Talk:Fresnel equations

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Form of equations

There seems to be an incoherence on this page: while in the linked articles the transmission and reflection coeficient are defined as ratios of Amplitudes (in this case of the E-Field), the equations in this article seem to describe a ratio of intensities.

I see two possibilities to correct this:

  • make clear, that the coefficients describe intensities (perhaps remove the links to the other articles or change these):

with the intensities of the incident light and if the reflected light.

  • change the equations to fit Amplitudes (I don't know wether this would correspond to the graphs any more):

with the corresponding amplitudes of the electric field

I would prefer the second possibility since it corresponds to the majority of the literature I have at hand - but I don't have time to check and replace the graphs right now - I'll be back when I have more time and take care of this if nobody else has until then...—The preceding unsigned comment was added by Nanomage (talkcontribs) .

  • This is probably a conflict between the optical usage, which cares mostly about intensity, and general electromagnetic wave usage, which cares more about amplitude. I've clarified that the equations refer to intensity coefficients. (Note: most literature I have uses small r and t for amplitude coefficients, and large R and T for intensity coefficients, if you decide to change it.) -- DrBob 16:48, 8 September 2005 (UTC)

I also think that the equations for the reflection and transmission coefficients of the electric field should be given, further more their relation to the intensities should be emphasized and not only linked to. Some people care about phase, for example when going from low index to high index there is a 180 phase change at reflection, but when going from high index to low there isn't. Also r^2+t^2 is not equal to 1. Unlike the sum of the intensities which take into account the different velocities of the fields in different media. I think that ignoring amplitudes is simpler but will lead to misunderstandings as the texts and many people refer to amplitudes. —Preceding unsigned comment added by Eranus (talkcontribs) 16:16, 21 July 2008 (UTC)

Adding the amplitude form in addition to the intensity form seems fine to me, but it has to be very carefully written. See my comment further down in this section. The formalism needs to be made very clear, since there are incompatible sign conventions and other differences in the mathematical formalism between different presentations of the amplitude equations.--Srleffler (talk) 16:31, 21 July 2008 (UTC)

-Shouldn't also be pointed out that the refractive index used for these equations is a complex number? --an anonymous Raptor 10:00,07/10/05 CET

I think the equation for p-polarization is wrong. From what I've seen in the literature, the following should be the equation:

—The preceding unsigned comment was added by (talkcontribs) .

I haven't checked the math here, but be aware that the form of the equations depends on an arbitrary choice of sign convention. Different authors make different choices. --Srleffler 01:01, 18 February 2006 (UTC)

Why not include the equations for both intensity and amplitude and make clear which is which and where each is used?- 17:06, 7 August 2006 (UTC)

That could work, but the amplitude form is a little more complicated pedagogically. The exact form of the equations depends on a choice of sign convention, and in addition I have seen at least three incompatible mathematical treatments, that essentially define "amplitude" differently. Which treatment is most common depends on what field you're in. The intensity formulism has the advantage of avoiding these difficulties. Differences in sign conventions and definitions often leads to Wiki articles with outright errors, as different editors modify equations without understanding the formalism or realizing that what appears in their textbook might not be correct given the definitions used in the wikipedia article.--Srleffler 00:32, 8 August 2006 (UTC)

Power ratio or intensity ratio? They are differnt!--Antonysigma (talk) 16:55, 1 May 2011 (UTC)

spelling polarise or polarize

All references at my disposal use z not s in all forms. -- 16:52, 7 August 2006 (UTC)

"Polarise" is an accepted British Commonwealth spelling. Wikipedia's Manual of Style prescribes that all varieties of English are accepted, although they should not be mixed in a single article. If a particular subject is not directly tied to a particular country, it stays in whichever variety of English was first used in that article.--Srleffler 00:32, 8 August 2006 (UTC)

Imprecise physics

Some physics in the article are imprecise. Approximations or assumptions made for some formulae are not clearly spelled out (e.g. the paragraph on reflectivity of a dual-surface window and the next paragraph on Fabry-Perot interference directly contradict each other), and other assumptions may be misleading (i.e. mentioning dielectric materials, but neglecting parelectric materials). Contrary what is claimed in the article, Fabry-Perot interferometers cannot be used to create perfect mirrors or reflection-free lenses (causality, i.e. the Kramers-Kronig relations, dictates that there is always some absorption.)

