Inverse Square Law Conundrums in Red Light Therapy: Can Intensity Increase With Distance?

What is the optimal distance to use a Red Light Therapy panel?

After years of consideration, it feels more like a philosophical question than a technical question. Firstly, assuming we are properly informed about the skin contact method and choose use devices at a distance anyway. 

The real goal to to obtain the appropriate intensity landing on the skin, and we can adjust the distance accordingly. The "gold standard" distance of 6 inches away is not evidence based, but was a convenient guideline for the LED panels back in 2017-2019. Now, modern LED panels are more powerful and complex. 

The relationship between intensity and distance is famously connected via the Inverse Square Law. The fallacy is invoking the Inverse Square Law without considering the underlying assumptions and unknown complexity of the situation.

In this blog we will discuss many of the conundrums related to trying to calculate intensity based on beam angles and the inverse square law. While we cannot find a universal equation, there are many considerations that would aid our understanding of the relationship between distance and intensity for Red Light Therapy panels.  

Some of the points we will cover in this blog:

• The basic version of the Inverse Square Law does not apply to LED panels because LED panels have multiple sources over a wide area.
• The theoretical "point source" may exist far behind the face of the panel, which is one big aspect of why calculations with the Inverse Square Law are failing when applied to LED Panels.
• Small spot sizes from skin contact devices and lasers will give a high intensity measurement even when the total power output is low. 
• The Lens Angle (the beam angle of each individual LED lens) is different than the Beam Angle of the entire LED panel
• In some cases: the intensity from an LED Panel is higher at 12 inches away than it is at 6 inches away, a paradigm shift from the old recommendation to always move closer to the panel for more intensity. 

Revenge of the Inverse Square Law:

Isaac Asimov gives a simple example of the Inverse Square Law in his 1966 book Understanding Physics :

"A book held near a candle may be read easily; held farther away it becomes first difficult and then impossible to read." [1] [Vol. 2, Ch. Light, pg. 9]

As you move further away from a lamp or candle, it becomes increasingly harder to read as the brightness reaching the book diminishes. 

The intensity of light from the Sun reaching Mercury is about 912.6 mW/cm^2, but reaching Earth is only 136.7 mW/cm^2. Earth being about 3 times further away from the sun receives about 9 times less intensity than Mercury. [Wiki]

The foundation of this concept is called the Inverse Square Law. It states that electromagnetic radiation (i.e. light) will decay in intensity in a way that is proportional to the inverse of the distance squared. 

https://en.wikipedia.org/wiki/Inverse-square_law

To simplify the law: every time the distance is increased by 2x, then the intensity is decreased by 4x. 

As we go from 3 inches away to 6, to 12, to 24 - each increment theoretically means a reduction of 1/4. The resulting intensity at 24 inches away would theoretically be 1/64 of the intensity measured at 3 inches away.

If we measured at 100mW/cm^2 at 3 inches away, then we would calculate only (100 ÷ 64) 1.56mW/cm^2 at 24 inches away based on the simplest interpretation of the Inverse Square Law. 

This would be of massive concern for Light Therapies. Adequate intensity must reach the cells for the biological effects. Clearly, the intensity landing on the body is greatly affected by distance away from the device. 

An LED Panel does not abide by the Inverse Square Law in such an simplified way. The exclusions and complexities are:

• The Inverse Square Law assumes a single source of light, where an LED panel has many LEDs as it's light source. 
• The Inverse Square Law assumes the light is coming from an infinitesimally small "point source", where an LED panel has a very large area as it's source. 

The intensity of a single LED could be roughly approximated by the Inverse Square Law, the optics of an LED Panel becomes much more complex. 

Similar observations can be found on the cinematography lighting blogs. They discuss how the large box soft lighting does not abide by the Inverse Square law for similar reasons. 

https://neiloseman.com/inverse-square-law/

An LED panel can have dozens or even hundreds of LEDs. The number of LEDs, the spacing between the LEDs, the beam angle of the LEDs, the size and geometry of the panel - all of those will play a role in the optical measurements. 

