Overlap Integrals: Understand & Learn from a Source

In summary, the conversation discusses the concept of "overlap integral" in optics. This expression is used to calculate how much light from a focused beam will go into a fiber. It is a correlation of the light field and the mode field of the fiber. The conversation also explains the mathematical process of calculating the overlap integral using normalized functions and integration. Additionally, the conversation touches on the importance of using field coupling and taking the absolute square of the expression to represent intensity.
  • #1
Ahmed123
6
1
Hi
I study optics and many times i found a term called (Overlap integral ) as attached pictures .. I can't understand from where these expression comes (mathematically) and what these functions means in particular ( even from mathematical point of view)
I can't understand the nature of multiplying the two functions then integrating or making the integrals of individual functions then multiplying.. So I ask if anyone can help me to find a source for reading to understand .. or explain these expressions for me .. thank you in advance .
 

Attachments

  • (1).png
    (1).png
    86.5 KB · Views: 627
  • (2).png
    (2).png
    26.4 KB · Views: 749
Physics news on Phys.org
  • #2
If you have light focused down so it is imaged into a small spot on the tip of a fiber, this expression answers the question “how much of the light will go into the fiber?” (However this isn’t the only place in physics you will find an “overlap integral”). This function is a correlation of how well the field of the imaged spot (“light field”) overlaps the shape of the field the fiber can propagate (“mode field”).

The easiest way to describe it is to start with something simpler. Suppose you had a beam of light that made a perfectly circularly uniform spot on the wall. (Uniform intensity “top hat” profile). And suppose you had a circular window. In this scenario, all the light that hits the window goes through. If the circular beam partially overlaps the circular window, part of the beam goes through.
Mathematically how would you calculate how much gets through? One way to do it would be to write a function that describes the beam in some coordinate system: a constant value inside the circle, zero outside. Normalize this to one (divide by the integral of the function). Now in the same coordinates describe the window with a similar normalized function. If you multiply the two functions together and integrate you get the fraction of light that goes through the window. It’s how much gets through because the multiplied function is nonzero only where both functions are nonzero. (the “overlap”) It’s the fraction because the two functions are normalized to 1.
Of course you might have come up with a simpler geometric expression for that particular case, but this recipe is more generally useful. Suppose the beam isn’t uniform, but has a Gaussian intensity distribution. Same recipe: describe the intensity shape, normalize, multiply by the normalized window shape and integrate. Finally suppose the window isn’t completely transparent everywhere but has some pattern of transparency. You still use the same recipe.

One caveat, the simplified description above contemplates using the intensity throughout, but, really, when it comes to mode coupling, you overlap the fields. The intensity is proportional to the square of the field, and that is why your expression has all those conjugate squares.

So now let’s apply this to your expression. The light incident on the fiber tip has some field profile. Integrating the conjugate square of the field gives the total intensity. Dividing the field by the square root of the integrated conjugate square gives the normalized field. The fiber has a mode distribution that it can propagate. Dividing the description of the mode field by its integrated conjugate square gives the normalized mode field. Multiplying the two fields gives the field coupling: how much of the incident field couples into the fiber mode. Taking the absolute square of the whole expression puts this in terms of coupled intensity. The normalizers are real, so they just get squared and are once again intensities rather than fields.
 
  • Like
  • Love
Likes Bart Bas, Nazon, Ahmed123 and 1 other person
  • #3
Cutter Ketch said:
If you have light focused down so it is imaged into a small spot on the tip of a fiber, this expression answers the question “how much of the light will go into the fiber?” (However this isn’t the only place in physics you will find an “overlap integral”). This function is a correlation of how well the field of the imaged spot (“light field”) overlaps the shape of the field the fiber can propagate (“mode field”).

The easiest way to describe it is to start with something simpler. Suppose you had a beam of light that made a perfectly circularly uniform spot on the wall. (Uniform intensity “top hat” profile). And suppose you had a circular window. In this scenario, all the light that hits the window goes through. If the circular beam partially overlaps the circular window, part of the beam goes through.
Mathematically how would you calculate how much gets through? One way to do it would be to write a function that describes the beam in some coordinate system: a constant value inside the circle, zero outside. Normalize this to one (divide by the integral of the function). Now in the same coordinates describe the window with a similar normalized function. If you multiply the two functions together and integrate you get the fraction of light that goes through the window. It’s how much gets through because the multiplied function is nonzero only where both functions are nonzero. (the “overlap”) It’s the fraction because the two functions are normalized to 1.
Of course you might have come up with a simpler geometric expression for that particular case, but this recipe is more generally useful. Suppose the beam isn’t uniform, but has a Gaussian intensity distribution. Same recipe: describe the intensity shape, normalize, multiply by the normalized window shape and integrate. Finally suppose the window isn’t completely transparent everywhere but has some pattern of transparency. You still use the same recipe.

One caveat, the simplified description above contemplates using the intensity throughout, but, really, when it comes to mode coupling, you overlap the fields. The intensity is proportional to the square of the field, and that is why your expression has all those conjugate squares.

So now let’s apply this to your expression. The light incident on the fiber tip has some field profile. Integrating the conjugate square of the field gives the total intensity. Dividing the field by the square root of the integrated conjugate square gives the normalized field. The fiber has a mode distribution that it can propagate. Dividing the description of the mode field by its integrated conjugate square gives the normalized mode field. Multiplying the two fields gives the field coupling: how much of the incident field couples into the fiber mode. Taking the absolute square of the whole expression puts this in terms of coupled intensity. The normalizers are real, so they just get squared and are once again intensities rather than fields.
Woooow ,, then these denominator comes from normalization of these functions !
I can't describe how your answer is great and helpful ..
 
  • Like
Likes Cutter Ketch
  • #4
Hi, can you provide a reference for the first image you provided? Thank you so much for your help
 

FAQ: Overlap Integrals: Understand & Learn from a Source

What are overlap integrals?

Overlap integrals are mathematical calculations used to determine the degree of overlap between two atomic orbitals. They are an important concept in quantum mechanics and are used to understand the bonding between atoms in molecules.

How are overlap integrals calculated?

Overlap integrals are calculated using the wave functions of the two orbitals being considered. These wave functions are multiplied together and integrated over all space to give a numerical value representing the degree of overlap between the two orbitals.

What is the significance of overlap integrals?

Overlap integrals play a crucial role in understanding the bonding between atoms in molecules. They provide information about the strength and type of bonding, such as whether it is covalent or non-covalent. They also help in predicting the physical and chemical properties of molecules.

How do overlap integrals affect molecular geometry?

The value of overlap integrals can impact the shape and orientation of molecules. In molecules with high overlap integrals, the orbitals are more likely to overlap, resulting in a more compact and stable molecular geometry. On the other hand, molecules with low overlap integrals may have a more distorted or less stable geometry.

Can overlap integrals be used to compare different molecules?

Yes, overlap integrals can be used to compare the bonding strength and stability of different molecules. By calculating the overlap integrals for different molecules, scientists can gain insight into the similarities and differences in their bonding and predict their properties and reactivity.

Similar threads

Replies
1
Views
2K
Replies
5
Views
1K
Replies
9
Views
3K
Replies
14
Views
1K
Replies
3
Views
2K
Replies
5
Views
2K
Back
Top