical to that for addition.

Practice Problems

What is the magnitude of a vector whose x -component is 4 cm and whose y -component is 3 cm?

What is the magnitude of a vector that makes an angle of 30° to the horizontal and whose x -component is 3 units?

Links To Physics

Atmospheric science.

Atmospheric science is a physical science , meaning that it is a science based heavily on physics. Atmospheric science includes meteorology (the study of weather) and climatology (the study of climate). Climate is basically the average weather over a longer time scale. Weather changes quickly over time, whereas the climate changes more gradually.

The movement of air, water and heat is vitally important to climatology and meteorology. Since motion is such a major factor in weather and climate, this field uses vectors for much of its math.

Vectors are used to represent currents in the ocean, wind velocity and forces acting on a parcel of air. You have probably seen a weather map using vectors to show the strength (magnitude) and direction of the wind.

Vectors used in atmospheric science are often three-dimensional. We won’t cover three-dimensional motion in this text, but to go from two-dimensions to three-dimensions, you simply add a third vector component. Three-dimensional motion is represented as a combination of x -, y - and z components, where z is the altitude.

Vector calculus combines vector math with calculus, and is often used to find the rates of change in temperature, pressure or wind speed over time or distance. This is useful information, since atmospheric motion is driven by changes in pressure or temperature. The greater the variation in pressure over a given distance, the stronger the wind to try to correct that imbalance. Cold air tends to be more dense and therefore has higher pressure than warm air. Higher pressure air rushes into a region of lower pressure and gets deflected by the spinning of the Earth, and friction slows the wind at Earth’s surface.

Finding how wind changes over distance and multiplying vectors lets meteorologists, like the one shown in Figure 5.27 , figure out how much rotation (spin) there is in the atmosphere at any given time and location. This is an important tool for tornado prediction. Conditions with greater rotation are more likely to produce tornadoes.

• Vectors have magnitude as well as direction and can be quickly solved through scalar algebraic operations.
• Vectors have magnitude but no direction, so it becomes easy to express physical quantities involved in the atmospheric science.
• Vectors can be solved very accurately through geometry, which helps to make better predictions in atmospheric science.
• Vectors have magnitude as well as direction and are used in equations that describe the three dimensional motion of the atmosphere.

Check Your Understanding

Between the analytical and graphical methods of vector additions, which is more accurate? Why?

• The analytical method is less accurate than the graphical method, because the former involves geometry and trigonometry.
• The analytical method is more accurate than the graphical method, because the latter involves some extensive calculations.
• The analytical method is less accurate than the graphical method, because the former includes drawing all figures to the right scale.
• The analytical method is more accurate than the graphical method, because the latter is limited by the precision of the drawing.

What is a component of a two dimensional vector?

• A component is a piece of a vector that points in either the x or y direction.
• A component is a piece of a vector that has half of the magnitude of the original vector.
• A component is a piece of a vector that points in the direction opposite to the original vector.
• A component is a piece of a vector that points in the same direction as original vector but with double of its magnitude.
• θ = cos − 1 ⁡ A y A x
• θ = cot − 1 ⁡ A y A x
• θ = sin − 1 ⁡ A y A x
• θ = tan − 1 ⁡ A y A x

How can we determine the magnitude of a vector if we know the magnitudes of its components?

• | A → | = A x + A y | A → | = A x + A y
• | A → | = A x 2 + A y 2 | A → | = A x 2 + A y 2
• | A → | = ( A x 2 + A y 2 ) 2 | A → | = ( A x 2 + A y 2 ) 2

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• Published: 08 April 2024

A wideband, high-resolution vector spectrum analyzer for integrated photonics

• Yi-Han Luo 1 , 2   na1 ,
• Baoqi Shi   ORCID: orcid.org/0000-0002-3828-5131 1 , 3   na1 ,
• Wei Sun 1 ,
• Ruiyang Chen 1 , 2 ,
• Sanli Huang 1 , 4 ,
• Zhongkai Wang 1 ,
• Jinbao Long 1 ,
• Chen Shen 1 ,
• Zhichao Ye   ORCID: orcid.org/0000-0002-4708-3582 5 ,
• Hairun Guo   ORCID: orcid.org/0000-0002-9913-2817 6 &
• Junqiu Liu   ORCID: orcid.org/0000-0003-2405-6028 1 , 4  

Light: Science & Applications volume  13 , Article number:  83 ( 2024 ) Cite this article

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• Infrared spectroscopy
• Integrated optics
• Microresonators
• Silicon photonics

The analysis of optical spectra—emission or absorption—has been arguably the most powerful approach for discovering and understanding matter. The invention and development of many kinds of spectrometers have equipped us with versatile yet ultra-sensitive diagnostic tools for trace gas detection, isotope analysis, and resolving hyperfine structures of atoms and molecules. With proliferating data and information, urgent and demanding requirements have been placed today on spectrum analysis with ever-increasing spectral bandwidth and frequency resolution. These requirements are especially stringent for broadband laser sources that carry massive information and for dispersive devices used in information processing systems. In addition, spectrum analyzers are expected to probe the device’s phase response where extra information is encoded. Here we demonstrate a novel vector spectrum analyzer (VSA) that is capable of characterizing passive devices and active laser sources in one setup. Such a dual-mode VSA can measure loss, phase response, and dispersion properties of passive devices. It also can coherently map a broadband laser spectrum into the RF domain. The VSA features a bandwidth of 55.1 THz (1260–1640 nm), a frequency resolution of 471 kHz, and a dynamic range of 56 dB. Meanwhile, our fiber-based VSA is compact and robust. It requires neither high-speed modulators and photodetectors nor any active feedback control. Finally, we employ our VSA for applications including characterization of integrated dispersive waveguides, mapping frequency comb spectra, and coherent light detection and ranging (LiDAR). Our VSA presents an innovative approach for device analysis and laser spectroscopy, and can play a critical role in future photonic systems and applications for sensing, communication, imaging, and quantum information processing.

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Introduction.

The analysis of light and its propagation in media is fundamental in our information society. The discovery of light refraction and dispersion in media has resulted in the invention of prisms and gratings that are ubiquitously used in today’s optical systems for imaging, sensing, and communication. Key enabling building blocks to these applications are dispersive elements that separate light components of different colors (i.e. frequencies) either spatially or temporally 1 , with precisely calibrated chromatic dispersion. With these elements, modern optical spectrum analyzers (OSA) and spectrometers can deliver unrivaled frequency resolution, large dynamic range, and wide spectral bandwidth of hundreds of nanometers. Time-stretched systems 2 can probe ultrafast and rare events in complex nonlinear systems.