So, in reality the intensity of an LED Panel will decay in a much more complicated way than using the simplified form of the Inverse Square Law, as the beams from hundreds of LEDs will first converge together and then diverge. 

Intensity from a Point Source:

A light bulb or candle will emit a fixed amount of brightness or optical power. In terms of brightness a lamp will be quantified in Lumens, and in power it is Watts. 

As the light spreads in a nearly 360 degree sphere from the source, that initial brightness or power gets spread over an increasingly wider area. 

The area of the sphere is calculated as 4πr^{2 .} In English, the area of a sphere is four times Pi (3.14159) times the Radius Squared.

As we increase distance from the candle, the radius will increase, thus increasing the area of the sphere. 

The Optical Power emitted from a candle or standard light bulb is fixed. If a light bulb emits 700 lumens, then it is always emitting the same total brightness regardless of your distance away from it. What changes is that this fixed amount of brightness is getting spread over a rapidly increasing area of a sphere.

Isaac Asimov describes: "Suppose a fixed amount of light is emerging from the candle flame. As it spreads out in all directions, that fixed amount must be stretched over a larger and larger areas." [1, Light, pg. 9]

The intensity is the Power Density, the power divided by an area. So we divide the Watts of the lamp by 4πr² to ascertain the intensity at a given distance (r) away. If we multiply the distance or radius by 2x, then the equation squares it, thus leading to a 4x of the area increase. 

Thus, we can see the origin of the Inverse Square Law in action. As now we established that Intensity is related to the inverse proportion of the radius squared. Where the Radius also represents the distance we are away from the source. 

Point Source Problems:

If we adhere to the "point source" definition of the Inverse Square Law, this can become problematic in several ways.

If we use Red Light Therapy directly on the skin at 0 inches away, then the point source is infinitesimally small. Thus, the intensity would be infinitely large. Which of course is not the case for several reasons. 

Isaac Asimov Explains: "In the previous section I have been considering the source of light to be a point. Actually, of course, it is not really a point. Suppose the source of light is a candle flame which, naturally, covers an area." [1] [Vol. 2, Ch. Light, pg. 19]

The laws of science often rely on assumptions of idealized conditions. Rarely do such perfect conditions exist in the real world. In this case, there are no true "point sources" of light. 

Typically the LED or Laser emitter does have some size to it to it. In addition there is often a primary or secondary lens over the emitter, which adds spacing and more width to what we recognize as the face of the LED panel. A typical secondary lens seen on an LED panel is about 2 cm diameter. 

One technique to reconcile the Inverse Square Law for LEDs and LASERs is that we draw the point source as originating behind the actual device. As seen in the diagram above. And confirmed in the cinematography blogs.

If we use this model for an LED panel, it can help reconcile some confusion around how people are improperly applying the Inverse Square Law for Red Light Therapy. 

Lets say the the point source is 36 inches behind the face of the panel in the example above. Being 12 inches away is no longer double the distance compared to 6 inches away when we take into account the point source is far behind the panel. 

• 6 inches away from the panel is now (36+6) 42 inches away from the point source
• 12 inches away from the panel is now (36+12) 48 inches away from the point source

Instead of being 2x further away, now 12 inches from the face of the panel is only 1.14x away from the point source compared to 6 inches. So, when using the Inverse Square Law we don't decrease the intensity by 4x, it would only theoretically decrease by about 1.30x.

Note that this is just a rough example and this model is not perfect as we will find later, but this is a much better way to consider how the intensity of LED Panels would be modeled with the Inverse Square Law. Many people have been falsely assuming the face of the panel is the zero distance point from the source, when the theoretical point source is likely far behind the panel face. 

Small Spot Size Problems:

Regardless, the intensity from a small source used close to the skin is challenging to reconcile in one's mind. Small LEDs used in skin contact devices will be only be several millimeters in a square. Lasers often have even smaller spot sizes. 