I spent a while editing the article, but Wikipedia lost the edited version on trying to preview it. I'm not sure I want to go through that frustration again, so someone else has to correct the article. —Preceding unsigned comment added by (talkcontribs)

Sorry about the lost edit. That rarely happens, although the wiki has been a bit flaky the last four days or so. When you do get an error screen on "preview" or "save" always try your browser's back button. This generally returns you to the edit screen with no loss of text. Thanks for the suggestions.--Srleffler 19:58, 6 November 2006 (UTC)

This article needs to include the phase shifts associated with reflection and transmission from these surfaces. —Preceding unsigned comment added by (talk) 17:17, 27 November 2009 (UTC)

I think the Fresnel equations should be expressed in terms of the amplitude instead of intensity. This will bring out important features such as the phase shift upon reflection when going from a low to a high index material. —Preceding unsigned comment added by (talk) 19:34, 27 November 2009 (UTC)


Would it be okay to redirect a Fresnel's law page to this page? —The preceding unsigned comment was added by Wk muriithi (talkcontribs) 13:03, 24 January 2007 (UTC).

Sounds good to me. Dicklyon 04:18, 25 January 2007 (UTC)

Metals absorb light?

The article says "For materials which absorb light, like metals and semiconductors, n is complex, and Rp does not generally go to zero." But don't clean metallic surfaces essentially not absorb light? Is this right? —Ben FrantzDale (talk) 21:56, 30 April 2008 (UTC)

It is correct: metals are absorptive. A lot of light is reflected at the surface, but what gets into the metal is absorbed over a very short propagation distance. I think silver and aluminum reflect 90–95% in the visible. The rest is absorbed (assuming the metal is thick enough.)--Srleffler (talk) 23:22, 30 April 2008 (UTC)
Interesting. Obviously copper absorbs in the blue end of the visible spectrum. This leads me to another question: how does mirror reflection work at an atomic scale? That is, when the continuum assumption breaks down, materials can't have a smooth surface yet light behaves as though surfaces are smooth. If it didn't, one couldn't make a lens and a polished crystal, which has atomic-scale star steps between crystal planes, wouldn't reflect like a mirror but would instead reflect like, well, stair steps. —Ben FrantzDale (talk) 01:16, 1 May 2008 (UTC)
Atoms are about 0.1 nm wide. Atomic spacing in solids is maybe a few tenths of a nanometer. Visible light has a wavelength of ~400–700 nm. The light doesn't see roughness below about the 100 nm scale. X-rays have wavelengths comparable to atomic radii and/or lattice spacings, which is why reflecting x-rays off of crystalline solids produces x-ray diffraction.--Srleffler (talk) 01:58, 1 May 2008 (UTC)
Thanks. That's basically what I assumed, but have never had it stated explicitly. Do you know where I would find the quantum-mechanical equations describing light interacting like that with a "bumpy" surface? It seems like one aught to be able to find an "effective surface geometry" for nanoscale-rough surfaces such that physics "acts" like that's where the surface is. —Ben FrantzDale (talk) 11:34, 12 May 2008 (UTC)

Optical coupling

I've heard the term "optical coupling" and "wetting out" to describe what happens when optical films come in contact. I assume this is the transition from a film-air-film sandwich to a film-film sandwich as the air thickness starts to get well below the wavelength of the light. Could someone point me to more information on that? (talk) 14:16, 19 May 2008 (UTC)

It may be related not to the air film being less than a wavelength, but rather to the gap becoming small enough that intermolecular forces (van der Wahls??) pull the surfaces together. This effect is used in optical contacting to bond optical surfaces together. The surfaces are rubbed together with a fluid in between. Surface tension/capillary forces in the fluid pull the surfaces close enough together that van der Wahls or something similar takes over. It can be used to make both temporary and permanent bonds, but I don't know much more about it than that.--Srleffler (talk) 17:10, 22 July 2008 (UTC)
I think the original poster is onto something with regard to this being a wave-like effect. Fyneman has a great description of how light acts in QED in which you add up all possible routes a photon could take; this explains how a photon can appear to "decide" not to reflect off the first surface of an AR coating. As I understand it, optical coupling should work like this: Suppose you have two pieces of glass with a combined four surfaces. If you bring them close together you'll start getting Newton's rings. At one wavelength apart, you'll get constructive interference from the path that bounces off the 3rd surface. At 1/4 wavelength you get destructive interference because light bouncing off the 3rd surface is 180° out of phase with light hitting the 2nd surface. At less than 1/4 wavelength of air gap, I would guess that the two surfaces would begin to get less and less meaningful. I think that means the evanescent wave from would-be total internal reflection somehow connects ("couples"?) to the second piece of glass (3rd surface). The ~1/4-wavelength size scale seems consistent with the "200nm" quoted in total internal reflection fluorescence microscope. This looks relevant: evanescent wave coupling. —Ben FrantzDale (talk) 00:39, 23 July 2008 (UTC)
See also frustrated total internal reflection.--Srleffler (talk) 00:50, 23 July 2008 (UTC)

