Relative position of diffracted orders and incident beam in direction cosine space for a reflection grating. Diffracted orders outside the unit circle are evanescent.

Beyond 1,000 nm, the story changes. Infrared saunas often use wavelengths longer than 1,000 nm as part of infrared light therapy — although this is another area where caution is advised due to the possibility of thermal damage.

The diffraction efficiency is defined as the fraction of the incident optical power that appears in a given diffracted order (usually the +1 order) of the grating. Integrating the irradiance distribution representing a given diffracted order and dividing by the incident optical power Po=E0w2 gives the diffracted efficiency for that order. Since, for any b and xo Eq. (20)∫−∞∞∫−∞∞1b2 sinc2(x−xob,yb)=1,it is clear that the efficiencies are just the coefficients of the three sinc2 terms in the curly brackets of Eq. (18). These efficiencies are tabulated in Table 1.

Using Eq. (69) to calculate at what value of λ/d the +2 diffracted order goes evanescent, we obtain Eq. (75)sin(−π/2)=−1=−(2−12)λdorλ/d=2/3.

As mentioned earlier, all lights in the PlatinumLED Therapy Lights BIOMAX series do away with the need to choose: all five wavelengths are delivered simultaneously.

The irradiance of the Fraunhofer diffraction pattern in the x2−y2 observation plane a distance z from the grating is proportional to the squared modulus of the Fourier transform of the complex amplitude distribution emerging from the grating: Eq. (56)E2(x2)∝1λz|F{tA(x1)}|ξ=x2/λz|2,Eq. (57)E2(x2)∝sinc2[x2−(nBλB/λ)λz/dλz/d]1λz/dcomb(x2λz/d).

You could compare the process to photosynthesis, whereby plants absorb sunlight and convert it into energy. In red light therapy, our cells absorb the energy of the red and near-infrared light photons, which enhances our cellular potential by promoting oxygen utilization within the cell and generating ATP.

The optical power contained in the m’th diffraction order is obtained by integrating the above irradiance distribution over all space in the x2−y2 plane; however, due to Eq. (20), the integral of the quantity in curly brackets is just unity, and we simply obtain Eq. (29)Pm(x2,y2)=E0w2b2d2 sinc2(mbd).

As the light photons enter the skin, all five wavelengths interact with the tissues they pass through. It’s very “bright” in the irradiated area, and this five-wavelength combination has a significant impact on the cells in the treatment area.

A troublesome aspect of the multiple order behavior of diffraction gratings is that adjacent higher order spectra frequently overlap. In fact, in Fig. 3, one can see the third-order principle maximum for blue light almost overlapping the second-order red principle maximum. One can readily show that the second order for wavelengths 100, 200, and 300 nm is diffracted into the same directions as the first order for wavelengths 200, 400, and 600 nm.

Learn more about the most powerful red light therapy devices on the market, the PlatinumLED BIOMAX series, here, and browse the Learn page to discover the many and often surprising ways that red light therapy can enhance your health and well-being.

Note that in all cases, the zero order is diametrically opposite to the origin from the incident beam and the diffracted orders remain equally spaced in a straight line. However, this line is rotated about the zero order such that it is always perpendicular to the grating grooves. This simple behavior of conical diffraction from linear gratings when expressed in direction cosine space provides understanding and insight not provided by most textbook treatments. It is interesting to note that Rowland expressed the grating equation in terms of direction cosines in a paper published over 125 years ago.26

Grating

Figure 18 illustrates the diffracted intensity distribution as a function of diffraction angle θx and groove depth h, for a sinusoidal “reflection” grating with period d=20λ operating at normal incidence.

We thus have three discrete diffracted waves or “orders,” each of which is a scaled replica of the Fraunhofer diffraction pattern of the square aperture bounding the grating. The central diffraction lobe is called the “zero order,” and the two side lobes are called the plus and minus “first orders.” The spatial separation of the first orders from the zero order is λf/d, whereas the width of the main lobe of all orders is 2λf/w as shown in Fig. 14.

Let’s start with the red wavelengths. If you use 630 nm light along with 660 nm light, the shorter 630 nm wavelengths will, of course, have a shallower absorption depth.

However, more power output increases the number of light photons that can penetrate the targeted tissue to spark biological processes in the cells. An underpowered device won’t deliver the light photons with the intensity needed to push the light deep into your body.

The TRUMPF basic mode fiber lasers are the lasers with the best beam quality. They are particularly well-suited for precision processing of thin sheets.

Diffraction gratingformula

Above 1,000 nm, the body begins to perceive the wavelengths as heat, not light. This is important for anyone treating their eyes, or trying to boost testosterone levels, because certain parts of the body, particularly the eyes and the testicles, are incredibly sensitive to thermal damage.

Monochromatic light of wavelength λ incident upon a refractive transmission grating (interface between two dielectric media exhibiting a periodic surface relief pattern) of spatial period d at an angle of incidence θi will be diffracted into the discrete angles θm according to the following (planar) grating equation:3,4,16–18 Eq. (1)n′ sin θm−n sin θi=−mλ/d,m=0,±1,±2,±3,where n is the refractive index of the media on the incident side of the diffracting surface, n′ is the refractive index of the media containing transmitted diffracted light, and m is an integer called the order of diffraction. The sign of m is arbitrary and determines the sign convention for labeling diffracted orders.

With the wide-spread availability of rigorous electromagnetic (vector) analysis codes for describing the diffraction of electromagnetic waves by specific periodic grating structures, the insight and understanding of nonparaxial parametric diffraction grating behavior afforded by approximate methods (i.e., scalar diffraction theory) is being ignored in the education of most optical engineers today. Elementary diffraction grating behavior is reviewed, the importance of maintaining consistency in the sign convention for the planar diffraction grating equation is emphasized, and the advantages of discussing “conical” diffraction grating behavior in terms of the direction cosines of the incident and diffracted angles are demonstrated. Paraxial grating behavior for coarse gratings (d  ≫  λ) is then derived and displayed graphically for five elementary grating types: sinusoidal amplitude gratings, square-wave amplitude gratings, sinusoidal phase gratings, square-wave phase gratings, and classical blazed gratings. Paraxial diffraction efficiencies are calculated, tabulated, and compared for these five elementary grating types. Since much of the grating community erroneously believes that scalar diffraction theory is only valid in the paraxial regime, the recently developed linear systems formulation of nonparaxial scalar diffraction theory is briefly reviewed, then used to predict the nonparaxial behavior (for transverse electric polarization) of both the sinusoidal and the square-wave amplitude gratings when the +1 diffracted order is maintained in the Littrow condition. This nonparaxial behavior includes the well-known Rayleigh (Wood’s) anomaly effects that are usually thought to only be predicted by rigorous (vector) electromagnetic theory.

The Fraunhofer diffraction pattern of an array of equally spaced narrow slits is illustrated as the number of slits is increased: (a) two slits, (b) three slits, (c) five slits, and (d) eleven slits.

Figure 29 schematically illustrates the dispersive behavior over the visible spectrum of a grating blazed for the first order at a wavelength 500 nm. The seven classical discrete colors: red (λ1=650  nm), orange (λ2=600  nm), yellow (λ3=550  nm), green (λ4=500  nm), blue (λ5=450  nm), indigo (λ6=400  nm), and violet (λ7=350  nm) are obtained by replacing the integral in the above equation by a discrete summation: Eq. (63)E2(x2)∝∑m=−∞∞∑λ=λ1λ7sinc2[m−(nBλB/λ+θ0d/λ)λz/dλz/d]⁢δ[x2−(θ0d/λ)λz/d].

A linear systems approach to modeling nonparaxial scalar diffraction phenomena has been developed by normalizing the spatial variables by the wavelength of light:20–23 Eq. (65)x^=x/λ,y^=y/λ,z^=z/λ,etc.

This behavior is even more evident in Fig. 7, in which the location of the incident beam and the diffracted orders are displayed in direction cosine space for a reflection grating whose grooves are parallel to the y or β axis. The diffracted orders are always exactly equally spaced in direction cosine space and lie in a straight line perpendicular to the orientation of the grating grooves. From Eq. (4), this equidistant spacing of diffracted orders is readily shown to be equal to the nondimensional quantity λ/d. The diffracted orders that lie inside the unit circle are real and propagate, and the diffracted orders that lie outside the unit circle are evanescent (and thus do not propagate).

Some joints, like the knees, are quite large, however, as are the major leg muscles. Consider the thickness of these tissues, along with the thickness of the skin, to get an idea of just how far the light therapy wavelengths need to reach to be effective.

Finally, there’s just the 850 nm wavelength left, traveling to its maximum absorption depth of slightly more than 5 mm. Even if a percentage of the light photons from this wavelength have dissipated, some will reach their target tissue deeper in the body.

Since red light therapy works at the cellular level, it’s important to be patient, be consistent with the treatment, and give your body a chance to heal.

The classical blazed grating is thus a reflection grating with a sawtooth groove profile as shown in Fig. 25. Such gratings have been manufactured for over 150 years by scribing, or burnishing, a series of grooves upon a good optical surface. Originally, this surface was one of highly polished speculum metal.

The conservation of energy is easily shown for this perfectly conducting paraxial (d≫λ) reflection grating at normal incidence because the sum over m from −∞ to ∞ of the squared Bessel function in Eq. (33) is equal to unity.

The BIOMAX panels deliver these widely studied wavelengths together, which means you never have to wonder which one to choose. If you’re treating more than one condition, for example, such as fine lines and wrinkles along with a deep muscle bruise, you will get more profound and faster results by using all five wavelengths at once.