For spectrum analysis, precise and broadband frequency-calibration of dispersive elements is pivotal. Due to the ultimate need for spectrometers with reduced size, weight, cost, and power consumption, extensive efforts have been made to create miniaturized spectrometers 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 and broadband laser sources 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 based on integrated waveguides. For these devices, frequency-calibration is particularly crucial yet challenging since the dispersion of integrated waveguides can be significantly altered by the structures and sizes 31 , 32 . Meanwhile, stationary phase approximation for time-stretch dispersive Fourier transform 33 necessitates carefully frequency-calibrated elements that are strongly dispersive. For these purposes, optical vector network analyzers (OVNA) are viable tools. Analog to an electrical VNA, an OVNA enables direct characterization of the linear transfer function (LTF) of passive devices, therefore allowing simultaneous measurement of transmission (i.e. loss), phase response, and dispersion 34 , 35 , 36 . Previously demonstrated OVNAs are based on interferometry 35 , 37 , optical channel estimation 36 , 38 , single-sideband modulation 39 , 40 , and frequency-comb-assisted asymmetric double sidebands 41 . Despite, all these methods have limited measurement bandwidth of sub-terahertz to a few terahertz. Therefore for booming demands to understand and to engineer devices used for broadband laser sources that span over tens of terahertz, including optical frequency combs 13 , 14 , 15 , parametric oscillators 16 , 18 , 19 , quantum frequency translators 20 , 21 , supercontinua 22 , 23 , 24 , 25 , and parametric amplifiers 26 , 27 , 28 , 29 , all these methods fail.

Here we demonstrate a new paradigm of vector spectrum analysis that unites OVNA for passive devices and OSA for active laser sources in one setup. Our vector spectrum analyzer (VSA) can measure LTF and dispersion properties of passive devices, or coherently map an optical spectrum into the RF domain.

The principle of our VSA is illustrated in Fig. 1a . A continuous-wave (CW), widely chirping laser, is sent to and transmits through a device under test (DUT), or interferes with a light under test (LUT). During laser chirping, for the DUT, the frequency-dependent LTF containing its loss and phase information is photodetected and recorded. Particularly, the phase is extracted by interfering the light exiting the DUT with a reference branch. For the LUT, the chirping laser beats progressively with different frequency components of the optical spectrum, and the beatnote signal is photodetected, recorded in the RF domain, and further processed with a digital narrow-band-pass filter offline. In either case, the VSA outputs a time-domain trace, with each data point corresponding to the DUT’s instantaneous response, or the LUT’s instantaneous beatnote from interference, at a particular frequency during laser chirping. In short, the chirping laser coherently maps the frequency-domain response into the time domain, which is digitally processed to retrieve the frequency-domain response. Since the laser cannot chirp perfectly linearly, critical to this frequency-time mapping is precise and accurate calibration of the instantaneous laser frequency at any given time. This requires referring the chirping laser to a calibrated “frequency ruler”.

a The principle of our VSA is based on a chirping CW laser that is sent to and transmits through a device under test (DUT), or interferes with any light under test (LUT). The DUT can be any passive device, and the LUT can be any broadband laser source. The transmission spectrum of the chirping laser through the DUT, and the beatnote signal generated from the interference between the chirping laser and the LUT, are both time-domain traces. Together with a “frequency ruler” for calibration, the chirping laser coherently maps the time-domain trace into the frequency domain. This trace carries the information of the DUT’s loss, phase and dispersion over the chirp bandwidth (blue). For LUT, the chirping laser beats progressively with different frequency components of the optical spectrum, thus, analyzing the beat signal in the RF domain allows extraction of the spectral information (purple). Critical to this frequency-time mapping is precise and accurate calibration of the instantaneous laser frequency during chirping. This requires referring the chirping laser to the frequency ruler (red). b Experimental setup. The chirping laser unit can be a single laser, or multiple lasers that are bandwidth-cascaded. The latter allows extension of the full spectral bandwidth by seamless stitching of individual laser traces into one trace. The chirping laser power is then split into two branches. The upper branch is directed to the frequency-calibration unit (i.e. the “frequency ruler”), which in our case, is a phase-stable fiber cavity of 55.58 MHz FSR. The lower branch is sent to the DUT or the LUT. For the two branches, the photodetector signals are recorded with an oscilloscope and digitally processed offline. PD photodetector, OSC oscilloscope

Following this principle, we construct the setup as shown in Fig. 1b . A widely tunable, mode-hop-free, external-cavity diode laser (ECDL, Santec TSL) is used as the chirping laser. Cascading multiple ECDLs covering different spectral ranges allows the extension of full spectral bandwidth, which is 1260–1640 nm (55.1 THz) in our VSA with three ECDLs (see Note 1 in Supplementary Materials).

The ECDL’s CW output is split into two branches. One branch is sent to the DUT or the LUT, selected by an optical switch. The other branch is a frequency-calibration unit based on a fiber cavity. Such frequency calibration involves relative- (i.e. the frequency change relative to the starting laser frequency) and absolute-frequency calibration (i.e. accurately measured starting laser frequency). The absolute-frequency calibration is performed by referring to a built-in wavelength meter with an accuracy of 200 MHz (see Note 1 in Supplementary Materials). The relative-frequency calibration is described in the following.

Relative frequency calibration of the VSA

Figure 2 illustrates the principle of relative-frequency calibration. We use a fiber cavity with an equidistant grid of resonances as the frequency ruler 42 , 43 . By counting the number of resonances passed by the chirping laser and multiplying the number with the fiber cavity’s free spectral range (FSR, f fsr ), the laser frequency excursion is calculated. Extrapolation of laser frequency between two neighboring fiber cavity’s resonances further improves frequency resolution, precision and accuracy, which will be discussed later. Therefore, critical to this method is the measurement precision of f fsr and compensation of fiber dispersion to account f fsr variation over the 55.1 THz spectral range.

a Experimental setup. PM phase modulator, PC polarization controller, FFT fast Fourier Transformation. b–d Principle of the frequency-calibration process of the fiber cavity’s FSR. Charts compare the differences when \({f}_{\mathrm{mod}}\ne N\cdot {f}_{\text{fsr}}\) (red arrows and curves) and \({f}_{\mathrm{mod}}=N\cdot {f}_{\text{fsr}}\) (blue arrows and curves). The three red/blue arrows in Panel b mark the incident CW laser with the two sidebands generated from it. From the experimental data (with blue background) and simulation (with red background), the differences between \({f}_{\mathrm{mod}}\ne N\cdot {f}_{\text{fsr}}\) and \({f}_{\mathrm{mod}}=N\cdot {f}_{\text{fsr}}\) are illustrated by: 1. The envelope modulation on the time-domain trace (Panel b bottom); 2. The resonance profile (Panel c ); 3. The Fourier peaks in the RF domain via FFT (Panel d ). e Measured fiber cavity’s FSR variation over the 55.1 THz frequency range with fitted dispersion. We perform the measurement at two different temperatures \({T}_{0}\) and \({T}_{0}+\Delta T\) , where T 0  = 23.5 °C and Δ T  = 9.3 °C. f For fiber cavities made of single-mode fibers (SMF) or phase-stable fibers (PSF), the measured cavity FSR drifts versus relative temperature change, as well as the linear fit. g Totally 150 measurements of the fiber cavity’s FSR show a standard deviation (STD) of 112.5 Hz

The experimental setup to calibrate f fsr is shown in Fig. 2a . The ECDL’s CW output is phase-modulated by an RF signal generator to create sidebands. Though consisting of multiple CW components, i.e. the carrier and sidebands, the amplitude is constant in the time domain. The carrier and both sidebands are together sent into the fiber cavity with maintained polarization. The transmitted signal through the fiber cavity is probed by a 125-MHz-bandwidth photodetector, recorded by an oscilloscope, digitally processed, and fed back to the RF signal generator. Based on the fiber cavity length, an initial value of the fiber cavity’s FSR, Δ f 0  = 55.58 MHz, is estimated. The RF driving frequency \({f}_{\mathrm{mod}}\) of the phase modulator is set to \({f}_{\mathrm{mod}}=N\cdot \Delta {f}_{0}\) , where N is an integer ( \(N=3\) in our case).