First picture

Since the whole article deals with intensities, it is confusing that the picture deals with amplitudes (there is no negative intensity) I added the word amplitude in the caption of the article, I think the article requires more info on amplitudes. Eranus (talk) 08:26, 22 July 2008 (UTC)

Changing the caption was a good call. --Srleffler (talk) 00:50, 23 July 2008 (UTC)

Sentence ends in a period?

I don't think this is the case for an equation. I mean, none of the other non-inline equations in this article have period after them.--Zipspeed (talk) 14:02, 17 May 2009 (UTC)

Good point, that I should have followed the style in the rest of the article. Punctuation around equations can be handled in several ways. The style I personally prefer, which is commonly used in physics articles, is to treat the equations as if they were part of the sentence, and punctuate them accordingly. If the equation falls at the end of a sentence, the equation ends with a period. If the sentence continues on after the equation, the equation may be followed by a comma or nothing, depending on the context.
I added the period partly to separate the equation from the reference. I'll space it out instead.--Srleffler (talk) 02:14, 18 May 2009 (UTC)

Accuracy of Fresnel Equations

I'm concerned that there appears to be a sign error in the numerator of the expression for Rp. Hecht (pp 114, 115, 4th ed) and Born and Wolf (p40, 3rd ed) both agree that the form should be:

Rp = n2Cos(θi) - n1cos(θt) / n2cos(θi) - n1cos(θt)

I'll attempt to make the change to make the formula for Rp agree with Hecht and Born and Wolf if I don't get any comments to the contrary.

Patrick (talk) 04:23, 8 June 2009 (UTC)

DO NOT make any changes based simply on sign disagreements with some book. As noted in edit comments in the article, and in the discussion above, there is an arbitrary sign convention involved here. Different authors make different choices for the convention they use, so the exact form of the equations varies between different books, but all are correct as long as you use the definitions of the variables that go with the equations you are using. --Srleffler (talk) 04:39, 8 June 2009 (UTC)
You forgot to square the RHS of that formula. So it doesn't matter what the sign is. OTOH, the lower-case-r formula, which isn't given in the article, does depend on the sign. The formula should be included in the article. One of the sign conventions is overwhelmingly more common for that...the one where r>0 if E doesn't flip directions and B does, and r<0 if E does flip directions and B doesn't. --Steve (talk) 05:08, 8 June 2009 (UTC)
See above. Including formulas for the amplitudes has been discussed before. There are multiple conventions (not merely sign conventions) in common use, in different fields.--Srleffler (talk) 05:41, 8 June 2009 (UTC)

I think someone needs to take a look at the Fresnel equations, with respect to definitions of amplitude coefficients and intensity, as it has the potential to confuse. I think the amplitude coefficents should be quoted, rather than the intensity, and then a note made that the intensity is simply the square of the amplitude. —Preceding unsigned comment added by Nheath555 (talkcontribs) 09:28, 22 October 2009 (UTC)

See above. Treatment of this subject in amplitude is more complicated, because of dependence on arbitrary choices of formalism (including sign conventions).
Note that intensity is not in general simply the square of the amplitude. Whether this is true depends on the formalism chosen.--Srleffler (talk) 16:48, 22 October 2009 (UTC)


I think a history section would be a nice addition. I can't find the original form of Fresnel equations when did Fresnel derive them, when where did he publish them? An hour on Google and Google scholar did not help me much in contrast to History of Young's double slit experiment, Snell's law, Fresnel propagation... which are well documented, I would appreciate any refferance anyone may have.Eranus (talk) 17:48, 28 August 2009 (UTC)

Normal incidence problems

I'm wondering if I'm the first one noticing that the formula with the difference of angles fails at normal incidence:


both resolve to zero for , i.e. normal incidence. This is clearly wrong.

The problem lies in the careless simplification of the correct expression (for , but similar arguments hold for the case of )

using Snell's law . Multiplying the numerator and denominator of with (e.g.) is only allowed if is not zero, which it of course is in the case of normal incidence.

The correct result for normal incidence is given further down on the page. My suggestion is to remove the terms with the difference of angles and just keep the following expressions, i.e. change


and correspondingly for .