Efficiency of the first few diffracted orders produced by a rectangular phase grating with a duty cycle of 0.5 as a function of the phase step a.

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We have demonstrated that when the grating equation is expressed in terms of the direction cosines of the propagation vectors of the incident beam and the diffracted orders, even wide-angle diffraction phenomena (including conical diffraction from arbitrarily oriented gratings) is shift invariant with respect to variations in the incident angle. New insight and an intuitive understanding of diffraction behavior for arbitrary grating orientation were then shown to result from the use of a simple direction cosine diagram.

Figure 30 illustrates that the dispersion is indeed doubled if the grating is blazed for the second diffracted order. Note also that the diffraction efficiency is substantially reduced for all wavelengths other than the blaze wavelength.

The maximum value of J12(a/2) is 0.3386 and occurs for a=3.68, corresponding to a groove depth of h=0.293λ . The diffraction efficiency of the first few orders for this value of a is tabulated in Table 3. Note that the energy falls off rapidly, with 99.88% of the diffracted radiant power contained in diffracted orders |m|≤3. This paraxial model is accurate only for very coarse gratings (d≫λ).

As valuable as these results are, it bears repeating that better results could be expected by combining wavelengths in the red and NIR spectrum. Let’s explore that now.

Red wavelengths penetrate the skin and sebaceous glands to rejuvenate the skin’s tone and texture. The 630 and 660 nm wavelengths are the two most widely studied wavelengths of the red light spectrum. Here’s a small sampling of some of the powerful benefits:

Only one device on the market features this five-wavelength spectral array: the BIOMAX series from PlatinumLED Therapy Lights.

The paraxial behavior of the square-wave amplitude grating was discussed in detail in Sec. 5.2. Equation (28) indicated that there is a myriad of diffracted orders produced; however, they are rapidly attenuated by a sinc2 envelope function. For a 50% duty cycle square-wave amplitude grating (d=2b), the zeros of the envelope function fall precisely on the even diffraction orders as illustrated in Fig. 33. We see from Eq. (28) and Fig. 33 that the diffraction efficiency of the m’th diffracted order is given by Eq. (74)ηm=14 sinc2(m2).

It has a very important role in imaging, as it forms the first magnified image of the sample. The numerical aperture (NA) of the objective indicates its ...

If your red light therapy device gives you a choice of wavelengths, choose specific wavelengths that are most relevant to your treatment goals: red light for “skin-deep” conditions, and near-infrared light for deep tissue concerns.

Any light that enters the body has to pass through the layers of the skin. Meanwhile, some RLT wavelengths pass through all the skin layers deeper into the body.

As the grating is rotated to increase λ/d, both the angle of incidence and the diffraction angles increase. If we use Eq. (71) to calculate at what value of λ/d the −1 diffracted order goes evanescent, θ−1=π/2, we obtain Eq. (72)λ/d=2/3=0.667.

A missing insight that we now take for granted was provided by John Anderson in 1916 while working at the Mt. Wilson Observatory. He demonstrated that superior gratings could be produced by “burnishing” (plastic deformation of the surface) rather than cutting the grooves into the substrate.37 The material thus had to be soft enough to accept local deformation and at the same time be highly polished.

NIR light in the low 800 nm range generates negligible amounts of heat. This could make the treatment suitable (and safe) for applications in which far-infrared light could cause cell damage.

Note that setting m=0 in Eq. (1) results in θ0 having the same sign as θi. Likewise, setting m=0 in Eq. (2) results in θ0 having the opposite sign as θi . We have thus adopted a sign convention that conforms to that used in geometrical optics whereby all angles are directional quantities measured from optical axes or surface normals to refracted or reflected rays. These directional angles are “positive if counterclockwise,” and “negative if clockwise.” An “angle” here is the smaller of the two angles that a ray forms with the axis or surface normal.

One example of a diffraction grating would be a periodic array of a large number of very narrow slits. This would be a binary amplitude grating (completely opaque or completely transparent). Consider the cylindrical Huygens’ wavelet produced at each narrow slit when the grating is illuminated by a normally incident plane wave as shown in Fig. 1. It is clear to see that there will be constructive interference only in those discrete directions where the optical path difference from adjacent slits is an integral number of wavelengths (i.e., phase differences in multiples of 2π). Every point P in the focal plane of the lens that satisfies this condition will exhibit a primary maximum. The angular width of this interference maximum depends upon the number of slits making up the grating. Figure 2 illustrates the one-dimensional profile of the Fraunhofer diffraction pattern of an array of slits as we progress from two slits (Young’s interference pattern) to three slits, to five slits, and to eleven slits.

Note in Fig. 35, the incremental increase in diffraction efficiency of both the zero and the +1 diffracted order as successive pairs of diffracted orders go evanescent.41 A major increase is observed at λ/d>0.667 when the −1 order goes evanescent, after which the renormalization factor has a value of Eq. (78)K=0.5η0+η1=0.50.25+0.1013=1.4233.

The diffraction efficiency of the m’th diffracted order is just the above optical power divided by the optical power in the incident beam, Po=E0w2, or Eq. (30)efficiency≡Pm(x2,y2)Po=b2d2 sinc2(mbd).

Evaluating at ξ=x2/λz and substituting into Eq. (56) yields the following expression for the diffraction pattern projected onto a screen at a distance z from the grating: Eq. (60)E2(x2)∝sinc2[x2−(nBλBλ+θ0dλ)λzdλzd]1λzdcomb[x2−(θ0dλ)λzdλzd].

The equation for a reflection grating can be obtained by setting n′=−n, just as we do when tracing rays from a reflecting surface:4 Eq. (2)sin θm+sin θi=mλ/nd,m=0,±1,±2,±3.

Illustration of the 100% efficiency achieved by a perfectly reflecting blazed grating satisfying the Littrow condition for the second diffracted order.

Today, many manufacturers only offer two wavelengths: the 660-nanometer (nm) red wavelength and the 850 nm near-infrared (NIR) wavelength. But if you just limit yourself to those two, standard options, you may be losing out on powerful benefits that other wavelengths provide.

Again, there’s that ripple effect throughout the neighboring cells, the system the cells belong to, and to a lesser degree, the body as a whole since all systems are interconnected.

This paraxial requirement obviously places strong limitations on the applicability of the results of this section concerning the grating period-to-wavelength ratio d/λ. The paraxial expressions in Eq. (15) are accurate to within 5% if the angle does not exceed about 18 deg. Although scalar diffraction theory is known to predict diffraction grating performance for TE-polarized light, not transverse magnetic (TM) or unpolarized light,22 at these paraxial angles there will be very little difference between the diffraction efficiency for the two orthogonal polarizations.

Since many individual measurements are required to completely characterize the efficiency behavior of a given grating, it has become commonplace to make diffraction efficiency measurements with a given diffracted order in the Littrow condition.19 For transmission gratings, a given diffracted order satisfies the Littrow condition if θm=−θi. For reflection gratings, the Littrow condition is satisfied if the given diffracted order is antiparallel to the incident beam, i.e., θm=θi. This allows the experimenter to leave the detector and the source in a fixed location and merely rotate the grating between measurements.

The paraxial behavior described by Eq. (36) above leads to the common misconception that it is impossible to get more than 33.86% of the incident energy into the first diffracted order with a sinusoidal phase grating. “Nothing could be further from the truth!” In fact, if you decrease the grating period, the diffracted angles increase and the higher orders eventually go evanescent. When only the zero and ±1 diffracted orders remain, changing the incident angle will cause the −1 order to go evanescent. Then one can vary the groove depth to squelch the energy in the zero order. For a perfectly conducting sinusoidal reflectance grating, we can thus get 100% of the incident energy in the +1 diffracted order!33

Once the 660 nm wavelength departs the group, the 810, 830, and 850 nm wavelengths continue passing through tissue together in straight and scattered formations.

We can now readily calculate the diffraction efficiencies for a paraxial rectangular phase grating with an arbitrary phase step and duty cycle. Figure 23 graphically illustrates the efficiency of the first few diffracted orders produced by a rectangular phase grating with a phase step of π as a function of the duty cycle (b/d). Note that when b/d equals either zero or unity, that no phase variations exist, and all of the diffracted energy remains in the undiffracted beam (zero order). Also note that for b/d=0.5, we obtain the same results as those tabulated in Table 4.

One of the disadvantages of amplitude gratings is that much of the incident optical power is lost through absorption, whereas phase gratings can be made with virtually no absorption losses. Transmission phase gratings can consist of periodic index of refraction variations, or of a periodic surface relief structure, in a thin transparent optical material. Reflection phase gratings are merely a surface relief grating covered with some highly reflective material.

Although the above expression might at first appear to be rather unwieldy, it is rather easily solved numerically with the array operations provided with the MATLAB software package. In fact, the above operation results in an array of delta functions that represents the diffracted orders produced by the rectangular phase grating. The squared moduli of the coefficients of those terms are the efficiencies of the diffracted orders.

The complex amplitude transmittance of a thin sinusoidal amplitude grating can be written as Eq. (16)tA(x1,y1)=[12+a2 cos(2πx1/d)]rect(x1w,y1w).

Due to the replication property of convolution with delta functions, we can now write the quantity in the curly bracket as an infinite series of shifted and scaled sinc functions, thus eliminating the convolution operation from the above equation: Eq. (25)F{tA(x1,y1)}=w2bd[∑m=−∞∞sinc(mbd)⁢sinc(ξ−m/d1/w)]sinc(wη).