Since \(\Delta {f}_{0}\ne {f}_{\text{fsr}}\) , as shown in Fig. 2b , the carrier and both sidebands are misaligned with the respective three resonances in the frequency domain. In this case, the three CW components experience different cavity responses (including loss and phase). Thus, the superposition of light fields varies accordingly, leading to a beatnote at the output. When \({f}_{\mathrm{mod}}\) is slightly varied such that \({f}_{\mathrm{mod}}=N\cdot {f}_{\text{fsr}}\) is satisfied, the carrier and both sidebands can be simultaneously aligned with the fiber cavity’s resonances, experiencing the same cavity’s response. Thus, the superposition of light fields of these CW components is unaffected at the output, producing a constant current on the photodetector.

Experimentally, the laser chirps across a resonance of the fiber cavity, and the time-domain interference at the fiber cavity’s output is photodetected and recorded by the oscilloscope. The time-domain trace is then normalized to the level of off-resonant signal power, as the transmission spectrum of a resonance shown in Fig. 2c . When \({f}_{\mathrm{mod}}\ne N\cdot {f}_{\text{fsr}}\) , the resonance profile is modulated at the frequency \({f}_{\mathrm{mod}}\) (red curves). As \({f}_{\mathrm{mod}}\) approaches \(N\cdot {f}_{\text{fsr}}\) , i.e., \(\left|{f}_{\mathrm{mod}}-N\cdot {f}_{\text{fsr}}\right|\to 0\) , the modulation amplitude decreases. When \({f}_{\mathrm{mod}}=N\cdot {f}_{\text{fsr}}\) , the resonance profile is unaffected as a normal Lorentzian profile probed by a single CW laser (blue curves). We simulate this modulation behavior (left red panels) which agrees with the experimental data (right blue panels). The modulation amplitude is extracted with fast Fourier transformation (FFT) as shown in Fig. 2d , where red curves represent \({f}_{\mathrm{mod}}\ne N\cdot {f}_{\text{fsr}}\) and blue curves represent \({f}_{\mathrm{mod}}=N\cdot {f}_{\text{fsr}}\) . When \({f}_{\mathrm{mod}}\ne N\cdot {f}_{\text{fsr}}\) , a binary search to minimize \(\left|{f}_{\mathrm{mod}}-N\cdot {f}_{\text{fsr}}\right|\) is performed until the modulation peaks vanish, signaling \({f}_{\mathrm{mod}}=N\cdot {f}_{\text{fsr}}\) . For more details on the calibration process and numerical simulations, see Note 2 in Supplementary Materials.

We apply this method to measure the fiber cavity’s \({f}_{\text{fsr}}\) from 1260 to 1640 nm wavelength (55.1 THz frequency range) with an interval of 10 nm, at an ambient temperature of T 0  = 23.5 °C. The fiber cavity is made of phase-stable fibers (PSF, described later). Figure 2e shows that plots and analysis of frequency-dependent \({f}_{\text{fsr}}\) enable extraction of the fiber dispersion using a cubic polynomial fit (see Note 3 in Supplementary Materials). This dispersion-calibrated fiber cavity’s resonance grid is used as the frequency ruler in our VSA and following experiments.

We further characterize the temperature stability of \({f}_{\text{fsr}}\) . The fiber cavity is heated and its \({f}_{\text{fsr}}\) shift versus the relative temperature change at 1490 nm is measured, as shown in Fig. 2f . In addition, we compare two types of fibers to construct the cavity: the normal single-mode fiber (SMF, blue data) and phase-stable fiber (PSF, red data). The linear fit shows that the PSF-based fiber cavity features a temperature sensitivity of \(\text{d}{f}_{\text{fsr}}/\text{d}T=-262\) Hz/K, in comparison to −676 Hz/K of the SMF. The lower \(\text{d}{f}_{\text{fsr}}/\text{d}T\) of PSF is the reason why we use PSF instead of SMF. Correspondingly, 1 K temperature change (the level of our ambient temperature stabilization and control) causes ~240 MHz cumulative error of the PSF-based fiber cavity over the entire 55.1 THz range.

We also measure the fiber cavity’s dispersion at elevated temperature \({T}_{0}+\Delta T\) , where Δ T  = 9.3 °C. Figure 2e shows that, the two measured fiber dispersion curves at different temperatures are nearly identical except with a global offset in the y -axis. This indicates that the temperature change only affects \({f}_{\text{fsr}}\) but not fiber dispersion. More details concerning the measurement are found in Note 3 in Supplementary Materials. Therefore, once the ambient temperature is known, the \({f}_{\text{fsr}}\) at 1490 nm can be calculated, as well as the \({f}_{\text{fsr}}\) variation over frequency.

Finally, to verify the measurement reproducibility, the \({f}_{\text{fsr}}\) value at 1490 nm is repeatedly measured 150 times. Figure 2g shows the occurrence histogram, with a standard deviation of 112.5 Hz.

Performance of the VSA

Here we use a dispersion-calibrated, phase-stable fiber cavity for relative-frequency calibration. We note that frequency comb spectrometers 15 , 44 , 45 with a precisely equidistant grid of frequency lines can also be used 46 , 47 . While frequency combs have been a proven technology for spectroscopy 48 with unparalleled accuracy, they have several limitations in the characterization of passive devices. First, in addition to being bulky and expensive, commercial fiber laser combs as spectrometers suffer from limited frequency resolution due to the RF-rate comb line spacing (typically above 100 MHz). Second, the simultaneous injection of more than 10 5 comb lines can saturate or blind the photodetector, yielding a severely deteriorated signal-to-noise ratio (SNR) and dynamic range.

Different from frequency combs, CW lasers featuring high photon flux and ever-increasing frequency tunability and agility are particularly advantageous for sensing 49 . In our method, after frequency-calibration by the fiber cavity, the chirping CW laser behaves as a frequency comb with a “moving” narrow-band-pass filter, where the filter selects only one comb line each time and rejects other lines. Therefore the nearly constant laser power during chirping provides a flat power envelope over the entire spectral bandwidth. Consequently our method avoids photodetector saturation and device damage. It also increases SNR and dynamic range.

To improve frequency resolution, the extrapolation of instantaneous laser frequency between two neighboring fiber cavity’s resonances is performed, which relies on the frequency linearity of the chirping laser. Such linearity is experimentally characterized in a parallel work 50 of ours, where the chirping ECDL (Santec TSL) is referenced to a commercial optical frequency comb. The result from ref. 50 evidences that the chirping linearity is better for a higher chirp rate. We experimentally test different laser chirp rates, including 20, 50, 100, and 200 nm/s. When combined with a 55-MHz-FSR fiber cavity, the accuracy of linear interpolation within the fiber cavity’s “dead zone” achieves a precision better than 200 kHz for a chirp rate exceeding 50 nm/s, which surpasses the laser linewidth. Meanwhile, if the chirp rate is too high, such that the time that the chirping laser sweeps across the DUT’s resonance is comparable to the resonance lifetime, the measured transmission curve is distorted due to the cavity’s ringdown effect. Taking all these issues into consideration, the chirp rate of 50 nm/s is the most appropriate value in our experiment. More details are elaborated in Note 4 in Supplementary Materials.

