Volume Holographic Gratings – Part 1
Diffraction Gratings and Volume Gratings
The origin of diffraction gratings dates back to the 19th century, when German physicist Joseph von Fraunhofer created what is recognized as the earliest artificial grating. He achieved this by using wire mesh and parallel lines engraved on glass plates. Later, American physicist Henry Augustus Rowland successfully fabricated high-precision concave ruled gratings, which enabled the broader application of gratings in spectroscopy. Since the 20th century, the use of two coherent laser beams for interference has enhanced grating resolution to the nanometer level, significantly expanding the scope of grating applications.
Figure 1: Schematic diagram of grating diffraction principles.
The fundamental principle of a diffraction grating lies in its ability to impose periodic spatial modulation on the amplitude or phase of incident light, thereby inducing constructive interference in the outgoing light at specific directions, as illustrated in Figure 1. An ideal diffraction grating can be conceptualized as an array of infinitely long, infinitely narrow slits with uniform spacing. In the xz-plane,
an incident light wave passing through such a grating generates equally spaced Huygens point sources separated by the grating period Λₛ
For monochromatic plane waves of wavelength λ incident from air at angle θi ，
relative to the z-axis, with the grating's average refractive index n₀, the effective wavelength inside the grating becomes λ/n₀. Constructive interference occurs when the optical path difference between adjacent point sources equals an integer multiple of the wavelength, defining the diffraction angle θd，The grating equation is derived as:
1)
From the above equation, we can see that the grating decomposes an incident plane wave into multiple plane waves propagating in different directions, where m represents the diffraction order of the light wave. Here, we can perform some analysis:
When m = 0, the diffracted light is simply the refracted light that passes directly through the grating medium. The exit angle of this diffracted light and the incident angle obey the law of refraction. In this case, the diffraction angle must be smaller than the critical angle for total internal reflection (TIR), so the diffracted light will transmit through the opposite side of the medium into air, making TIR propagation within the medium impossible.
When m ≠ 0, by carefully designing the grating period, we can make the exit angle of the diffracted light exceed the critical angle for TIR. Such diffracted light will undergo total internal reflection at the interface between the grating medium and air, resulting in back-and-forth reflective propagation within the medium.
This forms the fundamental operating principle of diffractive waveguide technology.
The aforementioned Huygens point sources exist only in the xy-plane, constituting what we call a "thin grating" with nearly negligible thickness. If we extend these point sources along the z-direction at a certain angle, they form a three-dimensional array of point sources. As shown in Figure 2, this is equivalent to continuously replicating and translating a two-dimensional thin grating along a specific direction, resulting in a volume grating with true three-dimensional characteristics.
For volume gratings, the conditions required to achieve maximum diffraction intensity become more stringent. Similar to thin gratings, the diffraction maxima of volume gratings must satisfy the constructive interference condition for all point sources - meaning all light waves emitted from the three-dimensional Huygens point source array must interfere constructively in a particular direction.
To analyze this, let's select any two point sources B and C. The phase difference or optical path difference between them can be calculated using an intermediate point source A. Since A and C lie on the same thin layer, the optical path difference for light passing through them is:
where θi' represents the angle between the incident light ray and the z-axis within the medium. Points A and B lie on the same array formed by translational extension of the point sources. The optical path difference for light passing through them is calculated as:
The condition for achieving constructive interference is:
（2）
For an arbitrary length AB, the only scenario that satisfies Equation (2) is when:
（3-1）
（3-2）
This equation demonstrates that diffraction in volume gratings must first satisfy the grating equation, meaning the exit angle of diffracted rays is determined by the grating equation. From Equation (3-2), we can derive that θd-SA=SA-θi or θd=2SA-θi，
This condition requires that the diffracted and incident light obey the law of reflection relative to the translation-extension plane. This characteristic is identical to that of blazed gratings - it selects specific diffraction orders, where only the "blazed" orders achieve maximum diffraction efficiency, while other orders
exhibit weak diffraction intensity due to not being "blazed". Substitutingθd= 2SA-θi' into Equation (3-1) yields the following expression
（4）
⋀=⋀s·cos(SA) represents the spatial period of the volume grating. Equation (4) is known as the Bragg matching condition, which constrains the incident angle of light beams on the volume grating for a specific wavelength. Only when the incident angle satisfies this condition will maximum diffraction efficiency be achieved.
Conversely, at a specific incident angle, it also restricts the light wavelength that can achieve maximum diffraction efficiency. This restrictive characteristic for both incident angle (or wavelength) and diffraction angle may be referred to as angular (or/and wavelength) selectivity. It reveals the volume grating's distinctive feature of selective diffraction towards light.
Figure 2: Schematic of volume grating diffraction principles.
Volume Holographic Gratings (VHGs)
In 1948, Gabor invented holography, earning him the Nobel Prize in Physics. Holography records both amplitude and phase information of light waves, enabling 3D visual reproduction—unlike traditional photography, which captures only intensity. In the 1960s, Soviet scientist Yuri Denisyuk combined holography with Lippmann's color photography to produce reflection volume holograms using thick recording media, now termed Volume Holographic Gratings (VHGs) (Figure 3).