The most beneficial red light wavelength for skin is commonly considered to be 660 nm, which is near the upper range of visible red light. This wavelength has deeper penetration than the shorter 630 nm wavelength, with similar effects.

James E. Harvey received his PhD in optical sciences from the University of Arizona. He is a retired associate professor from CREOL at the College of Optics and Photonics of the University of Central Florida, and currently a senior optical engineer with Photon Engineering, LLC, Tucson, Arizona, USA. He is credited with more than 220 publications and conference presentations in diverse areas of applied optics. He is a member of OSA and a fellow and past board member of SPIE.

Since the diffracted orders are distributed symmetrically about the grating normal, a positive and a negative order always go evanescent simultaneously. Figure 34 illustrates the situation for a transmission grating with λ/d=0.25 and the +1 diffracted order satisfying the Littrow condition (θ1=−θi).

The outermost layers of skin are actually made up of three different layers of tissue (seven layers if you include the four to five layers just within the epidermis).

Geometry for producing a Fraunhofer diffraction pattern of an aperture (or transmission grating) in the back focal plane of a lens.

It is thus possible to get a maximum diffraction efficiency of 0.1442 for the +1 order with a square-wave amplitude grating. This is a 42.3% increase over the paraxial value of 0.1013.

The reciprocal variables in Fourier transform space become the “direction cosines” of the propagation vectors of the plane wave components in the angular spectrum of plane waves discussed by Ratcliff,39 Goodman,27 and Gaskill:28 Eq. (66)α=x^/r^,β=y^/r^,andγ=z^/r^.

Here’s an example of how important irradiance is. If two red light devices deliver the same wavelengths, the penetration depth — or the maximum that a particular wavelength can travel — would theoretically be the same.

We will find later in Sec. 6 that a nonparaxial analysis indicates somewhat better performance for certain combinations of grating period and incident angle.

Since w≫d, there is again negligible overlap between the discrete diffracted orders, and there will be no cross terms in the squared modulus of this sum. The Fraunhofer diffraction pattern predicted by Eq. (13) for a square-wave amplitude grating is thus given by Eq. (27)E(x2,y2)=E0w4λ2f2b2d2[∑m=−∞∞sinc2(mbd)⁢sinc2(x2−mλf/dλf/w,y2λf/w)].

A companion paper, Understanding Diffraction Grating Behavior, Part II is currently in progress and will discuss in detail the limits of applicability of nonparaxial scalar diffraction theory to sinusoidal reflection (holographic) gratings as a function of the grating period to wavelength ratio.

The advantage of using multiple wavelengths has been confirmed in numerous studies, such as one chronicled in this 2012 paper in MedEsthetics. The study found that while both red and NIR wavelengths work in promoting collagen production, patients treated with a combination of either 630 nm/850 nm LED or 660 nm/830 nm LED showed superior results.

The more general phenomenon of “conical” diffraction that occurs with large obliquely incident angles will be discussed in Sec. 3 and the parametric behavior will be shown to be particularly simple and intuitive when formulated and displayed in terms of the direction cosines of the incident and diffracted angles. In Sec. 4, we will use the remarkably intuitive direction cosine diagram to portray the conical grating behavior exhibited in the presence of large obliquely incident beams and arbitrary orientation of the grating. Section 5 examines the paraxial diffraction efficiency behavior of several elementary grating types. Section 6 will review the underlying concepts of nonparaxial scalar diffraction theory and apply them to the sinusoidal and square-wave amplitude gratings when the +1 diffracted order is maintained in the “Littrow condition.” This nonparaxial behavior includes the well-known Rayleigh (Wood’s) anomaly effects that are usually thought to only be predicted by rigorous (vector) electromagnetic theory.16

You could get good results by using a two-way combination of the 660 nm and 850 nm wavelengths, but you may experience somewhat slower results.

Table 4 thus lists the efficiency for the first several orders for this special case of a rectangular phase grating. Note that the π phase step has eliminated the zero order, and the efficiency of all other even orders is identically zero because the zeros in the envelope function in Eq. (40) fall exactly upon the even diffracted orders. This thus maximizes the efficiency of the remaining orders.

Be sure to consult your doctor before starting red light therapy, especially if you are treating any chronic conditions, your eyes, or other very sensitive parts of the body.

Typically, a larger light therapy device will have more light power output. It’s important, however, to compare the output of any light therapy device you're considering to ensure you’re getting the most potent light intensity, and therefore, the most value.

For a rectangular phase grating with an arbitrary phase step, the complex amplitude transmittance can be written as Eq. (42)tA(x1,y1)=exp[iϕ(x1)],where the phase variation is given by Eq. (43)φ(x1)=a rect(x1b)(1)**1dcomb(x1d)δ(y1).

For a reflection grating, the undiffracted zero order always lies diametrically opposite the origin of the α−β coordinate system from the incident beam. As the incident angle is varied, the diffraction pattern (size, shape, separation, and orientation of diffracted orders) remains unchanged but merely shifts its position maintaining the above relationship between the zero order and the incident beam. Note also that when the plane of incidence is perpendicular to the grating grooves (ϕ0=0), Eq. (4) reduces to the familiar grating equation presented in Eq. (3).

The power output of the red light device is very important to deliver the maximum amount of light intensity to the skin. Even between similarly sized panels, the difference in power output (not power consumption) can be astonishing.

Making use of the replication properties of convolution with a comb function, the complex amplitude transmittance (or reflectance in this case) of a grating blazed for the n’th order and operating at the blaze wavelength can thus be written as Eq. (54)tA(x1)=rect(x1d)exp(−i2πnBλBx1/λd)*1dcomb(x1d).

This Canyon Zen 100 ML diffuser has its cover made of glass. The colored light makes this a beautiful and elegant diffuser, which is a nice addition to any ...

There is thus a myriad of diffracted orders produced by the square-wave amplitude grating as shown in Fig. 16. However, they are rapidly attenuated by the sinc2 envelope function. The irradiance distribution representing the m’th diffracted order is thus given by Eq. (28)Em(x2,y2)=E0w2b2d2 sinc2(mbd)[1(λf/w)2 ⁢sinc2(x2−mλf/dλf/w,y2λf/w)].

For any wavelengths to treat conditions deeper in the skin or beyond the skin, they must be long enough to get through the layers of skin: a combined total of anywhere from 2 mm (about the thickness of the eyelids) up to 3.5 cm (the thicker skin on the buttocks).

One example of this ripple effect is that when cells like fibroblasts are stimulated to synthesize proteins like collagen and elastin, this process can improve skin health and also the repair and health of the muscles and joints.

Elementary diffraction grating behavior (including diffraction efficiency and dispersion) was reviewed and early challenges in the development of diffraction grating fabrication technology were discussed. The importance of maintaining consistency in the sign convention for the planar diffraction grating equation was emphasized. The advantages of discussing conical diffraction grating behavior in terms of the direction cosines of the incident and diffracted angles were demonstrated, particularly for oblique incident angles and arbitrary grating orientation.

Another finding was that NIR light (810, 830, and 850 nm) can absorb into the tissue to a maximum depth somewhat greater than 5 mm, or a little over 2 inches.

Relative position of diffracted orders and incident beam in direction cosine space for a transmission grating. The zero order and incident beam are superposed.

Making use of the Bessel function identity34 Eq. (49)exp[iz cos(θ)]=∑m=−∞∞imJm(z)exp(imθ),we have an infinite product of infinite sums, which upon Fourier transforming results in an infinite array of convolutions of infinite sums of delta functions: Eq. (50)F{tA(x1,y1)}={[∑m=−∞∞Jm(cn)δ(ξ−n m/d)]n=1*[∑m=−∞∞Jm(cn)δ(ξ−n m/d)]n=2*[∑m=−∞∞Jm(cn)δ(ξ−n m/d)]n=3*⋯*[∑m=−∞∞Jm(cn)δ(ξ−n m/d)]n=∞}.

Many devices will also allow you to use both wavelengths together. This can be beneficial for skincare: You can increase collagen production and microcirculation, and address any underlying inflammation with the longer NIR light wavelengths.

Both Goodman27 and Gaskill28 discussed in some detail both the Fraunhofer and the Fresnel approximations and the geometrical criteria for each. Goodman, in particular, showed that the cosine obliquity factor in the more general Huygens–Fresnel principle must be approximately unity for both the Fraunhofer and the Fresnel approximations to be valid. It is this requirement that limits our diffraction angles to be paraxial angles.

For “skin-deep” conditions, research has shown red light wavelengths to be most useful. For deep tissue applications, opt for near-infrared light, which can pass through all skin layers and even tough connective tissue and bone.

The arrangement of the diffracted orders is the same for the two gratings except they are reflected about the plane of the reflection grating. Note also that the algebraic signs of two directional angles are different if they are measured on different sides of the grating normal. A final useful observation is that for both the transmission and the reflection grating, the positive diffracted orders lie on the same side of the grating normal as the incident beam; whereas the negative diffracted orders lie on the opposite side of the grating normal from the incident beam. A “plus” sign has thus been placed on the lower side of the grating normal in Fig. 5 and a “minus” sign has been placed on the upper side of the grating normal as an indicator of our sign convention. Some authors absorb the minus sign on the right side of Eq. (3) into the m, thus achieving a seemingly simpler equation. However, this results in a different sign convention for labeling the diffracted orders.