The ultimate frequency resolution of each individual time-domain trace is determined by the chirp range divided by the oscilloscope’s memory depth ( \(2\times {10}^{8}\) ). For the ECDL of the widest spectral range from 1480 to 1640 nm (19.8 THz), we estimate that the ultimate frequency sample resolution of our VSA is around 99 kHz, i.e. the frequency interval between two recorded neighboring data points. The actual resolution can be compromised by the chirping laser’s linewidth. We measure the dynamic laser linewidth using a self-delayed heterodyne setup. Experimental details are elaborated in Note 5 in Supplementary Materials. Within a 100 μs time scale, the ECDL’s dynamic linewidth at 50 nm/s chirp rate is averaged as 471 kHz. This linewidth is due to multiple reasons, including laser intrinsic linewidth, laser chirp nonlinearity, and the fiber delay-line’s instability in the heterodyne setup. The measured laser dynamic linewidth of 471 kHz sets the lower bound of our VSA’s frequency resolution.

Finally, we compare our method for relative frequency calibration using a fiber cavity with the commonly used method based on an unbalanced Mach–Zehnder interferometer (UMZI) 37 , 47 . The transmission of a UMZI versus frequency is a sinusoidal curve, which theoretically could provide frequency calibration at any instant. However, it is challenging to measure the local FSR of a UMZI with a precision better than 1 kHz. Thus, to reduce the accumulative error, an optical frequency comb is required 47 . On the other hand, practically, when the laser chirping nonlinearity heavily distorts the sinusoidal curve, extraction of the actual phase at arbitrary instants becomes infeasible. In comparison, it is easier and more reliable to obtain the center dip of a narrow resonance of ~1 MHz linewidth for a fiber cavity. These two points mark the advantages of the fiber cavity over the UMZI for relative frequency calibration.

Characterization of passive integrated devices

Next we demonstrate several applications using our VSA. We first use our VSA as an OVNA to characterize passive devices. We select two types of optical devices: an integrated optical microresonator and a meter-long spiral waveguide. Both devices, fabricated on silicon nitride (Si 3 N 4 , see Materials and Methods) 51 , have been extensively used in integrated nonlinear photonics 14 , 16 . For example, optical microresonators of high quality ( Q ) factors are central building blocks for microresonator-soliton-based optical frequency combs (“soliton microcomb”) 13 , 14 , 15 , ultralow-threshold optical parametric oscillators 16 , 18 , 19 , and quantum frequency translators 20 , 21 . Ultralow-loss, dispersion-flattened waveguides are cornerstones for multi-octave supercontinua 22 , 23 , 24 , 25 and continuous-traveling-wave optical parametric amplifiers 26 , 27 , 28 . All these applications require precisely characterized properties of integrated devices, such as loss, phase, and dispersion over a bandwidth spanning more than 100 nm.

Figure 3a shows an optical microscope image of a Si 3 N 4 microresonator, which is a microring side-coupled by a bus waveguide for optical input and output 51 . The resonance frequency \(\omega /2{\rm{\pi }}\) and linewidth \(\kappa /2{\rm{\pi }}\) are often the most critical parameters that reveal the waveguide’s dispersion, loss, and phase response. Experimentally, we measure these two parameters of each fundamental-mode resonance, ranging from 1260 nm (237.9 THz) to 1640 nm (182.8 THz) wavelength. First, we extract the integrated dispersion of the Si 3 N 4 microresonator, which is defined as

where \({\omega }_{\mu }/2{\rm{\pi }}\) is the μ th resonance frequency relative to the reference resonance frequency \({\omega }_{0}/2{\rm{\pi }}\) , \({D}_{1}/2{\rm{\pi }}\) is microresonator’s FSR, \({D}_{2}/2{\rm{\pi }}\) describes group velocity dispersion (GVD), and D n ≥3 are higher-order dispersion terms. Experimentally, as shown in Fig. 3a , half of the laser power is split for frequency calibration with the fiber cavity, and the other half is then split into two branches. In the lower branch, half of the light transmitted through the DUT is directly detected by a photodetector and recorded with an oscilloscope. The microresonator transmission over 55.1 THz optical bandwidth (from 1260 to 1640 nm) is shown in Fig. 3b . Exact resonant frequencies \({\omega }_{\mu }\) are determined with dips searching on the transmission data-trace.

a Experimental setup to measure the transmission (loss), dispersion and phase response of Si 3 N 4 microresonators. Inset is an optical microscope image showing a Si 3 N 4 microresonator coupled with a bus waveguide. BPD, balanced photodetector. b A typical broadband transmission spectrum of an integrated Si 3 N 4 microresonator. c Measured integrated microresonator dispersion profile and fit up to the fifth order. d Measured transmission and phase profiles of three resonances that are under-coupled (top), over-coupled (middle), or feature mode split (bottom, also under-coupled)

Figure 3c top plots the measured \({D}_{\mathrm{int}}\) profile, with each parameter extracted from the fit using Eq. ( 1 ). We note that, due to our 55.1 THz measurement bandwidth and 471 kHz frequency resolution, our method can measure higher-order dispersion 52 up to the fifth-order ( D 5 ) term. This is validated in Fig. 3c middle, where D 3 and D 4 terms are subtracted from \({D}_{\mathrm{int}}\) , and the residual dispersion is fitted with \({D}_{5}{\mu }^{5}/120\) . Figure 3c bottom shows that, after further subtraction of the D 5 term, no prominent residual dispersion is observed. Some data points deviate from the fit due to avoided mode crossings in the microresonator 53 .

Next, we fit each resonance to obtain their linewidths. For each resonance fit 54 , the intrinsic loss \({\kappa }_{0}/2{\rm{\pi }}\) , external coupling strength \({\kappa }_{\text{ex}}/2{\rm{\pi }}\) , and the total (loaded) linewidth \(\kappa /2{\rm{\pi }}=\left({\kappa }_{0}+{\kappa }_{\text{ex}}\right)/2{\rm{\pi }}\) , are extracted. Figure 3d shows three typical resonances with fit curves (blue), including one with visible mode split (bottom). Conventionally, based on a single resonance profile, it is impossible to judge whether the resonance is over-coupled ( \({\kappa }_{\text{ex}} > {\kappa }_{0}\) ) or under-coupled ( \({\kappa }_{\text{ex}} < {\kappa }_{0}\) ) 55 . The coupling condition can only be revealed by phase (vector) measurement. As shown in Fig. 3a , since the light exiting the DUT carries an additional phase shift φ , we interfere it with the light from the reference upper branch to extract φ . To improve the signal-to-noise ratio, a delay Δ τ caused by a 20-meter-long fiber is brought into the reference branch. The frequency difference \(\Delta f=\gamma \Delta \tau\) between the two branches corresponds to the frequency of the beatnote signal readout with the balanced photodetector, where γ is the laser chirp rate. The extra phase shift φ also applies to the beat signal, which can be extracted with Hilbert transformation 56 (see Note 6 in Supplementary Materials). The measured and fitted phases are shown in Fig. 3d red curves. The continuous phase transition across the resonance in Fig. 3d top and bottom represents under-coupling, while the phase jump by 2 π in Fig. 3d middle represents over-coupling. From top to bottom, the fitted loss values ( κ 0 /2 π , κ ex /2 π ) for each resonance are (23.8, 14.0), (19.9, 42.4), and (24.7, 12.8) MHz. The complex coupling coefficient 57 in the bottom is κ c /2 π  = 29.1 + 2.25 i MHz.

