Why Do Cross-Terms Average to Zero for Incoherent Radiation?

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In summary, the conversation discusses the process of proving that the cross-terms in a formula for calculating intensity and energy density average to zero. The person is struggling with understanding the purpose of the cross-terms and how to proceed with a double integral to find the result. They mention taking the dot product of ε(r,t) ^2 and summing up different frequencies or using a Fourier Transform, which results in the cross-terms averaging to zero.
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davon806
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Homework Statement


Please see the attached,this was extracted from my notes.(Left side is the 1st page).I was trying to prove the sentence underlined (on the right page). Given ε(r,t) , how can I prove that the cross-terms average to 0? Actually I don't really understand why there are cross-terms. Would be much appreciated if someone can elaborate the paragraph on right page in detail !
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Homework Equations

The Attempt at a Solution


To calculate the intensity and energy density, using the formula on left page, I suppose we need to find ε(r,t) ^2 by taking its dot product. However,this would give me a double integral which I don't know how to proceed...
 
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If you sum up and have a bunch of different frequencies (or take an F.T. and get a sum of ## E(\omega)exp^{i \omega t} ## over all frequencies), when you square the expression, the terms involving two frequencies (the cross terms) each time average to zero.
 

FAQ: Why Do Cross-Terms Average to Zero for Incoherent Radiation?

What is incoherent radiation?

Incoherent radiation is a type of electromagnetic radiation that is not organized or synchronized in phase. This means that the waves of the radiation do not have a specific relationship to each other, and therefore do not create a stable interference pattern.

What are the characteristics of incoherent radiation?

Incoherent radiation is characterized by its lack of coherence, meaning that the waves are not in sync with each other. It also has a broad range of frequencies and random fluctuations in intensity.

How is incoherent radiation different from coherent radiation?

Coherent radiation is organized and synchronized in phase, meaning that the waves have a specific relationship to each other and can create a stable interference pattern. Incoherent radiation, on the other hand, is not organized and does not create a stable interference pattern.

What is the cross term in incoherent radiation?

The cross term in incoherent radiation refers to the interference between two or more sources of incoherent radiation. This interference is not stable and can result in fluctuations in intensity.

How is incoherent radiation used in scientific research?

Incoherent radiation is commonly used in research to study the properties of materials and their interactions with electromagnetic radiation. It is also used in imaging techniques such as fluorescence microscopy and X-ray imaging.

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