The specularly reflected plane wavefront segments will then be out of phase by precisely 2π, thus producing constructive interference for that wavelength and diffracted order. Stated another way, the reflected phase variation over one period of the above grating can be written as Eq. (53)ϕ(x1)=2πλOPD(x1)=2πλ2hx1d=2πnBλBx1/(λd).

An especially exciting discovery of several studies is near-infrared light’s promise in enhancing brain health — including treatment of brain disorders and recovery from brain injuries.

As with the case of the reflection grating, the diffracted orders remain equally spaced and in a straight line as the incident angle is changed, i.e., the size, shape, separation, and orientation of diffracted orders again remains unchanged, merely shifting its position such that the zero order remains superposed upon the incident beam.

The spatial frequencies ξ and η are the reciprocal variables in Fourier transform space. Also the Fresnel diffraction integral is given by the Fourier transform of the product of the aperture function with a quadratic phase factor.27,28 Implicit in both the Fresnel and the Fraunhofer approximation is a “paraxial limitation” that restricts their use to small diffraction angles and small angles of incidence.27,28 This paraxial limitation severely restricts the conditions under which this conventional linear systems formulation of scalar diffraction theory adequately describes real diffraction phenomena.

To get the best results, we recommend using a spectrum of wavelengths together, delivered by a high-light-power energy red light device such as the BIOMAX 600. It is a versatile mid-sized red light device suitable for both targeted and larger applications.

In spite of the fact that it is almost universally believed that—“in no way can scalar theory deal with cut-off anomalies,”40 the renormalization factor K in Eq. (67) and defined by Eq. (68) enables this linear systems formulation of nonparaxial scalar diffraction theory to predict and model the well-known Wood’s (Rayleigh) anomalies16 that occur in diffraction efficiency behavior for simple cases of amplitude transmission gratings discussed in the following two sections of this paper.

Beneath the skin is muscles, bones, cartilage, various organs, blood, lymph, and interstitial fluid, which is the fluid that fills the spaces between cells.

The short answer is that you’ll get the best results by using red (610–660 nm) and near-infrared (820–850 nm) wavelengths simultaneously. By combining multiple wavelengths of red and NIR light, you’ll experience superior results.

Evaluating this function at spatial frequencies ξ=x2/λf and η=y2/λf, and again writing as a two-dimensional function, we obtain Eq. (26)F{tA(x1,y1)}|ξ=x2/λfη=y2/λf=w2bd[∑m=−∞∞sinc(mbd)sinc(x2−mλf/dλf/w,y2λf/w)].

This absorption depth extends to the bone, muscle, blood vessels, organs, abdominal fat, lymph nodes, and other tissue and fluids well beneath the skin.

The mitochondria are susceptible to damage from oxidative stress caused by inflammation. This can lead to a condition known as mitochondrial dysfunction, meaning the inability of the mitochondria to convert raw materials into energy. Emotional stress, injury, and disease can also interfere with proper mitochondrial functioning.

We acknowledge that this list of grating types is nonexhaustive and nonexclusive but none-the-less is useful for comparing and contrasting grating performance for different gratings types, characteristics, and manufacturing techniques.

Some bones, cartilage, tendons, muscles, and joints lie immediately beneath the surface of the skin (the knuckles, for example), so they are relatively easily within reach of red wavelengths.

Diffraction grating

Here Uo+(x1,y1)=Uo−(x1,y1)t1(x1,y1) is the complex amplitude distribution emerging from the diffracting aperture of complex amplitude transmittance t1(x1,y1), and Uo−(x1,y1) is the complex amplitude incident upon the lens.

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For polychromatic light, we can represent the resulting diffracted orders with a summation over the discrete diffracted orders of an integral over some spectral band Δλ=λ2−λ1: Eq. (62)E2(x2)∝∑m=−∞∞∫λ1λ2sinc2[m−(nBλB/λ+θ0d/λ)λz/dλz/d]⁢δ[x2−(θ0d/λ)λz/d].

Red light therapy works by stimulating energy production in cells to halt and even reverse mitochondrial functioning — but that’s not all. It also reduces the inflammation that could be causing the problem in the first place.

Along with using the right wavelengths, you'll want to use a high-power-output LED light therapy device; otherwise, your red light therapy treatment sessions could be much longer and may not yield the best results.

Diffracted intensity distribution as predicted by the above paraxial model for a sinusoidal reflection grating of period d=20λ operating at normal incidence.

For a cell to be healthy it needs enough energy. Producing that energy is the role of mitochondria, which are organelles inside cells that act like tiny power generators. The mitochondria produce adenosine triphosphate (ATP), the primary fuel that provides cells with the energy they need to function.

Now that you know which wavelengths are best for certain applications, here’s an overview of what actually happens when the light photons interact with your body.

The impact of the 660 nm wavelength penetration depth is amplified by the longer wavelengths traveling alongside and scattering together. It’s still four wavelengths working together to a depth of about 5 mm.

The paraxial diffraction efficiencies of the first 19 diffracted orders of a square-wave amplitude grating with a 50% duty cycle are listed in Table 6. Note that 25% of the incident energy is contained in the zero diffracted order, all even orders are identically zero, and the remaining diffracted orders contain another 25%. The remaining 50% of the energy in the incident beam is absorbed by the opaque strips making up the square-wave amplitude grating.

This means ample light photons are required to ensure that the maximum amount of light reaches the targeted tissue — and that requires a light therapy device with more power.

Hence, the +1 and −1 diffracted orders produced by a sinusoidal amplitude grating propagate at angles: Eq. (71)θ1=−sin−1(12λd)andθ−1=sin−1(32λd).

The first reported observation of diffraction grating effects was made in 1785 when Francis Hopkinson (one of the signers of the declaration of independence and George Washington’s first Secretary of the Navy) observed a distant street lamp through a fine silk handkerchief. He noticed that this produced multiple images, which to his astonishment did not change location with motion of the handkerchief. He mentioned his discovery to the astronomer David Rittenhouse. Rittenhouse recognized the observed phenomenon as a diffraction effect and promptly made a diffraction grating by wrapping fine wire around the threads of a pair of fine pitch screws. Knowing the pitch of his screws in terms of the Paris inch, he determined the approximate wavelength of light.7

Since this is an even function, it can be decomposed into a discrete cosine Fourier series. The Fourier series coefficients for the above periodic function can be shown to be given by Eq. (44)cn=2abdsinc(nbd),thus Eq. (45)ϕ(x1)=a2+∑n=1∞cn cos(2πnx1/d).

Illustration of diffraction orders for a transmission grating with λ/d=0.25 and the +1 diffracted order satisfying the Littrow condition.

Scattering doesn’t cause light photons to lose energy; it just redirects them, which means they may travel “sideways” as opposed to straight into the tissue.

Comparison of diffracted efficiency of a sinusoidal phase grating as predicted by Eq. (34) and the common approximation for shallow (smooth) gratings expressed in Eq. (37).

(a) Two-dimensional image of square-wave amplitude grating (b) and profile of amplitude transmittance in the x direction.

When we are stressed or ill, mitochondria begin to produce excess nitric oxide. This is problematic because nitric oxide interferes with the consumption of oxygen within cells, which can lead to oxidative stress, and ultimately, cease the production of ATP. Affected cells may die as a result.

Using the scaling theorem and the convolution theorem of Fourier transform theory, we can write Eq. (55)F{tA(x1)}=sinc[d(ξ−nBλB/λd)][d comb(dξ)].

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A major advance in the development of diffraction gratings was the discovery by John Strong in 1936 that vacuum deposited aluminum on glass is a far superior medium into which to rule grating grooves than speculum metal, which had been almost universally used for nearly a century.38 Therefore, in recent times, diffraction gratings have been ruled in thin layers of aluminum or gold deposited upon a glass substrate.

This two-wavelength combination will help reduce the loss of energy that occurs as light photons pass through the body — and when you add longer wavelengths to the mix, you exponentially increase the number of light photons interacting with your cells.

However, we can ignore the constant term resulting from the fact that ϕ(x1) as illustrated above does not have a zero mean. The rectangular phase variation is thus represented as a superposition of cosinusoidal phase variations: Eq. (46)ϕ(x1)=∑n=1∞cn cos(2πnx1/d).

The fundamental diffraction problem consists of two parts: (i) determining the effects of introducing the diffracting aperture (or grating) upon the field immediately behind the screen and (ii) determining how it affects the field downstream from the diffracting screen (i.e., what is the field immediately behind the grating and how does it propagate).

This renormalization process is also consistent with the law of conservation of energy. However, it is significant that this linear systems formulation of nonparaxial scalar diffraction theory has been derived by the application of Parseval’s theorem and not by merely heuristically imposing the law of conservation of energy.20–23

Diffraction gratings can be categorized according to several different criteria: their geometry, material, their efficiency behavior, their method of manufacture, or their intended application. We thus talk about:

Although researchers often use lasers in their studies, LED light therapy devices are now being used more frequently. This is thanks to advances in LED technology and the fact that these devices are suitable for consumer use, meaning the results can be replicated at home.

Red light therapy is more than just “red” light. It uses wavelengths in the visible (red) as well as the invisible (near-infrared) spectrum. And it also goes by a variety of names: You may have heard it called low-level light therapy (LLLT), photobiomodulation, or even low-level laser therapy, which are terms often used in scientific studies on red light therapy.

In this section, we have systematically described in detail the paraxial behavior of five different classical grating types: the sinusoidal amplitude grating, the square-wave amplitude grating, the sinusoidal phase grating, the square-wave phase grating, and the blazed reflection grating (sawtooth profile). The result of the paraxial diffraction efficiency analyses of these five grating types is summarized in Table 5.

