What Happens When Four Coherent Sources of Intensity I Are Superimposed?Understanding how light waves behave when multiple sources are involved can be both fascinating and essential, especially in the field of wave optics. A common question in this area is What happens when four coherent sources, each of the same intensity I, are superimposed? This topic explores the principles of coherence, interference, and intensity, and explains how they apply when four coherent light sources interact.
What Are Coherent Sources?
To begin, it’s important to understand the term coherent sources. Coherent sources are light sources that have a constant phase difference, the same frequency, and ideally the same waveform. When such sources interact, they produce stable and predictable interference patterns.
Coherent sources are crucial in applications like lasers, interference experiments, and holography because they allow constructive and destructive interference to occur in a regular and controlled manner.
The Principle of Superposition
The principle of superposition is a key idea in wave physics. It states that when two or more waves overlap, the resulting wave at any point is the algebraic sum of the individual wave displacements at that point.
When multiple coherent waves are superimposed, their amplitudes add together. The final intensity depends on how these amplitudes combine either constructively or destructively.
Intensity and Amplitude Relationship
Before diving into the effect of four sources, let’s revisit the relationship between intensity and amplitude
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Intensity (I) is proportional to the square of the amplitude (A) of a wave.
That is I propto A^2
So, if we know the intensity, we can find the amplitude by taking the square root of the intensity.
Superimposing Two Coherent Sources
To understand four sources, it helps to start with two. When two coherent sources of equal intensity I are superimposed
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If they interfere constructively, the resultant amplitude becomes A + A = 2A
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Resulting intensity becomes (2A)^2 = 4A^2 , which is 4I
So, constructive interference between two equal sources can result in four times the intensity of a single source.
What Happens with Four Coherent Sources?
Now let’s apply the same idea to four coherent sources, each having intensity I .
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Amplitude of each source = A = sqrt{I}
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If all four waves are perfectly in phase (constructive interference), total amplitude = $4A$
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Resulting intensity = (4A)^2 = 16A^2 = 16I
Therefore, when four coherent sources of equal intensity are superimposed and their waves are perfectly in phase, the maximum intensity observed is 16I.
Constructive and Destructive Interference
The actual observed intensity depends on the relative phase differences between the waves
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Perfect constructive interference leads to maximum intensity (16I)
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Perfect destructive interference, if the waves cancel each other out, leads to minimum or zero intensity
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Partial interference results in intermediate intensities
This is why interference patterns consist of bright and dark fringes. Bright fringes represent regions of constructive interference, and dark fringes indicate destructive interference.
Real-World Factors
In practice, achieving perfect constructive interference from four separate coherent sources is very difficult. Some challenges include
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Phase alignment All sources must maintain fixed phase differences
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Identical frequency and wavelength Required for coherence
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Equal path lengths Necessary to avoid phase shifts that disrupt constructive interference
Even small variations can lead to phase mismatches and reduce the resulting intensity.
Applications of Multiple Coherent Sources
Superimposing multiple coherent light sources has many practical applications in modern science and technology
1. Interferometry
Interferometers use the principle of coherent source superposition to measure tiny changes in distance, temperature, or pressure. The interference pattern generated reveals precise information about the system being studied.
2. Holography
In holography, multiple coherent beams of light are used to create three-dimensional images. The interference of these beams captures depth information not present in ordinary photographs.
3. Optical Communication
Coherent light sources are used in fiber optics for high-speed communication. Combining multiple coherent sources can increase data transmission capacity and signal clarity.
Intensity Distribution and Diffraction Effects
When multiple coherent sources are arranged in space (such as in an array), the interference pattern can show a more complex distribution, depending on
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Distance between sources
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Wavelength of the light
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Observation angle
This is known as diffraction, and the resulting diffraction pattern may consist of multiple bright and dark spots depending on the number and arrangement of sources.
How Does This Differ from Incoherent Sources?
If the four sources were incoherent (random phase differences), they would not produce a stable interference pattern. In that case, the resulting intensity is simply the sum of individual intensities
- Total intensity = I + I + I + I = 4I
This is much lower than the 16I maximum produced by constructive interference from coherent sources. That’s why coherence is so valuable in wave-based technologies.
Summary of Key Points
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Coherent sources have constant phase relationships and produce predictable interference patterns.
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Intensity is proportional to the square of the wave’s amplitude.
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Four coherent sources, when perfectly in phase, can produce a maximum intensity of 16I.
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The actual observed intensity depends on the phase relationships between the waves.
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Coherent wave superposition is fundamental in many optical and physics-based applications.
The superposition of four coherent light sources demonstrates how wave behavior can lead to surprisingly high intensities due to constructive interference. Understanding these principles helps us grasp the deeper nature of light and its applications in science and technology. Whether in research labs or advanced communication systems, the concept of coherent source interference plays a crucial role in how we harness the power of light.