Oliver Jennrich (talk) 08:05, 23 April 2010 (UTC)

Saying the formulas "fail" is maybe a bit strong. When I hear "fail", I think "you plug into the formula and get a prediction which is wrong". That's not what happens. Instead you plug into the formula, and get zero divided by zero which is undefined. So I would say the formulas "don't apply" at exactly zero. But anyway it should be clearer for sure than it is now. My preference would be:

(The last expression is valid at non-normal incidence, but at normal incidence it equals 0/0.)

or something like that :-) --Steve (talk) 09:00, 23 April 2010 (UTC)

I removed the offending forms. They weren't really wrong. As Steve notes, they are technically undefined for the case of normal incidence. I haven't checked, but I expect that they are in fact correct in the limit as the angle of incidence goes to zero. Still, they didn't really add much value to the article; better to remove them to avoid confusion. Normal incidence is kind of an important case.--Srleffler (talk) 02:11, 24 April 2010 (UTC)
Fine with me :-) --Steve (talk) 05:55, 24 April 2010 (UTC)
I think they should be put back since they are a lot simpler than the other formulas, and therefore the form one preferably uses. They are not incorrect, just not well defined for normal incidence, and they do converge to the correct values in the limit as the angle of incidence goes to zero. Ulflund (talk) 08:24, 15 September 2011 (UTC)

Reflection coefficients graphic

I suggest, if possible, that the autor of the graphics modifies them by plotting the coefficients for a refraction index of 1.5, more typical af a glass, which is the main material dealt with in the article. --GianniG46 (talk) 07:52, 15 June 2010 (UTC)

Transmittance to/through a medium

I just wanted to direct your attention to a recent disambiguation implemented in transmittance involving Fresnel equations: see here. Thanks. Fgnievinski (talk) 22:30, 9 March 2011 (UTC)

R+T must not equal to 1

There is a sentence in the article:

As a consequence of the conservation of energy, the transmission coefficient in each case is given by Ts = 1 − Rs and Tp = 1 − Rp.[1]

This should be true *only if* R and T are defined in amplitudes.

However, the formula given in the article is in ratio of intensities:


Although conservation of energy still applies (), the areas ,, are not the same. This means R+T should not be equal to 1! (except at normal incidence)

I wish to know what you think about it.--Antonysigma (talk) 16:05, 1 May 2011 (UTC)

The article says "The fraction of the incident power that is reflected from the interface is given by the reflectance R and the fraction that is refracted is given by the transmittance T." So R + T = 1; these are not the intensities of the reflected and refracted light, but the fractions of incident light, which has no area difference effect. Maybe the issue is that transmission coefficient has a slightly different definition than the T here. It would not make sense for amplitudes, nor would it be quite right for intensities, but in the case of R, the incident and reflected beams have the same angle from the normal so they have the same areas, so the intensity ratio is the power ratio as specified. Dicklyon (talk) 16:12, 1 May 2011 (UTC)
Thanks for your reply. So the equation won't be true unless the Fresnel equation is defined in power ratio, not amplitude ratio. Even this, there's still a conflict to formula of in the article, which is defined as intensity ratio. —Preceding unsigned comment added by Antonysigma (talkcontribs) 16:53, 1 May 2011 (UTC)
Actually r + t ≠ 1 for electric field amplitude, because there is a change of dielectric permittivity when you change from one medium to another (Ei ≠ Er + Et). The relation R + T = 1 is correct for total power. It is also correct for intensity at the interface, since then there is no change in area, because the relevant area for all three beams is in the plane of the surface.
It was wrong of you to delete that paragraph without discussing it first. The paragraph is supported by a citation to a reliable source. A reliable source always trumps any one editor's opinion, and usually for good reason.--Srleffler (talk) 01:17, 2 May 2011 (UTC)
There is no conflict. The larger the incidence area is, the smaller the component of the speed perpendicular to that area will be. Dauto (talk) 13:18, 6 May 2011 (UTC)
A wrote an explicit caveat in the article about this source of confusion.Fgnievinski (talk) 08:20, 7 January 2012 (UTC)


The graphs shown are useful, but an index difference of n=1 to n=2 isn't very practical in real world situations. Practical examples with meaning to most readers will be 1/1.33 (air/water) or 1/1.5 (air/glass). Note that most practically common glasses occupy the narrow refractive index range between 1.45 and 1.55. In general, materials with refractive indices above 2 are very uncommon optical materials (though they of course exist). I suggest to redo the plots for 1/1.5 index contrast, if possible. — Preceding unsigned comment added by (talk) 18:00, 14 June 2011 (UTC)