The paraxial grating behavior for coarse gratings (d≫λ) was derived and displayed graphically for five elementary grating types: the sinusoidal amplitude grating, the square-wave amplitude grating, the sinusoidal phase grating, the square-wave phase grating, and the classical blazed grating (sawtooth groove profile). Paraxial diffraction efficiencies for various diffracted orders were calculated, tabulated, and compared for these five elementary grating types.

In this article, we discussed the unique benefits of red and near-infrared wavelengths: how using them together creates a synergy that can amplify the benefits of each wavelength and ensure more comprehensive treatment.

Joseph von Fraunhofer began his detailed study of diffraction gratings about 1821. He built the first ruling engine for fabricating reflection gratings on metallic substrates. His insight into the diffraction process led him to predict that diffraction efficiency behavior would “strain even the cleverest of physicists,” which it did for the next 150 years. Many of Fraunhofer’s findings were written up in great detail, so we are entirely justified in calling him the father of diffraction grating technology.8,9

When operating at the blaze wavelength λ=λB, the peak of the sinc2 function is centered on the nB’th diffracted order and all of the other delta functions (diffracted orders) fall on the zeros of the sinc2 function. All of the reflected energy is thus diffracted into the nB’th diffracted order. Figure 26 shows a plot of diffraction efficiency versus x2×λz/d for a coarse grating blazed to operate in the second order at normal incidence for a wavelength of 550 nm. If d≫nBλB, we can be assured, from the planar grating equation, Eq. (3), that the nB’th order will be diffracted at a paraxial angle and this predicted behavior will be accurate.

Phase variation for a special case of a rectangular phase grating with a peak-to-peak phase step of π and a 50% duty cycle.

The +1 diffracted order thus contains at most (if the quantity a is equal to unity) 6.25% of the optical power incident upon a sinusoidal amplitude grating. This very low diffraction efficiency is not adequate for many applications. As seen in Table 1, the sum of the efficiencies of all three orders is only equal to 1/4+a2/8. The rest of the incident optical power is lost through absorption by the grating.

The net will also be bigger when you use a larger light therapy device; but for now, we’ll stay focused on how the individual light photons behave in the body.

Again, applying the scaling theorem and the convolution theorem of Fourier transform theory,28 we can write Eq. (22)F{tA(x1,y1)}=[b sinc(bξ)δ(η)][comb(dξ)(1)]**w2 sinc(wξ,wη).

Clearly, the total amount of energy transmitted through this thin grating does not vary as the angle of incidence of the narrow beam is increased. Thus when the −1 diffracted order goes evanescent, the energy that was contained in it (6.25% of the incident energy) is redistributed into the two remaining propagating orders (the Rayleigh anomaly phenomenon).

Schematic illustration of diffraction orders for a 50% duty cycle square-wave amplitude grating. Note that all even orders are absent.

This can be beneficial if you're an athlete striving to maintain peak physical conditioning; someone who wants to turn back the clock on just your face and regain a more youthful appearance; or someone who is suffering from a chronic skin disorder.

Illustration of the 100% efficiency achieved by a perfectly reflecting blazed grating designed to operate at normal incidence in the second diffracted order.

The most common and profound effects of both red and NIR light include increased cellular energy, reduced inflammation, increased collagen production, and increased blood flow.

If the incident angle is nonzero, there would be an additional linear phase variation over the entire grating (not each facet individually). Equation (54) describing the complex amplitude distribution emerging from the reflecting blazed grating would thus have to be modified as follows: Eq. (58)tA(x1)=[rect(x1d)exp(−i2πnBλBx1/λd)*1dcomb(x1d)]exp(−i2πθ0λx1),where the diffraction angle of the zero order (angle of reflection) is merely the negative of the incident angle, i.e., θ0=−θi. Again, using the scaling theorem and the convolution theorem of Fourier transform theory, we obtain Eq. (59)F{tA(x1)}={sinc[d(ξ−nBλB/λd)][dcomb(dξ)]}*δ(ξ−θ0/λ).

Illustration of Rayleigh anomalies from a square-wave amplitude transmission grating with the +1 order satisfying the Littrow condition.

If the +1 diffracted order is in the Littrow condition (θ1=−θi) as shown in Fig. 31, the grating equation expressed in Eq. (3) results in the following expression for the incident angle Eq. (69)θi=sin−1(0.5λ/d).

In this article, we explore the benefits of these two popular wavelengths along with other red and near-infrared (NIR) varieties. We’ll also discuss why using multiple wavelengths together could be the most effective way to get maximum results from this safe and natural therapy.

John Strong, quoting G. R. Harrison, stated in a JOSA article in 1960—It is difficult to point to another single device that has brought more important experimental information to every field of science than the diffraction grating. The physicist, the astronomer, the chemist, the biologist, the metallurgist, all use it as a routine tool of unsurpassed accuracy and precision, as a detector of atomic species to determine the characteristics of heavenly bodies and the presence of atmospheres in the planets, to study the structures of molecules and atoms, and to obtain a thousand and one items of information without which modern science would be greatly handicapped.”11

As with the sinusoidal amplitude grating, the total amount of energy transmitted through a square-wave amplitude grating does not vary as the angle of the incident beam is increased. Thus as each pair of diffracted orders goes evanescent, the energy that was contained by them is redistributed into the remaining propagating orders (again the Rayleigh grating anomaly phenomenon) according to the nonparaxial scalar diffraction theory summarized earlier in this section. The renormalization constant K is equal to Eq. (77)K=∑m=−∞∞ηm∑prop.ordersηm=0.5∑prop.ordersηm,where ηm is the diffraction efficiency of the m’th diffracted order.

Diffraction efficiency of the first several orders produced by a square-wave amplitude grating as a function of the width of the transparent slits relative to the grating period.

There’s no question that light therapy in general creates positive biological effects; in fact, the body needs light to be healthy. Natural sunlight has been used for centuries as a healing modality, although today we know that certain wavelengths can be beneficial as well as harmful.

Mitochondrial dysfunction is considered a major contributing factor or cause of poor physical functioning and even disease.

Two generalizations to the behavior of gratings must now be discussed. First, if the individual slits making up the grating have significant width (in order to transmit more light), the Fraunhofer diffraction pattern of an individual slit will form an envelope function modulating the strength of the discrete diffracted orders.12–15 For the case illustrated in Fig. 4, we have chosen the width of the slits to be one-third of the slit separation. You will note that every third diffracted order is absent. This is caused by the envelope function going to zero at those locations.

The epidermis, or the outermost part of the skin, provides a waterproof protective barrier for the body. This layer of skin is composed of four or five layers of epithelial cells.

Note the 20% increase in diffraction efficiency of both the zero and the +1 diffracted order at λ/d>0.667.41 It is thus possible to get a maximum diffraction efficiency of 0.075 for the +1 order with a sinusoidal amplitude grating. In spite of this increase over the paraxial prediction of Sec. 5.1, this low diffraction efficiency combined with the fact that precision sinusoidal amplitude gratings are difficult to fabricate explains why they are rarely used for practical applications.

As previously shown in Table 1 of Sec. 5.1, for a narrow beam normally incident upon a paraxial sinusoidal amplitude grating with modulation of unity, five-eighths of the incident energy is absorbed and three-eights of it is transmitted. Twenty-five percent of the total incident energy is contained in the zero order and six and one-quarter percent is contained in both the +1 and the −1 orders.

Efficiency of the first few diffracted orders produced by a rectangular phase grating with a phase step of π as a function of the duty cycle (b/d).

For a transmission grating, with our sign convention, the diffraction angle for the zero order is equal to the incident angle (θ0=θi). Thus the coordinates of the location in the direction cosine diagram representing the zero order and the incident beam are superposed as illustrated as shown in Fig. 8.

Since blue light kills bacteria in skin tissue, blue light therapy is a popular acne treatment. The regular use of computers, smartphones, and other electronics can be a concern, but therapeutic blue light is a different story.

Due to the replication property of convolution with a delta function, and since the two-dimensional function is separable into the product of two one-dimensional functions:28 Eq. (18)F{tA(x1,y1)}|ξ=x2/λfη=y2/λf=w2 sinc(y2λf/w)[12sinc(x2λf/w)+a4 sinc(x2+λf/dλf/w)+a4 sinc(x2−λf/dλf/w)].

Recalling our definitions of radiometric quantities, it is clear that the diffracted intensity distribution (radiant power per unit solid angle) emanating from the grating is thus proportional to the diffracted irradiance distribution (radiant power per unit area) incident upon the focal plane as given by Eq. (33): Eq. (36)I(θx,θy)=I0∑m=−∞∞Jm2(a2)[1(λf/w)2 ⁢sinc2(x2−mλf/dλf/w,y2λf/w)].

When we are unwell, stressed, or injured, the ability of cellular mitochondria to function at full capacity becomes impaired. In the fast-paced, stress-saturated context of the modern world, most of us are uncomfortably aware of the effects of constant underlying tension.

Figure 19 compares the predicted diffraction efficiency of this approximation with the results of Eq. (34) for a perfectly conducting surface (R=1) and illustrates how shallow the grating must be to satisfy various error tolerances. Note that the above approximation exhibits only a 1% error in the prediction of diffraction efficiency of the +1 diffracted order at h=0.0318λ, a 5% error at h=0.0702λ, and a 10% error at h=0.098λ.