VHGs are fabricated via two-beam interference (object and reference beams) in photosensitive materials, creating a 3D periodic modulation of refractive index or absorption (grating fringes). Illuminating the VHG with the reference beam reconstructs the object beam.
VHGs are categorized into transmission and reflection types based on the relative directions of reference/reconstruction beams (Figure 3). Transmission VHGs have small slant angles (SA) between fringes and the surface, while reflection VHGs exhibit large SAs. This difference in slant angle leads to distinct properties. For instance, overall shrinkage of the grating thickness along the z-direction has a more pronounced effect on the fringe spacing (spatial period) of reflection VHGs. Consequently, the Bragg matching angle for reflection gratings is more susceptible to deviation due to shrinkage-induced changes.
Figure 3: Recording and reconstruction process of VHGs.
VHGs exhibit the fundamental properties of volumetric gratings, namely angular and wavelength selectivity. This enables diverse applications, including volume AR waveguide couplers, Volume Holographic Optical Elements (VHOEs), high-density volume holographic storage, volume holographic correlation recognition, and optical neural networks based on volume holographic weighting.
Analytical Tool for VHGs: k-Vector Sphere
Any complex light wave can be decomposed into a superposition of plane waves, each described by its wave vector (k-vector). The k-vector's direction and magnitude correspond to the wave's propagation direction and wavenumber Interference between two plane waves creates a periodic grating structure, characterized by a grating vector Kg (|Kg| = 2π/Λ,
perpendicular to the grating fringes). As shown in Fig. 4, a sphere of radius k = n0·2π/λ is constructed, known as the k-vector sphere or Ewald sphere. Its intersection with the xz-plane forms a circle, the k-vector circle. The k-vector of any light ray propagating within the medium originates at the coordinate origin, with its tip lying on the k-sphere surface. Rays propagating in the xz-plane have k-vector tips on the k-circle. Considering only ±1st order diffraction, the Bragg matching condition can be expressed vectorially as:
，（5）
where, k₁ (incident) and k₂ (diffracted) must lie on the k-vector sphere (Ewald sphere), forming an isosceles triangle with Kg. Special cases include Kg spanning the sphere's diameter, yielding counter-propagating beams. Plotting all valid Kg vectors generates a cylindrical surface, with corresponding k₁ and k₂ forming conical surfaces—only wave pairs on these cones satisfy Bragg matching.
Figure 4: k -vector sphere and grating vector representation.
Does diffraction vanish completely when the incident angle deviates from the Bragg match?
Of course not. Here, we must introduce the concept of the "grating vector cloud".
A VHG is fundamentally a periodic spatial modulation of the extinction coefficient or refractive index within a medium. Its grating vector Kg represents the spatial frequency of this modulation (differing by a factor of 2π). Taking the refractive index distribution as an example: Suppose the refractive index of a VHG follows a cosine modulation and is confined within a finite volume of dimensions X, Y and Z, while being surrounded by a medium of refractive index n0. The refractive index distribution near the grating can be expressed as:
，（6）
Δn represents the refractive index modulation depth of the volume holographic grating (VHG), which is the primary factor governing diffraction efficiency. By applying a three-dimensional Fourier transform to decompose this into a superposition of multiple gratings with distinct grating vectors, we obtain the grating's wavevector spectrum [2]:
,（7）
where ⊗ denotes the convolution operation. This equation reveals that the tip of the grating vector spreads outward from its ideal position, forming a blurred, cloud-like distribution—hence termed the "grating vector cloud", as illustrated in Figure 5.
This implies that for spatially confined VHGs, the effective grating vector is not singular but exhibits broadening around the central grating vector (Kg). In other words, it is equivalent to the coexistence of countless VHGs whose refractive index modulation depths follow a sinc-function decay relative to the central grating vector.
This answers our earlier question: even when the incident angle deviates from the Bragg condition for the central grating vector, diffraction may still occur if the angle matches other vectors within the "cloud." The diffraction efficiency depends on the refractive index modulation depth of the matched grating vector, which explains why efficiency attenuation at off-Bragg angles correlates closely with the sinc function's behavior.
Figure 5: Schematic diagram of the grating vector cloud for a finite-sized volume holographic grating (VHG), along with incident and diffracted wave vectors satisfying Bragg matching conditions with vectors within the "cloud".
Let us further analyze this: For volume holographic gratings (VHGs) used in AR waveguides, the dimensional constraints in the xy-plane primarily stem from the beam spot size, while the z-direction limitation is mainly determined by the grating thickness.
The thinner the VHG, the greater the z-directional broadening at the tip of its grating vector, resulting in a wider angular bandwidth. When the thickness becomes negligible, the angular bandwidth increases significantly - this corresponds to the "thin grating" concept introduced at the beginning of this article.
For VHGs operating within a certain range of refractive index modulation, their diffraction efficiency can be calculated using coupled-wave theory [3]. We will explore this methodology in detail in our next installment.
References
[1] Densisyuk Y N. Photographic reconstruction of the optical properties of an object in its own scattered radiation field.Sov. Phys. Dokl.,1962,7:543
[2] Tao S Q, et al. Optical Volume Holography: Techniques and Applications [M]. Science Press, 2013.
[3] Herwig K. Coupled wave theory for thick hologram gratings[J]. Bell System Technical Journal, 1969, 48(9): 2909-47.