A Formula for the number of reflections

[kwon0504]

Is there any formula for the number of reflections? I think I might have come up with a formula that identifies the number of lens reflections to a patient's eye. I am, however, not sure if there is an existed formula already. If there is, please let me know. Thank you.

[Laurie]

Hi There,

You may be thinking of "Fresnel's Equation", which is the formula to determine how much light is loss due to 'reflectance factor' in various materials (Ex., the higher the index of refraction, the higher the reflectance factor).

Fresnel's equation:

p = n-1/n+1, squared, times 100.

The higher the index (n), the higher the reflectance factor (p). This is why it is so important for high index lenses to be coupled with AR coating, to recapture the light loss due to reflectance.

: )

Laurie

[kwon0504]

Thank you, Laurie, but that is not what I have in mind.
I am looking for a formula to calculate the number of reflections. For example, how many reflections a patient may have while wearing a pair of glasses, or glasses with clip-ons. There are five types of reflections that reaches the patient's eye when wearing a pair of glasses. However, the number of reflections increases when he wears clip-ons, for example. Is there a formula to calculate the number of reflections as described in my example?

Sung

[HarryChiling]

Thank you, Laurie, but that is not what I have in mind.
I am looking for a formula to calculate the number of reflections. For example, how many reflections a patient may have while wearing a pair of glasses, or glasses with clip-ons. There are five types of reflections that reaches the patient's eye when wearing a pair of glasses. However, the number of reflections increases when he wears clip-ons, for example. Is there a formula to calculate the number of reflections as described in my example?

Sung

Are you sure you are not looking for a scientific answer to an anecdotal problem?

[OPTIDONN]

I would think the intensity of the reflections would be more important to figure out rather than the number of reflections.

[Darryl Meister]

For a single lens, there are 5 unique reflections that are observed by the actual wearer. As you increase the number of lenses, you also increase the number of reflections observed by the wearer accordingly -- probably in an exponential fashion due to reflections between the two lenses, though the intensity of these "higher-order" reflections will be very small.

[kwon0504]

Are you sure you are not looking for a scientific answer to an anecdotal problem?

Thank you for your interest, Harry.
I am wondering if there is a formula that can determine the number of reflections to the eye. When a wearer has clip-ons or inserted lenses with safty glasses there is an increased number of reflections. For example, when a person wears glasses, there are five reflections, but with clip-ons, the number of reflections increases to 14 (for non-polarized lenses). A formula that can determine the number of reflections could have the potential to determine the intensity of reflections with accuracy.

[kwon0504]

I would think the intensity of the reflections would be more important to figure out rather than the number of reflections.

The formula I have in mind will determine the number of reflections with greater ease and increase the accuracy of the intensity of reflections.

[kwon0504]

For a single lens, there are 5 unique reflections that are observed by the actual wearer. As you increase the number of lenses, you also increase the number of reflections observed by the wearer accordingly -- probably in an exponential fashion due to reflections between the two lenses, though the intensity of these "higher-order" reflections will be very small.

When the number of lenses increase, the number of reflections also increase, both first order and second order. The third order or higher is not considered. One pair of glasses has five different reflections. The first order reflections are 4.26% and 3.90%. The second order reflections are 0.17%, 0.09%, and 0.08%. The total intensity of reflections is 8.50%. However, with clip-ons, for example, the total number of reflections increases to 14 (for non-polarized lenses), 4 first orders and 10 second orders. The intensity of reflections with clip-ons is much higher than with a single lens. The formula I have in mind will determine the number of reflections with greater ease and increase the accuracy of the intensity of reflections.

Do you know if such a formula already exists?

[Darryl Meister]

I can't say that such a formula already exists, but someone with a sufficient mathematical aptitude could probably derive one with a little effort (as you may very well have already done). For instance, if you "solve" the problem for the case of two and/or three lenses, by determining each possible reflection, you could probably arrive at a mathematical series that solves the problem for any arbitrary number of lenses.

Do you have a mathematical expression that you would like to share with the group?

[Fezz]

Wouldn't index, thickness, space between each surface, curves, etc all add to the complexity of such a formula?

[Darryl Meister]

Refractive index would affect the intensity of the reflections, but not really the number of reflections. Surface curves and lens thickness would affect the size and clarity of the reflections, but not really the intensity or number of reflections.

[Fezz]

I understand, but, wouldn't they have to be taken into consideration?

Or, is it more a phenom of perception?

[Darryl Meister]

Yes, they would matter to the wearer, but if he is only referring to determining the number of reflections, which seems to be the case, those other factors wouldn't really matter.

[Fezz]

Thanks Darryl.

[rinselberg]

Is there any formula for the number of reflections? I think I might have come up with a formula that identifies the number of lens reflections to a patient's eye ...

Something to consider aside from a specific optical formula: The "brute force" of computerized Finite Element modeling and/or Monte Carlo simulations of photons. Or to put it in my own (humble) words, the world's smartest and most versatile ray-tracing software. Starting point:

Monte Carlo Methods and the Challenge of Photo-Realism in Computer Graphics
Steven Collins; Image Synthesis Group; Trinity College; Dublin.