Applying the scaling theorem and the convolution theorem of Fourier transform theory,28 we can write the Fourier transform of Eq. (16) as Eq. (17)F{tA(x1,y1)}=[12δ(ξ,η)+a4δ(ξ+1d,η)+a4δ(ξ−1d,η)]**w2sinc(wξ,wη),where ** is the symbolic notation for the two-dimensional convolution operation.28

A thin rectangular phase grating can thus be defined by the amplitude transmittance function: Eq. (47)tA(x1,y1)=exp[i∑n=1∞cn cos(2πnx1/d)].

For obliquely incident beams and arbitrarily oriented gratings, a complicated three-dimensional diagram is required to depict the diffraction behavior in real space.25 However, the direction cosine diagram provides a simple and intuitive means of determining the diffraction grating behavior even for these general cases. The general grating equation for a reflection grating with arbitrarily oriented lines (grooves) is given by24 Eq. (12)αm+αi=(mλd)sin ψβm+βi=(mλd)cos ψ,where ψ is the angle between the direction of the grating grooves and the α axis. Note that Eq. (12) still reduces to Eq. (3) when ψ=90  deg. Figure 11 illustrates the direction cosine diagram for a beam obliquely incident (αi=−0.3 and βi=−0.4) upon the same reflection grating discussed above (d=3λ) for different orientations of the grating.

As the light photons pass through these layers to reach their maximum absorption depths, they travel together, increasing the effect on the tissue that is within the range of both wavelengths.

Hence, when plotting diffraction efficiency versus λ/d, there can be at most only two propagating orders (the zero order and the +1 that is being maintained in the Littrow condition) for λ/d>2/3. All other orders are evanescent.

The trend is evident. In the limit of a large number of very narrow slits, the primary interference maxima (diffraction orders) become narrower and narrower, with more and more (n−2) small secondary maxima in between them.

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Figure 10 indicates that both Δx and Δθ asymptotically approaches infinity for diffracted angles of 90 deg, whereas Δβ remains constant for all diffracted angles. When projected upon a plane screen, the spacing of adjacent diffracted orders increases by a factor of two (100% increase) at a diffraction angle of merely 38 deg. The angular spacing of adjacent diffracted orders increases by a factor of two at a diffraction angle of 60 deg. If only a 5% increase in Δx were allowed, the diffraction angle would have to be held below 10 deg. For Δθ, a 5% increase is observed at a diffraction angle of 18 deg.

If there are many grating periods within the aperture, then w≫d, and there will be negligible overlap between the three sinc functions; hence, there will be no cross terms in the squared modulus of this sum. Substituting this into Eq. (13) thus yields the diffracted irradiance distribution in the focal plane of the lens: Eq. (19)E(x2,y2)=E0w4λ2f2 sinc2(y2λf/w)[14 sinc2(x2λf/w)⏟m=0+a216 sinc2(x2+λf/dλf/w)⏟m=+1+a216 sinc2(x2−λf/dλf/w)⏟m=−1].

Richard N. Pfisterer received his bachelor’s and master’s degrees in optical engineering from the Institute of Optics at the University of Rochester in 1979 and 1980, respectively. He is a co-founder and a president of Photon Engineering at LLC. Previously, he was the head of optical design at TRW (now Northup-Grumman) and a senior optical engineer at Breault Research Organization. He is credited with more than 20 articles and conference presentations in the areas of optical design, stray light analysis, and phenomenology. He is a member of OSA and SPIE.

The below illustration, from the 2017 study by UK researchers previously referenced, shows how light penetrates deep into the body’s tissues. This may help you visualize how the scattering effect works:

The product of a sinc2 function with a comb function can be written as an infinite sum of shifted and scaled delta functions,28 each of which represents a different diffracted order. Equation (60) can, therefore, be rewritten as Eq. (61)E2(x2)∝∑m=−∞∞sinc2[m−(nBλB/λ+θ0d/λ)λz/dλz/d]⁢δ(x2−(θ0d/λ)λz/d).

Diffraction gratingexperiment

Why? Because, as we illustrated earlier, light loses energy as it passes through tissue, absorbing and scattering depending on the material it encounters.

Introducing an arbitrary incident angle will thus shift both the sinc2 envelope function and the diffracted orders by precisely the same amount. Therefore, under “paraxial” conditions, the diffraction efficiency does not change with incident angle. For example, if we illuminate the above grating blazed for the second order with an incident angle equal to the blaze angle (θi=θB), the incident beam will strike the individual facets at normal incidence and the second order will be retroreflected as illustrated in Fig. 27. This situation (θi=θ2) is referred to as the Littrow condition for the second order,19 and the efficiency will remain at 100% as shown in Fig. 28. The zero order will of course be specularly reflected from the plane of the grating, and the +1 order will be diffracted normal to the plane of the grating.

A “diffraction grating” is an optical element that imposes a “periodic” variation in the amplitude and/or phase of an incident electromagnetic wave.1 It thus produces, through constructive interference, a number of discrete diffracted orders (or waves) which exhibit dispersion upon propagation. Diffraction gratings are thus widely used as dispersive elements in spectrographic instruments,2–5 although they can also be used as beam splitters or beam combiners in various laser devices or interferometers. Other applications include acousto-optic modulators or scanners.6

A 2020 scientific article in the publication Frontiers in Aging Neuroscience links mitochondrial dysfunction to Parkinson’s disease, which is a central nervous system disorder. The same article suggests that red light therapy could be a potential treatment for Parkinson’s.

Note that the sign of these two angles are consistent with the sign convention previously illustrated in Fig. 5. Figure 31 illustrates this situation for λ/d=0.4.

On their own, the 630 nm and 660 nm red light wavelengths are highly beneficial. But you’ll get the best results by combining shorter and longer wavelengths in the visible and invisible light spectrum.

As discussed briefly in Sec. 1–Sec. 5, it is well-known that the paraxial irradiance distribution on a plane in the far field (Fraunhofer region) of a diffracting aperture is given by the squared modulus of the Fourier transform of the complex amplitude distribution emerging from the diffracting aperture.27,28 A slight variation of Eq. (13), without the presence of the lens, can be written as Eq. (64)E(x2,y2)=E0λ2z2|F{Uo+(x1,y1)}|ξ=x2λz,η=y2λz|2.

The above grating equations are restricted to the special case where the grating grooves/lines are oriented perpendicular to the plane of incidence, i.e., the plane containing the incident beam and the normal to the grating surface. For this situation, all of the diffracted orders lie in the plane of incidence.

Following the discussion of the square-wave amplitude grating, we obtain a Fraunhofer diffraction pattern given by Eq. (40)E(x2,y2)=E0w4λ2f2[∑m=−∞∞sinc2(m2)⁢sinc2(x2−mλf/dλf/w,y2λf/w)],except that the zero diffracted order is absent. Continuing, we obtain Eq. (41)efficiency≡Pm(x2,y2)Po=sinc2(m2)for  m≠0.

According to numerous studies, the key to treating many physical conditions appears to be using red light therapy (both red and NIR light) to enhance cellular energy production in the mitochondria.

Here the minus signs describe a transmission grating and the plus signs describe a reflection grating as illustrated in Fig. 5. Note from this figure that the zero order corresponds to the directly transmitted or specularly reflected beam.

Also the product of a sinc function with a comb function can be written as an infinite sum of shifted and scaled delta functions,28 hence, Eq. (24)F{tA(x1,y1)}=w2bd{[∑m=−∞∞sinc(mbd)δ(ξ−m/d)]*sinc(wξ)}sinc(wη).

Blazedgrating

However, absorption into certain kinds of tissue (most notably, the tissue where a lot of water is present) can interfere with light photons passing through, and result in shallower tissue penetration.

Let us first look at a special case of a rectangular phase grating where the peak-to-peak phase step is equal to π (this should result in zero efficiency for the zero diffracted order) and a duty cycle of b/d=0.5 as illustrated in Fig. 20. From Euler’s equation Eq. (38)exp(iϕ)=cos(ϕ)+i sin(ϕ),we readily see that exp(iϕ) is equal to −1 when ϕ=π and +1 when ϕ=0 as illustrated in Fig. 21. The complex amplitude transmittance of this rectangular phase grating bounded by a square aperture of width w thus can be written as Eq. (39)tA(x1,y1)={[2 rect(x1d/2)(1)**1dcomb(x1d)δ(y1)]−1}⁢rect(x1w,y1w).

We have assumed that the grating is bounded by a square aperture of width w. The parameter a represents the peak-to-peak variation in amplitude transmittance and d is the spatial period of the grating. Figure 13(a) shows a two-dimensional image of the grating, and Fig. 13(b) illustrates a profile of the amplitude transmittance in the x direction.

We likewise discover that the −2 and +3 diffracted orders go evanescent when λ/d=2/5, and the −3 and +4 diffracted orders go evanescent when λ/d=2/7, etc.

A red light therapy device that delivers 183 mW/cm² is substantially more powerful than a device that delivers 60 mW/cm² at the same distance — meaning, the more powerful device will allow more light photons to absorb to their maximum possible depth in the body.

Echellegrating

Meanwhile, the SaunaMAX Pro has all the features of the BIOMAX Series, but can be used for in-sauna treatment. It's the ideal panel for red light therapy users who also have a home sauna.

All the cells in the human body perform their specialized functions, and our health is dependent on healthy cells. Whenever anything interferes with normal cellular functioning, there’s a ripple effect on the system the cells are part of and even on the body as a whole.

Keep in mind that these absorption depths for red and NIR light therapy are only general guidelines and refer to optimal conditions. The absorption depth of light photons can be influenced by the length of the treatment, any skincare products on the skin (which can block light), light-blockers such as hair or clothing, and the power of the LED light therapy device.