For example some SMD (Surface Mounted Diode) LEDs are named 2020 and 5050 - the name implies their dimensions. The first is 2.0mm by 2.0mm, and the next is 5.0mm by 5.0mm. That means their areas are 0.04 cm^2 and 0.25 cm^2.

But assume they only emit 20mW, then that becomes 500mW/cm^2 and 80mW/cm^2. So even though a power of 20mW is quite small, the intensity appears to be quite high. But this intensity is only treating a very small spot size area.

For this blog it will be important to understand that we will be referring to Power (mW) and how it is different from Intensity (mW/cm^2).

The Intensity is the Power divided by Area. Thus, the other word for Intensity is the Power Density. Similar to the traditional usage of the word Density describes the object's Mass (kg) divided by the Volume (m^3). 

When the denominator of the equation is less than 1, then that will make the intensity (mW/cm^2) higher than the power output (mW). And vice versa, a large area like a large LED Panel will spread out the power to make the intensity much lower. Even though the total power output of an LED Panel is often many times higher than a typical LLLT laser, the intensity is much lower. 

Similarly, the intensity seen in Laser studies can seem to be astronomically high due to the small spot size. A recent systemic review article on LLLT for muscle integrity for aged humans found the most common parameters were a 100mW laser with a 0.028 cm² spot size. Thus, the intensity they report was 3,571 mW/cm^2 due to the pin-point spot size. [2]

Another article describes this way of dividing the Power by a small Area:

"If the spot size is 2 mm x 1 mm, then the area will be (3.142 × 0.1 × 0.05) = 0.0157 cm². A 60 mW LD‐LLLT system with that spot size would thus have a power density of 3.82 W/cm². These examples give PDs (irradiances) on the high side, but which are still valid power densities to achieve athermal LLLT." [3]

As the quote confirms, the goal is to make sure Photobiomodulation is to be athermal (no heat). Even though the intensity is 3,820mW/cm^2, the total power of 60mW and spot size is so small that it does not make a significant heat effect. 

This is how influencers and brands will be able to cherry pick high intensity laser studies to justify their falsely high measured intensities. While these intensities may be safe on a small spot size, it is reckless to recommend such high intensities to cover a large area or an entire body. Thus, why experts like Dr. Hamblin often recommend considering the Total Joules for dosing, not just the J/cm^2. 

Beam Angle and Field Angle:

The Beam Angle is the angle at which the majority of the intensity is projected from the source. The official definition is that it is the angle of a light source's intensity until it drops below 50% of the centerpoint intensity.

There is some stray light emitted beyond the beam angle called Spill Light that can be additionally quantified by the Field Angle. The Field Angle is the measurement of a source's intensity that has fallen to 10%. 

Another good diagram is here:

https://www.renfrodesign.com/page/beam-angle-field-angle

So, many people will falsely assume the Beam Angle contains 100% of the light, but it only covers the angle which the intensity at the center point drops by 50%.

As well, we can now see the intensity in the center of the beam is the highest, then it drops down by a gradient. For example a laser beam often fits a Gaussian distribution curve, and only more recent lasers use a Flat-Top lens that makes the beam more uniform over it's entire area. 

This means that most of an LED's power will be projected in a straight line from it's source. Which is a large reason why the area of the LED panel itself is the primary "treatment area" of meaningful intensity, regardless of the lens angles used. 

But for simplicity of this blog and the points we are trying to make, we will assume the majority of a light beam is contained in it's Beam Angle and it forms a relatively uniform pattern. 

Rated Power Output:

The power output of a light source is fixed (unless it has multiple modes of operation like pulsing or dimming).

A 100mW rated LASER typically emits 100mW of power. This is nice. As this is much more intuitive than how LEDs are rated. 

LEDs are confusing because we have Rated Watts (i.e. 3W or 5W), then they are purposely driven less than the Rated Watts for efficiency and longevity. The Consumed Watts which is how much electricity the LED actually consumes. The conversion of electricity in the LED to light becomes it's actual Optical Power Output. Then the Intensity is that power divided by the area, We break this down thoroughly in an old blog. 

