What does the 'X' mean in the Mueller matrix rotation equation?

In summary, the conversation is about a question regarding the use of the "X" symbol in a formula for rotation of a polarization element with a Mueller matrix. The author clarifies that the "X" is simply used for matrix multiplication and was used in the equation because it continued onto a second line. The conversation also includes a request for clarification and a response from the author.
  • #1
islahna
7
0
Hi,
I have a basic question on Mueller matrix which I came across upon reading through the Handbook of Optic, chapter 22 polarimetry. It says that :-

when a polarization element with Mueller matrix M is rotated about the beam of light by an angle \theta such that the angle of incident is unchanged, the resulting Mueller matrix M is :

= M(\theta)
= R(\theta) M R(-\theta)
= [matrix elements of R(\theta)] [matrix elements of M] X [matrix elements of R(-theta)]

My question is, what does the "X" mean here..
From my reading, the cross product is used with vector. This time, it's a matrix.

Sorry if this is too basic, I've just started brushing up my matrix since college.

Appreciate any help.

regards,
--islahna
 
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  • #2
Welcome to PF!

Hi islahna! Welcome to PF! :smile:
islahna said:
… when a polarization element with Mueller matrix M is rotated about the beam of light by an angle \theta such that the angle of incident is unchanged, the resulting Mueller matrix M is :

= M(\theta)
= R(\theta) M R(-\theta)
= [matrix elements of R(\theta)] [matrix elements of M] X [matrix elements of R(-theta)]

My question is, what does the "X" mean here..

I don't know what that X is doing there …

in fact, I don't understand what that third line
[matrix elements of R(\theta)] [matrix elements of M] X [matrix elements of R(-theta)]

is supposed to mean at all. :confused:

The first and second lines, effectively M' = RMR-1, are just the standard formula for the effect of rotation R on matrix M …

it's ordinary matrix mulitplication. :smile:
 
  • #3
hi tiny-tim,
thank you so much for the insight. I'll take it as ordinary matrix mulitplication for now.
But I'm still curious, why they use that X sign, has got to be something ..mm.

thanks,
--islahna
 
  • #4
islahna said:
But I'm still curious, why they use that X sign, has got to be something ..mm.

Hi islahna! :smile:

dunno :confused:

i don't have a copy of that book …

can you scan it? :smile:
 
  • #5
hi tiny-tim,

I've contacted the author since and the following is the snippet from his response that I'd like to share.
Hope it clarifies any doubts ..

------- start -------

The x in Eq. 12 & 13 is just matrix multiplication and is there because the equation continued onto a second line. I understand how these little things can be so difficult to those starting who need the information the most.
------- end -----------

thanks you.

--islahna
 
  • #6
islahna said:
I've contacted the author …

Well done! :biggrin:

Duh :rolleyes: … an author trying to avoid one source of confusion by creating another one! :rolleyes:
 

FAQ: What does the 'X' mean in the Mueller matrix rotation equation?

What is a Mueller matrix?

A Mueller matrix is a mathematical representation of the optical response of a material or system. It is a 4x4 matrix that describes how polarized light is transformed after passing through or interacting with the material or system.

What does a rotation in the Mueller matrix represent?

A rotation in the Mueller matrix represents the change in polarization state of light after it has been rotated by a certain angle. This can occur when polarized light passes through a material with an oriented molecular structure or when it is reflected off a surface at a certain angle.

How is the Mueller matrix calculated in the case of a rotation?

The Mueller matrix for a rotation can be calculated by multiplying the Mueller matrix of the material or system by a rotation matrix that represents the rotation angle and direction. The resulting matrix represents the change in polarization state of light after the rotation.

What can be learned from analyzing a Mueller matrix - rotation?

Analyzing a Mueller matrix - rotation can provide information about the optical properties of a material or system, such as its birefringence, dichroism, and depolarization effects. This can be useful in characterizing and understanding the behavior of materials in various applications, such as in optics, remote sensing, and biomedical imaging.

Can Mueller matrix - rotation be used in real-world applications?

Yes, Mueller matrix - rotation has various practical applications, such as in polarimetry for material analysis, in remote sensing for measuring atmospheric or surface properties, and in biomedical imaging for detecting tissue birefringence and anisotropy. It is also used in the design and optimization of optical systems and devices.

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