The light energy output of an LED light therapy device is measured in milliwatts per square centimeter, or mW/cm². This is known as irradiance, and it’s the number you’ll want to compare to understand which red light device will give you the best results.

Since much of the grating community erroneously believes that scalar diffraction theory is only valid in the paraxial regime (d≫λ), it was emphasized that this limitation is due to an “unnecessary” paraxial approximation in the traditional Fourier treatment of scalar diffraction theory, not a limitation of scalar theory itself. The development of a linear systems formulation of “nonparaxial scalar diffraction theory”20–23 was thus briefly reviewed, then used to predict the nonparaxial behavior of both the sinusoidal and the square-wave amplitude transmission gratings when the +1 diffracted order is maintained in the Littrow condition. This nonparaxial behavior included the well-known Rayleigh anomaly effects that are usually thought to require rigorous (vector) electromagnetic theory.

In this section, we discuss the paraxial predictions of diffraction efficiency for five basic types of diffraction gratings: sinusoidal amplitude gratings, square-wave amplitude gratings, sinusoidal phase gratings, square-wave phase gratings, and the classical blazed grating (sawtooth groove profile). The paraxial diffraction efficiencies of various diffracted orders will then be tabulated and compared for these five elementary grating types. For all cases, transverse electric (TE) polarization for the incident beam has been assumed.

The 810 nm wavelength offers a unique array of neurological benefits. Many forward-thinking scientists share the belief that NIR light therapy will become a prominent medical treatment for brain disorders in the near future.

Fortunately, based on years of study, researchers have identified a “therapeutic window” of certain wavelengths — red and near-infrared light — that appear to have significant biological benefits without any known side effects.

A 2017 study by researchers from the UK found that visible red wavelengths — including 610, 630, 650, and 670 nm red light — can absorb into the skin to a maximum depth of between 4 and 5 mm.

Direction cosine diagrams for four orientations of a grating with period d=3λ illuminated with an obliquely incident beam (αi=−0.3 and βi=−0.4): (a) ψ=90  deg, (b) ψ=60  deg, ψ=30  deg, ψ=0  deg.

The “best” NIR wavelength for deep-tissue treatment is typically believed to be 850 nm, which is near the upper range of NIR light. Here’s a small sampling of the findings of various studies and clinical trials on the effects of the most widely studied wavelengths ranging from 810 to 850 nm.

If this grating is placed immediately behind an aberration-free positive lens of focal length f that is uniformly illuminated by a normally incident plane wave as illustrated in Fig. 12, the Fraunhofer diffraction pattern produced in the back focal plane of the lens is given by Eq. (13).

If the grating is placed immediately behind an aberration-free positive lens of focal length f that is uniformly illuminated by a normally incident plane wave as illustrated in Fig. 12, the Fraunhofer diffraction pattern produced in the back focal plane of the lens is given by27,28 Eq. (13)E2(x2,y2)=E0λ2f2|F{tA(x1,y1)}|ξ=x2/λfη=y2/λf|2,where E0 is the irradiance of the incident beam and F{} denotes the Fourier transform operation: Eq. (14)F{t(x1,y1)}=∫−∞∞∫−∞∞tA(x1,y1)exp[−i2π(x1ξ+y1η)]dx1 dy1.

The diffracted orders now propagate along the surface of a cone and will strike the observation hemisphere in a cross section that is not a great circle, but instead a latitude slice as illustrated for a reflection grating in Fig. 6. Note that the direction cosines are obtained by merely projecting the respective points on the hemisphere down onto the plane of the aperture and normalizing to a unit radius. Even for large angles of incidence and large diffracted angles, the various diffracted orders are equally spaced and lie on a straight line only in the direction cosine space.

Similarly, Fig. 24 graphically illustrates the efficiency of the first few diffracted orders produced by a rectangular phase grating with a duty cycle of 0.5 as a function of the phase step a. Note that the even orders are absent. Equation (50) and Figs. 23 and 24 combined constitute a rather unique and comprehensive graphical display of the parametric paraxial performance of square-wave phase gratings.

Since the diffraction angle for a given order varies with wavelength, a diffraction grating produces angular dispersion. This angular dispersion is illustrated in Fig. 3 for a grating with a period d=10  μm. Diffracted orders for wavelengths 450, 550, and 650 nm are plotted versus angle.

Conditions that can be successfully treated using red light therapy, as well as the optimal conditions needed to absorb the benefits of light therapy, are being studied with great interest as these natural remedies find more and more use in everyday healthcare.

Making use of the Bessel function identity27 Eq. (32)exp[ia2 sin(2πx1/d)]=∑m=−∞∞Jm(a2)exp(i2πmx1/d),where Jm is a Bessel function of the first kind, order m, and the fact that the exponential Fourier transforms into a shifted delta function,28 it is readily shown that, within the paraxial limitation, the irradiance distribution in the back focal plane of the lens is given by Eq. (33)E(x2,y2)=E0w2{∑m=−∞∞Jm2(a2)[1(λf/w)2 ⁢sinc2(x2−mλf/dλf/w,y2λf/w)]},and the diffraction efficiency of the m’th diffracted order of a perfectly conducting sinusoidal phase grating is given by the following well-known expression:1,22,27,31,32 Eq. (34)efficiency≡Pm(x2,y2)Po=Jm2(a2),where a=4πh/λ and h is the peak to peak groove depth of the sinusoidal reflection grating.

The above technique can also be used to calculate the paraxial diffraction efficiencies of a reflection grating with arbitrary groove shape by merely supplying the appropriate Fourier coefficients in Eq. (44).

For example, ultraviolet (UV) light therapy is used to effectively treat chronic skin conditions. Yet UV exposure should only be done in moderation since it is known to cause cell damage and even skin cancer.

The epidermis is about .05 mm thick in the “thin skin” portions of the skin, and up to 1.5 mm thick in the “thick skin” portions of the body (the feet and hands, and particularly the heels).

Some of the light photons scatter and change direction, creating a “net” effect in the treatment area in which all wavelengths are active. This net effect receives the light energy of five different wavelengths.

Fresnel Technologies designs and manufactures molded plastics, precision optics, Fresnel lenses, PIR lenses, and infrared-transmitting materials.

The diffraction efficiency of the zero order and the +1 order which is maintained in the Littrow condition for a square-wave amplitude diffraction grating is plotted versus λ/d in Fig. 35.

Studies have shown that NIR wavelengths ranging from 700 to 750 nm have limited biochemical activity and are therefore not often used. More research is needed; but in the meantime, light between 630–660 nm and 810–850 nm appears to have the most profound benefits; that’s the “therapeutic window” mentioned previously.

Illustration of the position of the diffracted orders in real space and direction cosine space for an arbitrary (skew) obliquely incident beam.

Illustration of the dispersion produced over the visible spectrum by a grating blazed for a wavelength of 500 nm in the first diffractive order.

The second generalization includes the situation where the light is incident upon the grating at an arbitrary angle θi rather than normal incidence. This situation will be taken care of by including the incident angle in the grating equation discussed in Sec. 2, where we will review the planar grating equation and the sign convention for numbering the various diffracted orders.

Illustration of paths of superposed optical disturbances that interfere constructively to produce discrete diffracted orders.

By incorporating sound radiometric principles into scalar diffraction theory, it becomes evident that the squared modulus of the Fourier transform of the complex amplitude distribution emerging from the diffracting aperture yields “diffracted radiance (not irradiance or intensity)20–23:” Eq. (67)L′(α,β−β0)=Kλ2As|F{Uo′(x^,y^;0)exp(i2πβ0y^)}|2for  α2+β2≤1L′(α,β−β0)=0for  α2+β2>1.

Diffraction configuration for a sinusoidal amplitude transmission grating with the +1 diffracted order satisfying the Littrow condition when λ/d=0.4.

For a paraxial grating designed to operate at normal incidence, the groove depth must be equal to Eq. (52)h=nBλB/2,where nB is the blaze (or design) order and λB is the blaze (or design) wavelength.

Consider diffraction from a conventional linear reflection grating. However, suppose the incident light strikes the grating at a large oblique angle (represented by direction cosines αi and βi) as illustrated in Fig. 6. The resulting diffraction behavior is described by the following grating equation written in terms of the direction cosines of the propagation vectors of the incident beam and the diffracted orders (the grooves are assumed to be parallel to the y axis):24 Eq. (4)αm+αi=mλ/d,βm+βi=0,where Eq. (5)αm=sin θm cos ϕo,αi=−sin θo cos ϕo,βi=−sin ϕo.

You could start with shorter red light therapy sessions of just a few minutes a day, gradually working up to 10- to 20-minute sessions three to five times a week.

We have specifically chosen the form of Eq. (3) not only to maintain the sign convention for directional angles used almost exclusively in geometrical optics and optical design ray trace codes (positive if counterclockwise and negative if clockwise), but also to be consistent with the sign convention for labeling diffraction grating order numbers used by the popular Diffraction Grating Handbook published and distributed free by the Newport Corporation (formerly Richardson Grating Laboratory).19

Since the Fraunhofer diffraction integral implicitly contains the paraxial approximation, the diffraction angle is proportional to displacement on the focal plane containing the Fraunhofer diffraction patterns Eq. (35)θx=tan−1(x2/f)≈x2/f,θy=tan−1(y2/f)≈y2/f.