In addition to microresonators as well as other resonant structures, our method can also characterize single-pass waveguides. Here we apply our method to measure the linear loss and frequency-dependent group refractive index of integrated waveguides. Figure 4a shows an optical microscope image of a Si 3 N 4 photonic chip containing a spiral waveguide of \({L}_{0}=1.6394\,{\rm {m}}\) physical length. We use our VSA as optical frequency-domain reflectometry (OFDR) 58 to characterize the waveguide loss and dispersion. The setup is shown in Fig. 4b . The laser is coupled into the DUT, and the reflected light from the DUT is collected by an AC-coupled photodetector through a circulator. The reflected light at distance l within the waveguide of physical length L 0 has an amplitude A l . The strong reflection signal from the waveguide front facet (i.e. \(l=0\) ) has a constant amplitude A 0 . Together these two fields A l and A 0 interfere and create a beat signal of frequency \(\Delta f=2\gamma {n}_{{\rm {g}}}l/c\) , where c is the speed of light in air, γ is the laser chirp rate, and n g is the group index. Thus, the photodetected beat signal is \({V}_{l}\left(t\right)={A}_{0}{A}_{l}\cos \left(2{\rm{\pi }}\Delta {ft}\right)\) . The instantaneous laser chirp rate γ is extracted by the relative frequency calibration using the fiber cavity. With FFT, the relative reflection power from different locations in the waveguide is calculated. Figure 4c plots the OFDR signal from the spiral waveguide. The prominent peak located at 1.6394 m physical length (3.4214 m optical length) is attributed to the light reflection at the rear facet of the chip, where the waveguide terminates. The difference in the physical and optical lengths indicates a group index of \({n}_{{\rm {g}}}=2.087\) at 192.681 THz.

a Optical microscope image showing a 1.6394-meter-long spiral waveguide contained in a photonic chip of 5 × 5 mm 2 size. The zoom-in shows the densely coiled waveguide. b Experimental setup of OFDR. The red solid and dashed arrows represent the incident light to and the reflected light from the DUT, respectively. c Measured OFDR data of the spiral waveguide. The major peak at 1.6394 m physical length (3.4214 m optical length) is attributed to the light reflection at the rear chip facet, where the waveguide terminates. This length difference indicates a group index of \({n}_{{\rm {g}}}=2.087\) at 192.681 THz. The loss rate \(\alpha =-3.0\) dB/m (physical length) is calculated with a linear fit of power decrease over distance (red line). d Measured group index n g (blue dots), the fit (black line), and loss α (red dots) of the waveguide over the 55.1 THz spectral range

In the presence of waveguide dispersion, the optical path length L op varies due to the frequency-dependent n g . This dispersion-induced optical path variation leads to deteriorated spatial resolution in broadband measurement 59 . By dividing the broadband measurement data into narrow-band segments 60 , 61 , the optical path length at different optical frequencies can be obtained, and thus the frequency-dependent n g over the 55.1 THz spectral range can be extracted. With the extracted n g , the waveguide dispersion can be de-embedded with a re-sample algorithm 61 , 62 .

Light traveling in the waveguide experiences attenuation following the Lambert–Beer Law \(I\left(L\right)={I}_{0}\cdot \exp \left(\alpha L\right)\) . In Fig. 4c , the average linear loss \(\alpha =-3.0\) dB m −1 (physical length) is extracted by applying a first-order polynomial fit of the power profile (red line) within the 19.8 THz bandwidth and centered at 192.681 THz. Figure 4d shows the frequency-dependent α (red dots) and n g (blue dots) extracted using segmented OFDR algorithm 60 , 63 . The n g is further fitted at 208.015 THz, and the dispersion parameters are extracted up to the fourth order as β 1  = 6955.0 fs/mm, β 2  = −74.09 fs 2 /mm, β 3  = 199 fs 3 /mm, and \({\beta }_{4}=2.4\times {10}^{2}\) fs 4 /mm. The loss fluctuation with varying frequency is likely due to multi-mode interference in the spiral waveguide 64 .

In OFDR, the resolution \(\delta {L}_{\text{op}}\) of optical path length is determined by the laser chirp bandwidth B as \(\delta {L}_{\text{op}}=c/2B\) , with c being the speed of light in vacuum. Our VSA can provide a maximum \(B=19.8\) THz in a single measurement, which enables \(\delta {L}_{\text{op}}=7.6\) μm. As shown in Fig. 4c , such a fine resolution allows unambiguous discrimination of scattering points in the waveguide, which are revealed by small peaks. Thus our VSA is proved as a useful diagnostic tool for integrated waveguides.

Characterization of a soliton spectrum

Next, we use our VSA as an OSA to characterize broadband laser spectra. While modern OSAs can achieve wide spectral bandwidth, they suffer from a limited frequency resolution ranging from sub-gigahertz to several gigahertz. This issue prohibits OSAs from resolving fine spectral features. For example, individual lines of mode-locked lasers or supercontinua with repetition rates in the RF domain cannot be resolved by OSAs. Soliton microcombs with terahertz-rate repetition rate can be useful for low-noise terahertz generation 65 , 66 , but their precise comb line spacing can neither be measured by normal photodetectors nor OSAs.

Here we demonstrate that our VSA can act as an OSA, which features 55.1 THz spectral range and megahertz frequency resolution. As an example, we measure the repetition rate (line spacing) of a soliton microcomb. The experimental setup is shown in Fig. 5a , where the generated soliton interferes with the chirping laser. The interference is then photodetected with a balanced photodiode and recorded with an oscilloscope. The schematic is depicted in Fig. 5b when the laser chirps across the entire soliton spectrum. Every time the laser passes through a comb line, it generates a moving beatnote. Using a digitally implemented finite impulse response (FIR) band-pass filter of 10 MHz center frequency and 3 MHz bandwidth, the beatnote creates a pair of marker signals when the laser frequency is ±10 MHz distant from the comb line. The polarization of the soliton spectrum is measured by varying the laser polarization until the beat signal with maximum intensity is observed. Since the instantaneous laser frequency is precisely calibrated, the comb line spacing is extracted by calculating the frequency distance from two adjacent pairs of marker signals. With the known laser power and measured marker signals’ intensity, the absolute power of each comb line can be calculated. To further measure the relative power of each comb line, we measured the output power of the chirping laser \({P}_{\text{c}}\left(f\right)\propto {\left|{A}_{\text{c}}\left(f\right)\right|}^{2}\) over its entire chirp range. Note that the amplitude of the beatnote signal \(\left|{A}_{\text{rf}}\left(f\right)\right|\) is proportional to \(\left|{A}_{\text{c}}\left(f\right){A}_{\text{LUT}}\left(f\right)\right|\) . Therefore the amplitude of the LUT \(\left|{A}_{\text{LUT}}\left(f\right)\right|\) can be calculated.

a Experimental setup. OSA, optical spectrum analyzer. b Principle of coherent detection of a broadband optical spectrum using a chirping laser. The laser beats progressively with different frequency components of the optical spectrum, which allows frequency detection in the RF domain and continuous information output in the time domain. c Single soliton spectra measured by our VSA (red) and a commercial OSA (blue). The spectral envelope of VSA data is fitted with a sech 2 function (green). Inset: Zoom-in of the comb line resolved by our VSA and the OSA, demonstrating a significant resolution enhancement by the VSA

Figure 5c compares the measured soliton microcomb spectra using our VSA and a commercial OSA. Both spectra are nearly identical, which validates our VSA measurement. The dynamic range of our VSA is found as 56 dB, on par with modern commercial OSAs with the finest resolution (e.g. 45–60 dB at 0.02 nm resolution for Yokogawa OSAs). Figure 5c inset evidences that our VSA indeed provides significantly finer frequency resolution than the OSA’s. The soliton repetition rate measured by the VSA is \(\left(100.307\pm 0.002\right)\) GHz.

