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Related Concept Videos

Vector Representation of Complex Numbers01:16

Vector Representation of Complex Numbers

Complex numbers, represented in Cartesian coordinates, can also be visualized as vectors. These vectors can be expressed in polar form, emphasizing their magnitude and angle. When a complex number is input into a function, the output is another complex number, highlighting the function's zero point from which the vector representation can originate.
Consider a function defined as the product of the complex factors in the numerator divided by the product of the complex factors in the denominator.
Complex Numbers01:29

Complex Numbers

The real number system cannot represent the square root of a negative number, which restricts solutions for certain equations, such as quadratics with negative discriminants. To address this, the complex number system was developed, introducing the imaginary unit i, where i = √(-1). This extension allows for the representation of all roots, including those involving negative radicands.A complex number is written in the form x + yi, where x and y are real numbers. Here, x represents the real...
Reconstruction of Signal using Interpolation01:10

Reconstruction of Signal using Interpolation

Signal processing techniques are essential for accurately converting continuous signals to digital formats and vice versa. When a continuous signal is sampled with a period T, the resulting sampled signal exhibits replicas of the original spectrum in the frequency domain, spaced at intervals equal to the sampling frequency. To handle this sampled signal, a zero-order hold method can be applied, which creates a piecewise constant signal by retaining each sample's value until the next sampling...
Aliasing01:18

Aliasing

Accurate signal sampling and reconstruction are crucial in various signal-processing applications. A time-domain signal's spectrum can be revealed using its Fourier transform. When this signal is sampled at a specific frequency, it results in multiple scaled replicas of the original spectrum in the frequency domain. The spacing of these replicas is determined by the sampling frequency.
If the sampling frequency is below the Nyquist rate, these replicas overlap, preventing the original signal...
Phasor Arithmetics01:13

Phasor Arithmetics

Phasors and their corresponding sinusoids are interrelated, offering unique insights into the behavior of alternating current (AC) circuits. One way to understand this relationship is through the operations of differentiation and integration in both the time and phasor domains.
When the derivative of a sinusoid is taken in the time domain, it transforms into its corresponding phasor multiplied by j-omega (jω) in the phasor domain, where j is the imaginary unit, and ω is the angular frequency.
Time and frequency -Domain Interpretation of Phase-lead Control01:24

Time and frequency -Domain Interpretation of Phase-lead Control

Phase-lead controllers are commonly used in various control systems to enhance response speed and stability. Adjusting the brightness on a television screen offers a practical example of phase-lead control. When contrast is enhanced, a phase-lead controller is employed. Mathematically, phase-lead control is identified when the first parameter is smaller than the second.
The design of phase-lead control involves the strategic placement of poles and zeros to balance steady-state error and system...

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Related Experiment Video

Updated: Jul 10, 2026

Shaping the Amplitude and Phase of Laser Beams by Using a Phase-only Spatial Light Modulator
08:39

Shaping the Amplitude and Phase of Laser Beams by Using a Phase-only Spatial Light Modulator

Published on: January 28, 2019

Pixelated phase computer holograms for the accurate encoding of scalar complex fields.

Victor Arrizón1, Ulises Ruiz, Rosibel Carrada

  • 1Instituto Nacional de Astrofísica, Optica y Electrónica, Apdo. Postal 51 y 216, Puebla PUE 72000, México. arrizon@inaoep.mx

Journal of the Optical Society of America. A, Optics, Image Science, and Vision
|November 3, 2007
PubMed
Summary

Researchers developed phase computer-generated holograms for encoding complex fields. These holograms enable high-quality reconstruction even with low-resolution modulators, offering practical applications in optics.

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Last Updated: Jul 10, 2026

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Area of Science:

  • Optics and Photonics
  • Digital Holography

Background:

  • Computer-generated holograms (CGHs) are crucial for optical information processing.
  • Encoding arbitrary complex fields presents challenges, especially with hardware limitations.

Purpose of the Study:

  • To introduce a class of phase CGHs for arbitrary scalar complex field encoding.
  • To demonstrate high-quality field reconstruction using these holograms with low-resolution modulators.

Main Methods:

  • Discussed a class of phase CGHs.
  • Described two specific hologram designs within this class.
  • Analyzed reconstruction quality with pixelated phase modulators.

Main Results:

  • Achieved high-quality reconstruction of encoded fields.
  • Demonstrated effectiveness even with low-resolution pixelated phase modulators.
  • Showed successful implementation with a phase modulator having reduced phase depth.

Conclusions:

  • The proposed phase CGHs are effective for encoding complex fields.
  • These holograms offer robust performance despite hardware constraints like low resolution and limited phase depth.