When operating in the Littrow condition, the diffracted orders are distributed symmetrically about the grating normal as shown in Fig. 34. For small λ/d, there are many diffracted orders, but they all have small diffraction angles. As λ/d is increased, both the angle of incidence and the diffraction angles increase, and the higher diffracted orders start going evanescent.

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However, since the sinc function is separable and the two-dimensional convolution of two separable functions can be written as the product of two one-dimensional convolutions, the above equation can be written as Eq. (23)F{tA(x1,y1)}=w2bd{[sinc(bξ)][dcomb(dξ)]*sinc(wξ)}sinc(wη).

Before we proceed to discuss the classical blazed grating, we want to derive the general solution for the diffraction behavior of an “arbitrary rectangular phase grating.” This derivation will lay the groundwork for studying the behavior of diffraction gratings with “arbitrary groove shapes.”

If you shine five wavelengths (630 nm, 660 nm, 810 nm, 830 nm, and 850 nm) on the target treatment area simultaneously, something incredible happens: the longer wavelengths amplify the effects of the shorter wavelengths.

As the light energy of the 630 nm wavelength is extinguished, the remaining four wavelengths continue passing through the tissues.

A whole new era of spectral analysis opened up with Rowland’s famous paper in 1882. He constructed sophisticated ruling engines and invented the “concave grating,” a device of spectacular value to modern spectroscopists.10

Figure 9 illustrates the propagating diffracted orders that would exist if a beam were normally incident upon a transmission diffraction grating with λ/d=0.08333. There would be precisely 25 propagating diffracted orders including the two at ±90  deg. The uniform diffracted order spacing in direction cosine space Δβ is contrasted with the increasing angular spacing Δθ , and the even more rapidly increasing linear spacing Δx, when the diffracted orders are projected upon a plane observation screen.

This 20x20cm single linear polarization film has a thickness of 0.7mm and is exactly the same we use for all our polarization filters in our shop.

In addition to being a paraxial (d≫λ) grating, if the sinusoidal reflection grating is also shallow (i.e., the groove depth is much less than a wavelength of the incident light), then the diffraction efficiency of the first orders of the sinusoidal reflection grating can be approximated by Eq. (37)efficiency≡J12(a/2)≈a2/16.

Spectral resolution and diffraction efficiency are quantities of practical interest in many diffraction grating applications. The diffraction efficiency is defined as the fraction of the incident optical power that appears in a given diffracted order of the grating. Note from Fig. 3 that the zero order exhibits no dispersion, and there is twice as much dispersion in the second order as there is in the first order.

The complex amplitude transmittance of a thin square-wave amplitude grating can be written as Eq. (21)tA(x1,y1)=[rect(x1b)(1)**1dcomb(x1d)δ(y1)]rect(x1w,y1w),where d is the period of the grating, and b

In spite of the fact that increasing the parameter b/d reduces the absorption of the grating, we see that for b/d>0.5, all of the additional transmitted power, plus some, goes into the zero order, with the efficiency of the +1 order actually diminishing with increasing b/d.

As they move through the tissue, both wavelengths will work together up to about 4 mm. After that, the 630 nm wavelengths are extinguished while the 660 nm wavelengths continue into slightly greater absorption depth before extinguishing.

According to this 2017 article by Hamblin, red and NIR light has been shown to reduce inflammation, which can protect cells from the damage that nitric oxide can cause. This suggests that reducing oxidative stress in the cells can support optimal mitochondrial functioning.

During a red light therapy treatment, chromophores within the mitochondria absorb red and NIR light photons, and in turn, are stimulated to produce more ATP. Just as depleted cells have a negative ripple effect on neighboring cells, their system, and even the body as a whole, the opposite is also true — energized cells have a positive ripple effect.

Setting Δm equal to unity, we obtain the following expression for the angular spacing of “adjacent” diffracted orders as a function of diffracted angle: Eq. (8)Δθm=λd cos θm.

The Fraunhofer diffraction pattern of an array of eleven equally spaced slits whose width is one-third of their spacing.

Blazed gratings can be designed for a particular wavelength, incident angle, and diffracted order. The blaze angle θB of the grating is given by Eq. (51)θB=tan−1(h/d),where h is the groove depth and d is the grating period.

The more general phenomenon of conical diffraction that occurs with large obliquely incident angles is rarely discussed in elementary optics or physics text books. However, the formulation of a nonparaxial scalar diffraction theory20–23 provides a simple and intuitive means of gaining additional insight into this nonparaxial diffraction grating behavior.

For a thin diffraction grating in air, we thus have n=n′=1, and the two grating equations can be combined to yield Eq. (3)sin θm∓sin θi=∓mλ/d,m=0,±1,±2,±3.

Laser Power Through Aperture Calculator (Gaussian Beam). Gaussian laser beams have power intensities in the shape of a bell curve (Gaussian). This means that ...

For large incident and/or diffracted angles, the diffracted radiance distribution function will be truncated by the unit circle in direction cosine space. Evanescent waves are then produced and the equation for diffracted radiance must be renormalized. The renormalization factor in Eq. (67) is given by20–23 Eq. (68)K=∫α=−∞∞∫β=−∞∞L(α,β−β0)dα dβ∫α=−11∫β=−1−α21−α2L(α,β−β0)dα dβand only differs from unity if the diffracted radiance distribution function extends beyond the unit circle in direction cosine space (i.e., only if evanescent waves are produced).

One can readily calculate that a square-wave amplitude grating with transparent and opaque strips of equal width (b=d/2) results in only 10% of the incident optical power being diffracted into the +1 order. This is a little better than we achieved with the sinusoidal amplitude grating, but still not adequate for many applications. Figure 17 illustrates the diffraction efficiency of the first several orders as a function of the parameter b/d.

Again setting Δm equal to unity yields an expression for the linear spacing of adjacent diffracted orders projected upon a plane observation screen as a function of diffracted angle Eq. (11)Δxm=λdL(1cos θm+sin2 θmcos3 θm).

Similarly, from Fig. 9, we can see that Eq. (9)xm=L tan θm,where L is the distance between the grating and the observation screen.

Illustration of Rayleigh anomalies from a sinusoidal amplitude transmission grating with the +1 diffracted order satisfying the Littrow condition.

This spectral output results in an unprecedented synergy that ensures each layer of tissue — within the skin and below the skin — receives the maximum light energy possible.

While the light energy does indeed dissipate as the light photons pass through the body, these distinct wavelengths work together to “saturate” the cells with more light energy.

The classical definition of a paraxial ray is that the ray must lie close to, and make a small angle with, the optical axis, i.e., 29,30 Eq. (15)sin θ∼θ,tan θ∼θ,andcos θ∼1.

Following Goodman,27 a thin sinusoidal phase grating can be defined by the amplitude transmittance function: Eq. (31)tA(x1,y1)=exp[ia2 sin(2πx1/d)]rect(x1w,y1w),where we have ignored a factor representing the average phase delay through the grating. The parameter a represents the peak-to-peak excursion of the sinusoidal phase variation. The grating, bounded by a square aperture of width w, is again placed immediately behind an aberration-free lens that is illuminated with a normally incident plane wave of uniform irradiance E0 as shown in Fig. 13.

Extensive research and development by our engineering and medical advisory staff led to the development of BIOMAX lights, which feature the patent-pending R+ | NIR+ spectrum. This spectrum merges five different wavelengths of red and near-infrared light: 630 nm, 660 nm, 810 nm, 830 nm, and 850 nm in a synergistic combination to deliver unmatched therapeutic value.

This could take anywhere from one to four months for most conditions as individual cells heal and resume normal functioning, and new healthy cells are born.

Fraunhoferdiffraction

The concept of a blazed grating is that each groove should be so formed that independently, by means of geometrical optics, it redirects the incident light in the direction of a chosen diffracted order, thus making it appear to “blaze” when viewed from that direction. Lord Rayleigh was first to describe the ideal groove shape in 1874.35 He wrote: “…the retardation should gradually alter by a wavelength in passing over each element of the grating and then fall back to its previous value, thus springing suddenly over a wavelength.” He was not very optimistic about achieving such geometry, but 36 years later, in 1910, Wood36 produced the first grating that we would call “blazed” for use in the infrared. He did this with a tool of carborundum, ruled into copper.

Plotting the expressions provided by Eqs. (8) and (11) provides a graphical comparison of the relative spacing between adjacent diffracted orders Δx, Δθ, and Δβ.

An extremely powerful red light therapy device such as the PlatinumLED Therapy Lights BIOMAX 900 could “push” more light photons deep into the body’s tissues.

We have thus seen that the maximum efficiency of the +1 diffracted order (in the paraxial limit) increases from 0.0625 for a sinusoidal amplitude grating, to 0.1013 for a rectangular amplitude grating, to 0.3386 for a sinusoidal phase grating, and to 0.4053 for a rectangular phase grating.

Table 2 lists the efficiency for the first several orders for b/d=0.5. Note that the efficiency of all even orders is identically zero because the zeros of the envelope function in Eq. (28) fall exactly upon the even diffracted orders. We can also see from Fig. 17 that the maximum efficiency that can be achieved for the second order is 0.025 for b/d=0.25 or 0.75.

According to Eq. (68), the renormalization constant K is equal to Eq. (73)K=η−1+η0+η1η0+η1=0.0625+0.25+0.06250.25+0.0625=1.2,where ηm is the diffraction efficiency of the m’th diffracted order. The diffraction efficiency of a sinusoidal amplitude diffraction grating is plotted versus λ/d in Fig. 32.

The difference lies in the ability of longer-wavelength NIR light to penetrate deeper into the body's tissues than red light.