We emphasize that, here the frequency resolution of our VSA as an OSA is limited by the bandwidth of FIR band-pass filters. In digital data processing, we find that 3 MHz FIR bandwidth yields the optimal resolution bandwidth of 3 MHz. Experimentally, we verify the resolution bandwidth by phase-modulating a low-noise fiber laser (NKT Koheras) to generate a pair of sidebands of 3 MHz difference to the carrier. The carrier and the sidebands are unambiguously resolved using our VSA (see Note 5 in Supplementary Materials). The 3 MHz resolution bandwidth is also consistent with the uncertainty of the measured soliton repetition rate of 100.307 GHz.

High-resolution LiDAR application

Finally, we note that the broadband, chirping, and interferometric nature of our VSA also enables coherent LiDAR. Frequency-modulated continuous-wave (FMCW) LiDAR is a ranging technique based on frequency-modulated interferometry 67 , as depicted in Fig. 6a . The chirping laser is split into two arms, with one arm to the reference and the other to the target with a path difference of d . When the reflected signals from both arms recombine at the photodetector, the detected beat frequency is determined as \(\Delta f=2d{\rm{\gamma }}/c\) , where c is the speed of light in air and γ is the chirp rate. Thus the measurement of Δ f in the RF domain allows distance measurement of d . The ranging resolution δ d , i.e. the minimum distance that the LiDAR can distinguish two nearby objects, is limited by the chirp bandwidth B as \(\delta d=c/2B\) . One advantage of our VSA as an FMCW LiDAR is that our laser can provide maximum \(B=19.8\) THz that enables \({\rm{\delta }}d=7.6\) μm.

a Principle of coherent LiDAR using a linearly chirping laser. With the known chirp rate γ , the heterodyne measurement of the beat frequency in the RF domain \(\Delta f=2d\gamma /c\) allows calculation of the time delay \(\Delta t=2d/c\) and thus the distance d . b The histogram showing the deviations of 4625 LiDAR measurements from their mean values. The LiDAR precision is revealed by the standard deviation of 20.3 nm. c LiDAR measurement of thermal expansion of our optical table using our VSA, in comparison with data from a digital ambient thermometer

In our LiDAR experiment, we set the linear chirp rate of \(\gamma =6.25\) THz/s and duration of \(T=0.4\) s. The experimental setup and data analysis procedure are found in Note 7 in Supplementary Materials. As a demonstration, we monitor the thermal expansion of our optical table due to ambient temperature drift, as shown in Fig. 6c . The distance difference between the target mirror and the reference mirror on the table is \(d=137.63128\) mm. The measured distance change Δ d within the 500 nm range agrees with the temperature decrease that causes the contraction of the optical table. After subtracting the global trend, Fig. 6b shows the histogram of the deviations of 4625 measurements from their mean values. Our LiDAR precision is revealed by the standard deviation of 20.3 nm. Such precision is provided by the careful relative-frequency calibration and long-term stability of our VSA.

In summary, we have demonstrated a dual-mode VSA featuring 55.1 THz spectral bandwidth, 471 kHz frequency resolution, and 56 dB dynamic range. The VSA can operate either as an OVNA to characterize the LTF and dispersion property of passive devices, or as an OSA to characterize broadband frequency comb spectra. A comparison of our VSA with other state-of-the-art OSAs and OVNAs is shown in Note 5 in Supplementary Material. Our VSA can also perform LiDAR with distance resolution of 7.6 μm and precision of 20.3 nm. Meanwhile, our VSA is fiber-based. It neither requires high-speed modulators and photodetectors, nor any active feedback control. Therefore the system is compact, robust, and transportable for field-deployable applications.

There are several aspects to further improve the performance of our VSA. First, the frequency accuracy can be improved by adding a highly stable reference laser in the system. When the ECDL scans through the reference laser, the two lasers beat and create a marker in the time-domain trace. The marker marks the point where the chirping ECDL has an instantaneous frequency as the reference laser’s frequency. Second, more ECDLs can be added to the system, allowing further extension of the spectral bandwidth and operation in other wavelength ranges, such as the visible and mid-infrared bands. Meanwhile, even ECDLs with mode hopping can be used in our VSA. The self-calibration and compensation of mode hopping can be realized by adding a calibrated, large-FSR cavity (e.g. a Si 3 N 4 microresonator of terahertz-rate FSR), in addition to the fine-tooth fiber cavity. By measuring the resonance-to-resonance frequency and referring to the previously calibrated local FSR of the microresonator, the exact mode hopping range and location can be inferred. Adding more calibrated cavities of different FSR values to form a Vernier structure can further enhance precision and accuracy.

Besides the characterization of passive elements and broadband laser sources for integrated photonics, our VSA can also be applied for time-stretched systems 2 , optimized optical coherent tomography (OCT) 68 , linearization of FMCW LiDAR 69 , and resolving fine structures in Doppler-free spectroscopy 70 . Therefore, our VSA presents an innovative approach for device analysis and laser spectroscopy, and can play a crucial role in future photonic systems and applications for sensing, communication, imaging, and quantum information processing.

Materials and methods

Device fabrication.

The Si 3 N 4 microresonator and meter-long spiral waveguides are fabricated using a foundry-standard process 46 . It is a subtractive process on 6-inch wafers. First, Si 3 N 4 and SiO 2 films are deposited on the thermal wet SiO 2 substrate via low-pressure chemical vapor deposition (LPCVD). Deep-ultraviolet (DUV) stepper photolithography (248 nm KrF) is used to expose the waveguide pattern. The pattern is subsequently transferred from the photoresist mask to the SiO 2 hard mask and then to the Si 3 N 4 layer via dry etching. Then the etched substrate is thermally annealed in a nitrogen atmosphere at 1200 °C to remove hydrogen contents in Si 3 N 4 . Top SiO 2 cladding is then deposited on the wafer, followed by another high-temperature annealing. Finally, contact photolithography and deep etching of SiO 2 and Si are used to divide a wafer into hundreds of chips, followed by dicing.

Data availability

The code and data that support the findings of this study are openly available in Zenodo ( https://doi.org/10.5281/zenodo.10803802 ).