But for simplicity, lets say our LEDs in the following examples are emitting 100mW of Optical Power. Which may be quite close to what a typical 3W Rated LED is actually emitting. 

Power vs Intensity vs Distance:

LEDs and Lasers does not emit light in a 360 degree area like a standard light bulb or candle. So does the Inverse Square Law still apply if the light is not being emitted in a spherical distribution?

The light from a single LED can be imagined to create a circle of a spot area. The area of a circle is Pi Times the Radius Squared (A = π * r^2) .

That radius gets larger depending on the distance from the LED and the beam angle. 

As the diagram above implies, we can use simple right-angle trigonometry to calculate the radius of the circle, then use that radius to discover the area. Without dragging this out, the area of the circle grows with distance according to this equation:

Area = pi * [TAN(Beam Angle / 2) * Distance]^2

As we can see, the Area still depends on the square of the distance. Thus, the Inverse Square Law still applies to a single LED source even when it is emitted from a narrower beam angle. 

It is important to remember the Power (mW) output is exactly the same at any distance from the source. Again, with the exception of devices with multiple modes like pulsing or dimming or turning off specific wavelengths. 

For example, in one advertisement an influencer claims a device emits 10.5 Watts at 6 inches away and 36 Watts at 1 inch away. This does not make sense, a device would not magically emit more power just because you moved closer to it. The intensity would be higher as you move closer as the area of spread decreases, the total power output remains the same regardless of distance. 

So what actually changes is that the fixed optical power is spread over a wider area. 

• 6 cm away the area is 36.7cm^2
• 12 cm away the area is 150.8 cm^2

So for both 6 cm away and 12 cm away, the Power is 100mW and that gets divided by the growing area of the circle. Thus we get Power Density, also known as the Intensity. 

• 6 cm away 100mW / 37.7cm^2 = 2.65 mW/cm^2
• 12 cm away 100mW / 150.8 cm^2 = 0.66 mW/cm^2

So yes, the intensity does decrease by 1/4th as the distance from the source doubles even from a 60 degree beam angle. Confirming that even with narrower beam angles that the Inverse Square Law does apply. 

60 Degrees vs 30 Degrees:

A wider beam angle (60) will spread it's light more rapidly with distance over a large area. A narrower beam angle (30) will project it's light to a narrower area and thus retain it's intensity over longer distances. 

So if we have two identical LEDs emitting 100mW, the intensity decays differently depending on the beam angle.

We can see the graph below, the intensity decreases rapidly with distance from the source. With the 30 degree remaining relatively more intense with distance.

All 3 beam angles do obey the inverse square law (assuming a basic point source and uniform distribution and initial power of 100mW). The difference of each beam angle is where the proportionality of the relationship comes into play. Each equation would have a different constant of proportionality to the relationship. That is why the graphs are not identical.

What is the Best Beam Angle?

Which beam angle has better therapeutic value or benefits? One study investigated, and did note the narrower beam angles delivered better penetration. 

"we can conclude that controlling the light beam to a smaller angle is beneficial for improving the penetration ability of light in tissues." [4]

Another article makes this comment:

"The reflection intensity of 660 nm and 825 nm lasers with incident angles ranging from 1° to 60° on the skin were simulated (Figure 3a,b). When the incidence angles of 660 nm and 825 lasers on the skin are increased, higher reflection intensity can be predicted. " [5]

These two recent 2022 and 2023 articles are some of the first and only to consider the effects of different beam angles. Knowing basic optics, a narrower beam angle would logically reduce reflection losses both externally and reduce scattering internally. So it is good to start to see experiments that confirm this logic is correct. 

However, since LED panels are an aggregation of multiple LEDs and beams overlapping, it would be more challenging to extrapolate if a particular beam angle is superior than another. This will need to be a bigger area of study for the future, particularly comparing similar panels but different beam angles. At this point, the few studies that do use non-contact LED panels do not mention the beam angle at all. Which should be reported in future studies. 