The only props (above) are numbers: This image is 100 percent computer generated using Maxon's Cinema4D professional online animation software. Look closely: I think that I see eight different specular reflections on the surface of the glass vase. Source: Creating Glass Material for Cinema4D.

Idea: kwon0504 (and anyone else interested in this topic) might try their hand with an online animation package like this Cinema4D. Would you be able to model a prescription eyeware frame with lenses and then add clip-ons in front of or behind the lenses? Would it generate an accurate number of reflections - or any reflections at all? Just for grins ... take a quick look at this Cinema4D Tutorials page.

Anyone could solve this problem using a high end computer, a PhD level of expertise in optical engineering and computer graphics and - most of the rest of their natural life span.


"You're gonna need a bigger retinascope!" Meet the world's biggest and baddest pair of eyeballs ...

[Darryl Meister]

anyone could solve this problem using a high end computer, a PhD level of expertise in optical engineering and computer graphics and - most of the rest of their natural life span.

Or a junior-level education in college mathematics... ;)

[rinselberg]

Is there any formula for the number of reflections? I think I might have come up with a formula that identifies the number of lens reflections to a patient's eye ...

Wouldn't it be necessary to specify the ambient lighting conditions to make any sense of this? Such as the location, spectral characteristics and other relevant optical criteria for every significant source of light? Or is there an implicit assumption (above) that we are talking about diffuse sunlight from the sky or from indirect lighting indoors?

If my posts here seem not to make sense - OK. Please don't beat me up (too hard). But I am trying to get my head around the idea that there would be some significance or advantage to deriving the number of all possible reflection types (which is what I think everyone else has been talking about) without going all the way to the end and specifying an exact, realistic scenario with definite ambient lighting conditions and then using a computer to generate a photo-realistic image (like the one I posted above) with an accurate presentation of how the reflections would actually appear to someone who is behind the lenses and clip-ons.

If it seems that I am just "way out in left field" about this - no harm.

Just scroll right past my "nonsense".


"You're gonna need a bigger retinascope!" Meet the world's biggest and baddest pair of eyeballs ...

[kwon0504]

Something to consider aside from a specific optical formula: The "brute force" of computerized Finite Element modeling and/or Monte Carlo simulations of photons. Or to put it in my own (humble) words, the world's smartest and most versatile ray-tracing software. Starting point:

Monte Carlo Methods and the Challenge of Photo-Realism in Computer Graphics
Steven Collins; Image Synthesis Group; Trinity College; Dublin.




The only props (above) are numbers: This image is 100 percent computer generated using Maxon's Cinema4D professional online animation software. Look closely: I think that I see eight different specular reflections on the surface of the glass vase. Source: Creating Glass Material for Cinema4D.

Idea: kwon0504 (and anyone else interested in this topic) might try their hand with an online animation package like this Cinema4D. Would you be able to model a prescription eyeware frame with lenses and then add clip-ons in front of or behind the lenses? Would it generate an accurate number of reflections - or any reflections at all? Just for grins ... take a quick look at this Cinema4D Tutorials page.

Anyone could solve this problem using a high end computer, a PhD level of expertise in optical engineering and computer graphics and - most of the rest of their natural life span.


"You're gonna need a bigger retinascope!" Meet the world's biggest and baddest pair of eyeballs ...

very intriguing... as a mechanical engineer and an optician...

[HarryChiling]

Lens elements, groups, reflections, and flare— some geeky math fun

Lens flare is closely related to the number of secondary reflections in a lens, i.e., light entering the lens that is reflected off one lens surface, then another, then back to the image plane. For uncoated air-to-glass surfaces, about 4% of the incident light is reflected. Simple coating reduces the reflection to around 2%; muti-coating reduces it to 1% or less. For a simple coating, the amount of the secondary reflection is 0.02 * 0.02 = 0.0004, which doesn't seem like much until you calculate the number of reflections.

A lens consists of N elements in M groups, where a group may consist of several elements cemented together. It's the number of groups M— actually the number of air-to-glass surfaces 2M— that really counts. The first surface has no secondary reflection. The second surface has 1: light that bounces off the second surface, then the first, then back to the image plane. Continuing with this reasoning, we see that the m(th) surface has m-1 secondary reflections, i.e.,

Total reflections = R = (2M-1) + (2M-2) + (2M-3) + ... + 1 = M (2M-1) = 2M²-M
If you add a filter to a lens with M groups, you increase the number of reflections R by 4M+1.

Groups
M
1
2
3
4
5
6
7
8
9
10
11
12
13
14Reflec-
tions R
1
6
15
28
45
66
91
120
153
190
231
276
325
378 Zooms typically have more elements than primes. Examples (easy to locate in the Canon Museum): The Canon 24-105mm f/4L IS USM has 18 elements in 13 groups. The Canon 50mm f/1.4 USM has 7 elements in 6 groups. The Canon 90mm f/2.8 TS-E has 6 elements in 5 groups. Now you know one reason primes are still used (others are large apertures, light weight, and excellent sharpness (MTF)), as well as why some photographers avoid using filters (though a UV filter is useful protection for field work).

This might help.