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Acknowledgements

We thank Ting Qing and Jijun He for the fruitful discussion on OVNA, Yuan Chen, Zhiyang Chen, and Huamin Zheng for assistance in the experiment, and Lan Gao for taking the sample photos. J. Liu is indebted to Dapeng Yu who provided critical support to this project. J. Liu acknowledges support from the National Natural Science Foundation of China (Grant No.12261131503), Innovation Program for Quantum Science and Technology (2023ZD0301500), Shenzhen-Hong Kong Cooperation Zone for Technology and Innovation (HZQB-KCZYB2020050), and the Guangdong Provincial Key Laboratory (2019B121203002). Y.-H.L. acknowledges support from the China Postdoctoral Science Foundation (Grant No. 2022M721482).

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These authors contributed equally: Yi-Han Luo, Baoqi Shi.

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International Quantum Academy, 518048, Shenzhen, China

Yi-Han Luo, Baoqi Shi, Wei Sun, Ruiyang Chen, Sanli Huang, Zhongkai Wang, Jinbao Long, Chen Shen & Junqiu Liu

Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology, 518055, Shenzhen, China

Yi-Han Luo & Ruiyang Chen

Department of Optics and Optical Engineering, University of Science and Technology of China, 230026, Hefei, China

Hefei National Laboratory, University of Science and Technology of China, 230088, Hefei, China

Sanli Huang & Junqiu Liu

Qaleido Photonics, 518048, Shenzhen, China

Key Laboratory of Specialty Fiber Optics and Optical Access Networks, Shanghai University, 200444, Shanghai, China

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Y.-H.L., B.S., W.S., Z.W. and J. Long built the experimental setup with assistance and advice from H.G. Y.-H.L., B.S., W.S. and R.C. performed the frequency-calibration. B.S., Y.-H.L., W.S. and S.H. performed the experiments on VSA applications. C.S. and Z.Y. fabricated the silicon nitride chips. Y.-H.L., B.S. and J. Liu analyzed the data and prepared the manuscript with input from others. J. Liu supervised the project.

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Stability analysis of Caputo fractional time-dependent systems with delay using vector lyapunov functions

• Jonas Ogar Achuobi 1,2 ,  ,  , 
• Edet Peter Akpan 1 , 
• Reny George 3 ,  ,  , 
• Austine Efut Ofem 4
• 1. Department of Mathematics, Akwa-Ibom State University, Ikot Akpaden, Akwa Ibom State, Nigeria
• 2. Department of Mathematics, University of Calabar, Calabar, Nigeria
• 3. Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharaj 11942, Saudi Arabia
• 4. School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa
• Received: 23 July 2024 Revised: 11 September 2024 Accepted: 20 September 2024 Published: 27 September 2024

MSC : 34A08, 34A34, 34D20, 34K37

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In this study, we investigate the stability and asymptotic stability properties of Caputo fractional time-dependent systems with delay by employing vector Lyapunov functions. Utilizing the Caputo fractional Dini derivative on Lyapunov-like functions, along with a new comparison theorem and differential inequalities, we derive and prove sufficient conditions for the stability and asymptotic stability of these complex systems. An example is included to showcase the method's practicality and to specifically illustrate its advantages over scalar Lyapunov functions. Our results improves, extends, and generalizes several existing findings in the literature.

• stability ,
• asymptotic stability ,
• Caputo derivative ,
• vector Lyapunov function ,
• fractional delay differential equation

Citation: Jonas Ogar Achuobi, Edet Peter Akpan, Reny George, Austine Efut Ofem. Stability analysis of Caputo fractional time-dependent systems with delay using vector lyapunov functions[J]. AIMS Mathematics, 2024, 9(10): 28079-28099. doi: 10.3934/math.20241362

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• Figure 1. Plot of $ u(t) = 2+10(t-2)^{\frac{1}{\Gamma(\alpha)}}u(t-2), \, \alpha = 0.2, 0.4, 0.6\, \text{and}\, 0.8 $