Lens Angle vs Panel Beam Angle:

It is important to note that most brands are advertising the Beam Angle from their individual lenses (assuming that the lens angle they claim is even accurate). However, as the LED Panel is comprised of tens or hundreds of individual LEDs with those lenses, the Beam Angle for the entire Panel

The amalgamation of all the different LEDs and Lens Angles overlapping will form a new "Beam Angle" to represent the entire panel. In our experience, typically this Panel Beam Angle is lower than the Lens Beam Angle. 

So, some people may erroneously be using the Lens Beam Angle for their Inverse Square Law calculations. When the Panel Beam Angle would be more appropriate to use if you want to more accurately model the intensity profile with the inverse square law. 

Knowing this, we still would not be able to create a universal model for the Panel Beam Angle. As different brands and panels will have different sizes, spacing between LEDs, LED arrangement patterns, etc. Typically, the beam angle of the panel is best measured with specialized equipment at at 3rd party lab.

Intensity from Multiple Sources:

When we add more lightbulbs or candles to a room, then the brightness of light in the room increases. 

When we have two 50 mW lasers both aimed at the same point, the resulting power landing on that point is 100mW. Thus, the power (and intensity) would be higher at the distance in which both beams intersect. Meaning, the power is higher at a distance rather than being closer to the individual lasers.

So, when we have converging beams from multiple sources, that can strengthen the resulting intensity even at a greater distance from the individual sources. Keep this in mind for later. 

Knowing that an individual LED were historically weaker than their predecessors, multiple LEDs are conjoined into a single device. Often called an "array" of LEDs, which is now called in the RLT industry a "panel". 

Can Intensity Increase with Distance?

Can the intensity of an LED panel increase with distance? With modern high intensity LED panels we are starting to find examples where the intensity measured at 12 inches away is paradoxically higher than at 6 inches away. 

According to an email from one influencer:

"The other problem/reason why i just ignored it - is because as you know, dosing is very complex. It's why I don't have any articles on dosing. It's too confusing with too many variables. Just look at the Rouge G3 panel I reviewed - they sent me their lab test numbers, the irradiance was HIGHER at 12 inches than it was at 6. This was from a lab. How the hell are you meant to come up with a dosing calculation when you have flawed numbers and unknown variables all over the place. " - Email, Alex Fergus, Nov 12 2023

I don't see how this complexity is a relevant motivation to condone false advertising of medical devices, but apparently it doesn't take much incentive for an influencer to promote false advertising these days. In contrast to the friendly persona they play on social media, this influencer becomes quite irritable when mentally taxed over technical and ethical quandaries. But lets see if we can help alleviate some of this convenient ignorance. 

We have already established the many ways that the inverse square law does not apply to Red Light Therapy panels. However, it may still be illogical that the intensity at a further distance away (12 inches) would be higher than being closer to the device (6 inches).

After all, the gold standard recommendation is to move closer to a panel to receive higher intensity. We hesitate to debunk yet another fallacy for this industry. 

One issue is that taking measurements too close to the panel will lead to hot spots and cold spots. Taking a measurement in the gap between LEDs will lead to a lower measurement than directly in line with the LED. 

So it is possible at only 6 inches away, the measurement could be taken at one of those "cold spot" gaps that are not in a direct line of a particular beam.

More importantly, another phenomenon occurs typically between 6 inches and 12 inches where the overlapping beams from multiple LEDs will strengthen the intensity.

According to the 3rd party data of the RedRush Pulse that was graciously leaked to us by their consultant Ari Whitten, we see a similar result:

*Note, the RedTherapyCo is still advertising 160mW/cm^2 at 6 inches away and 85mW/cm^2 at 18 inches away. Perhaps from advisement from their consultant. 

Even though this difference is very small, it does confirm another 3rd party measurement that the intensity at 12 inches away is indeed higher than 6 inches away.

We have also observed this in our own GembaRed Beacon panel 3rd party test report:

At 12 inches away the intensity is almost 1 mW/cm^2 higher than 6 inches away. Again, this is not a huge difference. But it may become an important consideration when choosing the ideal distance to use a Red Light Therapy panel.