They were made by User:DrBob, who seems to have stopped editing wikipedia two years ago. I agree that it's a worthwhile thing to do. It would take me a while to make it as pretty as DrBob did. I'm too busy now. Maybe someday. --Steve (talk) 19:39, 14 June 2011 (UTC)
I made new graphs similar to the old ones, but with n=1.5. If there are any other suggestions on changes please let me know. Ulflund (talk) 20:04, 20 September 2011 (UTC)
Nice job! :-) --Steve (talk) 02:17, 21 September 2011 (UTC)
Nice. I just toned down the saturation of the lines so they aren't vivid red and blue, but other than that it's great. I've always thought n=1.5 would make more sense as an example. —Ben FrantzDale (talk) 13:01, 21 September 2011 (UTC)
It looks better with those colors. Thanks. Ulflund (talk) 14:39, 21 September 2011 (UTC)
Not at all. When you are dealing with optical thin films there is usually a bigger change in the refractive index, e.g. air-Si (crystalline silicon 3.5), air-silicon monoxide (2.4), air-silicon dioxide,... or SU8 photoresist. --Javiucm (talk) 23:17, 30 October 2013 (UTC)
I disagree with Javiucm. The purpose of that graph is not to tabulate values that professionals can read off the graph and use in their calculations. The purpose is to help typical readers get a better understanding of the fresnel equations. Typical readers have never heard of SU8 or silicon monoxide, but they have seen a lot of air/glass interfaces in their lives. So using that as an example will help those readers more easily relate to the fresnel equations. So again, if we have to pick one example, I think that 1:1.5 is good. Of course we don't have to pick just one example ... maybe there could be another figure showing how the parameters change with different refractive indices. :-D --Steve (talk) 19:28, 31 October 2013 (UTC)


I cleaned up the new material on amplitude equations, and rearranged things a bit, but have not checked the equations themselves against a reference. It's not clear to me that the coefficients and the associated beam and field geometries are adequately defined, nor that the equations properly capture the phase of r and t. The construction has problems, in that it divides vectors to get a scalar quantity. As written and both equal , which is clearly wrong. I think the E's are just missing subscripts, as in the captions on the new images.

I don't have time to work on this any more right now, and more pairs of eyes looking at it is probably a good idea anyway. I am on record above as being opposed to incorporation of the amplitude equations into this article, but am willing to give it a chance if we can get them explained clearly and correctly, with the geometry and sign conventions fully defined.

Also on the to-do list: verify that these equations actually describe ratios of electric field amplitudes, as claimed in the text, and not the square root of intensity or some other quantity. It's been too long since I looked at the Fresnel equations for me to do that off the top of my head, but I've seen derivations that use square root of intensity and call it "E". --Srleffler (talk) 08:47, 18 December 2011 (UTC)

I tried to clarify the conventions, although it's difficult without making pictures. I added a reference (Sernelius) which is typo-free, clear, and open-access. The textbooks I have seen all use exactly the same conventions as Sernelius except the opposite sign convention for .
Anyway, people will presumably keep changing the formulas to agree with their textbook's conventions and definitions. With luck, I (or someone) will notice, and keep changing them back. :-/ --Steve (talk) 19:43, 18 December 2011 (UTC)
Thanks, that is a big help.--Srleffler (talk) 21:00, 18 December 2011 (UTC)

Template:Reqdiagram--Srleffler (talk) 21:03, 18 December 2011 (UTC)

Its late to reply after all this time - but I enhanced the notice (blanked out obviously) to reflect this fact in these edits [1][2]. -- F = q(E + v × B) 22:02, 23 February 2012 (UTC)
I left the all-caps, but otherwise reverted you. A smaller notice in two places seems more useful than a very long notice in one.--Srleffler (talk) 02:14, 24 February 2012 (UTC)
Sorry - I thought I copied and pasted the note in two places by accident, not realizing it was already there in two places, which is what I meant by my accidental duplication - I did NOT refer to it as someone else's mistake. I only tried to place more emphasis.-- F = q(E + v × B) 12:18, 24 February 2012 (UTC)
Becuase I would not like "people to keep changing the formulas to agree with their textbook's conventions and definitions", leaving it for others to keep changing them back, ":-/" (an insanley tedious process), why not request for an edit notice like this one for Maxwell's equations? It'll be more obvious than in the edit panel. At least its the first thing the editor will see anyway...-- F = q(E + v × B) 12:34, 27 February 2012 (UTC)
  1. Hecht (1987), p. 102.