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Speaker 1: In this video, we're going to look at the ever popular qualitative analysis method, thematic analysis. We'll unpack what thematic analysis is, explore its strengths and weaknesses, and explain when and when not to use it. By the end of the video, you'll have a clearer understanding of thematic analysis so that you can approach your research project with confidence. By the way, if you're currently working on a dissertation or thesis or research project, be sure to grab our free dissertation templates to help fast-track your write-up. These tried and tested templates provide a detailed roadmap to guide you through each chapter, section by section. If that sounds helpful, you can find the link in the description down below. So, first things first, what is thematic analysis? Well, as the name suggests, thematic analysis, or TA for short, is a qualitative analysis method focused on identifying patterns, themes, and meanings within a data set. Breaking that down a little, TA involves interpreting language-based data to uncover categories or themes that relate to the research aims and research questions of the study. This data could be taken from interview transcripts, open-ended survey responses, or even social media posts. In other words, thematic analysis can be used on both primary and secondary data. Let's look at an example to make things a little more tangible. Assume you're researching customer sentiment toward a newly launched product line. Using thematic analysis, you could review open-ended survey responses from a sample of consumers looking for similarities, patterns, and categories in the data. These patterns would form a foundation for the development of an initial set of themes. You'd then reduce and synthesize these themes by filtering them through the lens of your specific research aims until you have a small number of key themes that help answer your research questions. By the way, if you're not familiar with the concept of research aims and research questions, be sure to check out our primer video covering that. Link in the description. Now that we've defined what thematic analysis is, let's unpack the different forms that TA can take, specifically inductive and deductive. Your choice of approach will make a big difference to the analysis process, so it's important to understand the difference. Let's take a look at each of them. First up is inductive thematic analysis. This type of TA is completely bottom-up, inductive in terms of approach. In other words, the codes and themes will emerge exclusively from your analysis of the data as you work through it rather than being determined beforehand. This makes it a relatively flexible approach as you can adjust, remove, or add codes and themes as you become more familiar with your data. For example, you could use inductive TA to conduct research on staff experiences of a new office space. In this case, you'd conduct interviews and begin developing codes based on the initial patterns you observe. You could then adjust or change these codes on an iterative basis as you become more familiar with the full data set, following which you develop your themes. By the way, if you're not familiar with the process of qualitative coding, we've got a dedicated video covering that. As always, the link is in the description. Next up, we've got deductive thematic analysis. Contrasted to the inductive option, deductive TA uses predetermined, tightly defined codes. These codes, often referred to as a priori codes, are typically drawn from the study's theoretical framework, as well as empirical research and the researcher's knowledge of the situation. Typically, these codes would be compiled into a codebook where each code would be clearly defined and scoped. As an example, your research might aim to assess constituent opinions regarding local government policy. Applying deductive thematic analysis here would involve developing a list of tightly defined codes in advance based on existing theory and knowledge. Those codes would then be compiled into a codebook and applied to interview data collected from constituents. Importantly, throughout the coding and analysis process, those codes and their descriptions would remain fixed. It's worth mentioning that deductive thematic analysis can be undertaken both individually or by multiple researchers. The latter is referred to as coding reliability TA. As the name suggests, this approach aims to achieve a high level of reliability with regard to the application of codes. By having multiple researchers apply the same set of codes to the same data set, inconsistencies in interpretation can be ironed out and a higher level of reliability can be reached. By the way, qualitative coding is something that we regularly help students with here at Grad Coach, so if you'd like a helping hand with your research project, be sure to check out the link that's down in the description. All right, we've covered quite a lot here. To recap, thematic analysis can be conducted using either an inductive approach where your codes naturally emerge from the data or a deductive approach where your codes are independently or collaboratively developed before analyzing the data. So now that we've unpacked the different types of thematic analysis, it's important to understand the broader strengths and weaknesses of this method so that you know when and when not to use it. One of the main strengths of thematic analysis is the relative simplicity with which you can derive codes and themes and, by extension, conclusions. Whether you take an inductive or a deductive approach, identifying codes and themes can be an easier process with thematic analysis than with some other methods. Discourse analysis, for example, requires both an in-depth analysis of the data and a strong understanding of the context in which that data was collected, demanding a significant time investment. Flexibility is another major strength of thematic analysis. The relatively generic focus on identifying patterns and themes allows TA to be used on a broad range of research topics and data types. Whether you're undertaking a small sociological study with a handful of participants or a large market research project with hundreds of participants, thematic analysis can be equally effective. Given these attributes, thematic analysis is best used in projects where the research aims involve identifying similarities and patterns across a wide range of data. This makes it particularly useful for research topics centered on understanding patterns of meaning expressed in thoughts, beliefs, and opinions. For example, research focused on identifying the thoughts and feelings of an audience in response to a new ad campaign might utilize TA to find patterns in participant responses. All that said, just like any analysis method, thematic analysis has its shortcomings and isn't suitable for every project. First, the inherent flexibility of TA also means that results can at times be kind of vague and imprecise. In other words, the broad applicability of this method means that the patterns and themes you draw from your data can potentially lack the sensitivity to incorporate text and contradiction. Second is the problem of inconsistency and lack of rigor. Put another way, the simplicity of thematic analysis can sometimes mean it's a little too crude for more delicate research aims. Specifically, the focus on identifying patterns and themes can lead to results that lack nuance. For example, even an inductive thematic analysis applied to a sample of just 10 participants might overlook some of the subtle nuances of participant responses in favor of identifying generalized themes. It could also miss fine details in language and expression that might reveal counterintuitive but more accurate implications. All that said, thematic analysis is still a useful method in many cases, but it's important to assess whether it fits your needs. So think carefully about what you're trying to achieve with your research project. In other words, your research aims and research questions. And be sure to explore all the options before choosing an analysis method. If you need some inspiration, we've got a video that unpacks the most popular qualitative analysis methods. Link is in the description. If you're enjoying this video so far, please help us out by hitting that like button. You can also subscribe for loads of plain language actionable advice. If you're new to research, check out our free dissertation writing course, which covers everything you need to get started on your research project. As always, links in the description. Okay, that was a lot. So let's do a quick recap. Thematic analysis is a qualitative analysis method focused on identifying patterns of meaning as themes within data, whether primary or secondary. As we've discussed, there are two overarching types of thematic analysis. Inductive TA, in which the codes emerge from an initial review of the data itself and are revised as you become increasingly familiar with the data. And deductive TA, in which the codes are determined beforehand based on a combination of the theoretical and or conceptual framework, empirical studies, and prior knowledge. As with all things, thematic analysis has its strengths and weaknesses and based on those is generally most appropriate for research focused on identifying patterns in data and drawing conclusions in relation to those. If you liked the video, please hit that like button to help more students find this content. For more videos like this one, check out the Grad Coach channel and make sure you subscribe for plain language, actionable research tips and advice every week. Also, if you're looking for one-on-one support with your dissertation, thesis, or research project, be sure to check out our private coaching service where we hold your hand throughout the research process step by step. You can learn more about that and book a free initial consultation at gradcoach.com.

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Genetic Genealogy: DNA and Family History

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Genetic genealogy creates family history profiles (biological relationships between or among individuals) by using DNA test results in combination with traditional genealogical methods. By using genealogical DNA testing, genetic genealogy can determine the levels and types of biological relationships between or among individuals.

This branch of genetics became popular in recent years, as costs were drastically reduced and genealogical studies using molecular techniques became accessible to the general public. Advantages of including DNA, as opposed to traditional genealogical research alone, include the ability for researchers to extend their ancestry beyond the paperwork of recent centuries, and to construct ancient pedigrees through molecular evolutionary studies. Genealogists also use DNA to solve mysteries in their immediate families, such as to discover biological parents of adoptees or to determine the accurate male ancestor in a non-paternity event (NPE).

Autosomal, Mitochondrial, and Y-DNA: The Three DNA Tests Used by Genealogists

There are three sources of information in a DNA sample. Y-chromosomal DNA (Y-DNA) is present only in samples from males and gives information on patrilineal descent. Mitochondrial DNA (mtDNA), present in both male and females, gives information on matrilineal descent. Finally, autosomal DNA (atDNA) gives information on both matrilineal and patrilineal descent.

The signal of shared ancestry seen in autosomal DNA is highest in close relatives, but dilutes quickly so that by 5-7 generations of separation, it is difficult to distinguish exact relationships other than shared ethnic affinities. Thus, autosomal DNA (atDNA) is best to help identify ancestors within the most recent 5–7 generations of a family tree.

MtDNA and Y-DNA tests are limited to relationships along a strict female line and a strict male line, respectively. mtDNA evolves rapidly whereas Y-DNA (and atDNA) changes much more slowly. MtDNA and Y-DNA tests are utilized to identify archeological cultures and migration paths of a person's ancestors along a strict mother's line or a strict father's line. Based on MtDNA and Y-DNA, a person's haplogroup(s) can be identified. (A haplogroup is DNA or Chromosomal segments derived from a group of people who share a common genetic ancestor). The mtDNA test can be taken by both males and females, because everyone inherits their mtDNA from their mother, as the mitochondrial DNA is located in the egg cell. However, a Y-DNA test can only be taken by a male, as only males have a Y-chromosome.

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Fiber-reinforced recycled aggregate concrete (FR-RAC) has recently gained more popularity because of its advantages, high strength, eco-friendliness, and cost-effectiveness. This study uses an advanced machine-learning technique for forecasting the compressive strength of FR-RAC. In this study, an experimental database that contained pertinent data from several previous research was evaluated to train and test using machine learning (ML) techniques and models. To accurately represent the subtle interactions within the dataset, the multivariate analysis identifies and includes essential factors that impact the complicated behavior of FR-RAC in the model. This study presents a hybrid ML model for predicting concrete’s compressive strength by combining several machine learning algorithms in a novel way. To predict the reliability of machine learning models, several algorithms, such as adaptive boosting regressor, support vector regressor, KNN regressor, gradient boosting, and random forest, were developed to help find the interrelated behaviors of parameters. Among all the models used in this study, the Light Gradient-Boosting Machine (GBM) outperforms (R 2  = 0.90) other models, each of which was fitted to a different portion of the training dataset. Additionally, the SHAP analysis revealed that recycled coarse aggregate has an inverse impact on the strength of FR-RAC. Overall, the outcomes of this study can significantly contribute to cost and material reduction by predicting the compressive strength of FR-RAC without the need for extensive laboratory testing and promoting more efficient use of resources.

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Alsharari, F. Predicting the compressive strength of fiber-reinforced recycled aggregate concrete: A machine-learning modeling with SHAP analysis. Asian J Civ Eng (2024). https://doi.org/10.1007/s42107-024-01183-w

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