This phenomenon is becoming increasingly more common in in modern LED panels. It appears to be more common with 30 degree (or narrower) beam angle panels, as the most favorable overlap of beams is likely occurring around 12 inches from the panel. 

For example our 60 degree beam angle panels the OverClocked and Reboot models both clearly show higher intensity at 6 inches away than 12 inches. Older versions of name brand panels used 90 degrees or 60 degrees also had clearly higher intensities at 6 inches than 12 inches.. In contrast, our Beacon model panel uses 30 degree beam angle lenses, so that is why it shows higher intensity at 12 inches away instead. 

This phenomenon has indeed been observed and documented in published Photobiomodulation literature:

https://www.intechopen.com/chapters/54924

So these observations are not surprising as it has already been documented that the intensity can be higher at a certain distance away rather than being closer to the source.  

This is a paradigm shift from the general recommendation to move closer to the panel to receive higher intensity.

As is now the case with some devices will have slightly higher intensity at around 12 inches than the former "gold standard" of 6 inches away. So this will be an important consideration for making distance recommendations.

When does the Inverse Square Law Apply?

It is important to recognize that between 0 and 24 inches away from most LED panels, the optics will be very complex and challenging to rationalize with simple optical models. As within this range there are areas of hot spots, then overlap of beams, then divergence of the beams. 

Perhaps beyond 24 inches as all the beams blend into a more unified light distribution, then we could assume the simplified Inverse Square Law applies. As is described in the below quote from a PBM article:

"From a certain distance between the array and the tissue: the array is not seen by the tissue as individual LEDs, but as a fairly homogeneous single irradiator." [3]

https://www.intechopen.com/chapters/54924

In fact, the cinematography blogs discuss a general rule that a light box will abide by the Inverse Square Law only after a distance of 5 times the widest side of the box.

https://theasc.com/articles/revisiting-and-updating-inverse-square-law

https://neiloseman.com/inverse-square-law/

If a standard LED panel is 36 inches tall, then it would abide by the inverse square law starting at 180 inches away. That is 15 feet or 4.57 meters. 

So, within the range of distances that LED panels are typically used, it would be difficult to calculate or extrapolate or interpolate the intensity at various distances if we are only given limited measurements.

This means that the intensity profile of panels is best left up to being empirically measured, rather than theoretically modeled.

At least with this blog we can update our mental models to better understand the complex optical physics of how light is being emitted from LED panels. 

 

References:

[1]

Asimov, Isaac (1966), Understanding Physics, Walker and Company. 1988 reprint, New York: Buccaneer Books; ISBN 0-88029-251-2.

[2]

Kumar P, Umakanth S, N G. Photobiomodulation therapy as an adjunct to resistance exercises on muscle metrics, functional balance, functional capacity, and physical performance among older adults: A systematic scoping review. Lasers Med Sci. 2024 Sep 3;39(1):232. doi: 10.1007/s10103-024-04177-x. PMID: 39225877; PMCID: PMC11371873.

[3]

Calderhead, Robert Glen, and Yohei Tanaka. ‘Photobiological Basics and Clinical Indications of Phototherapy for Skin Rejuvenation’. Photomedicine - Advances in Clinical Practice, InTech, 17 May 2017. Crossref, doi:10.5772/intechopen.68723.

https://www.intechopen.com/chapters/54924

[4]

Feng Z, Wang P, Song Y, Wang H, Jin Z, Xiong D. Photobiomodulation for knee osteoarthritis: a model-based dosimetry study. Biomed Opt Express. 2023 Mar 30;14(4):1800-1817. doi: 10.1364/BOE.484865. PMID: 37078045; PMCID: PMC10110300.

[5]

Su, C.-T.; Chiu, F.-C.; Ma, S.-H.; Wu, J.-H. Optimization of Photobiomodulation Dose in Biological Tissue by Adjusting the Focal Point of Lens. Photonics 2022, 9, 350. https://doi.org/10.3390/photonics9050350