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Divergence and Curl

Divergence and Curl

The divergence of a vector field at a point is the net outward flow of the flux out of a small volume through a closed surface enclosing the volume, as the volume tends to zero. More practically, divergence measures how much a vector field spreads out or diverges from a given point. For an outgoing flux, conventionally, the divergence is positive. The diverging point is often called the "source" of the field. Meanwhile, the negative divergence of a vector field at a point means that the vector...

Divergence and Stokes' Theorems

Divergence and Stokes' Theorems

The divergence and Stokes' theorems are a variation of Green's theorem in a higher dimension. They are also a generalization of the fundamental theorem of calculus. The divergence theorem and Stokes' theorem are in a way similar to each other; The divergence theorem relates to the dot product of a vector, while Stokes' theorem relates to the curl of a vector. Many applications in physics and engineering make use of the divergence and Stokes' theorems, enabling us to write...

Gene Duplication and Divergence

Gene Duplication and Divergence

The seminal work of Ohno in 1970 popularized the idea of gene duplication and divergence. DNA sequence comparison studies reveal that a large portion of the genes in bacteria, archaebacteria, and eukaryotes was generated by gene duplication and divergence, indicating its critical role in evolution.
The duplicated copies of the gene are called Paralogs. Paralogs with similar sequences and functions form a gene family. Across several species, a large number of gene families are...

The Wave Nature of Light

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The nature of light has been a subject of inquiry since antiquity. In the seventeenth century, Isaac Newton performed experiments with lenses and prisms and was able to demonstrate that white light consists of the individual colors of the rainbow combined together. Newton explained his optics findings in terms of a "corpuscular" view of light, in which light was composed of streams of extremely tiny particles traveling at high speeds according to Newton's laws of motion.

Divergence and Curl of Electric Field

Divergence and Curl of Electric Field

The divergence of a vector is a measure of how much the vector spreads out (diverges) from a point. For example, an electric field vector diverges from the positive charge and converges at the negative charge. The divergence of an electric field is derived using Gauss's law and is equal to the charge density divided by the permittivity of space. Mathematically, it is expressed as

Divergence and Curl of Magnetic Field

Divergence and Curl of Magnetic Field

The magnetic field due to a volume current distribution given by the Biot–Savart Law can be expressed as follows:

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Related Experiment Video

Updated: Feb 13, 2026

Continuous-wave Thulium Laser for Heating Cultured Cells to Investigate Cellular Thermal Effects

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Published on: June 30, 2017

Continuous-wave highly-efficient low-divergence terahertz wire lasers.

Simone Biasco¹, Katia Garrasi¹, Fabrizio Castellano¹

¹NEST, CNR-Istituto Nanoscienze and Scuola Normale Superiore, Piazza San Silvestro 12, 56127, Pisa, Italy.

Nature Communications

|March 18, 2018

Summary

New wire distributed feedback (DFB) terahertz (THz) quantum cascade lasers (QCLs) achieve stable continuous-wave operation with low-divergent beams and single-mode emission, overcoming previous fabrication challenges.

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Area of Science:

Optics and Photonics
Semiconductor Devices
Quantum Electronics

Background:

Terahertz (THz) quantum cascade lasers (QCLs) offer high power and broad tunability but achieving continuous-wave (CW) operation with controlled beam profiles and spectral output is challenging.
Existing lithographic patterning methods for THz QCLs, such as distributed feedback (DFB) lasers, photonic crystals, and microcavities, often require demanding fabrication processes to optimize beam divergence or emission frequency.

Purpose of the Study:

To develop a novel architecture for THz QCLs that enables simultaneous optimization of beam divergence and spectral control.
To achieve stable continuous-wave (CW) operation with improved beam quality and precise frequency selection in THz QCLs.

Main Methods:

Demonstration of wire DFB THz QCLs utilizing a sinusoidal corrugation for frequency feedback and an array of surface holes for light extraction.
Fabrication of a new THz QCL architecture extendable to various far-infrared frequencies.

Main Results:

Achieved low-divergent beams with an angle of 10°.
Demonstrated single-mode emission.
Obtained high slope efficiencies of 250 mW/A.
Confirmed stable continuous-wave (CW) operation.

Conclusions:

The novel wire DFB THz QCL architecture effectively addresses the challenges of achieving controlled beam divergence and spectral output.
This design offers a promising pathway for realizing high-performance THz QCLs for various applications requiring stable CW operation and precise spectral control.