Simulating diffraction efficiency of a volume holographic grating using Kogelnik’s method

This article introduces the native volume hologram capabilities to fully simulate and analyze holographic gratings in Sequential and Non-Sequential Mode  considering their physical properties. These analyses are important for designing systems such as head-up displays (HUD) and head-mounted displays (HMD) for uses in virtual reality (VR) and augmented reality (AR). This article explains the theory and parameters used in the model, and 5 example systems are introduced.

Volume holograms are available in all versions of OpticStudio however Diffraction Efficiency analysis in Sequential Mode is a Subscription only feature.

Authored By Michael Cheng

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Introduction

Volume holograms are popular in many types of optical systems, such as: head-up displays (HUD), augmented reality (AR) and virtual reality (VR) head-mounted displays (HMD). The ability of holograms to diffract rays to any desired angle, and their wavelength and angular selectivity allows the creation of lighter and more compact optical systems.

OpticStudio has supported simulation of ideal holograms for a long time. However, to accurately account for the characteristics of volume holograms it is important to consider factors such as diffraction efficiency, material shrinkage, or index shift, in addition to the propagation direction of the diffracted rays. Considering diffraction efficiency allows advanced analyses like Image Simulation and comprehensive optimization.

Surface-Relief Grating vs. Volume Holographic Grating

Before introducing this model, let’s explain briefly the difference between a Surface-Relief Grating (SRG) and a Volume Holographic Grating (VHG). These two gratings are almost the same in terms of their role in the optical system, but are quite different in terms of their fabrication and simulation.

 

Figure 1. (a) Surface-Relief Grating (b) Volume Holographic Grating

  • The VHG, shown in Figure 1 (b), is fabricated by exposing two or more beams on a light-sensitive emulsion film. The film is then chemically or thermally developed: that’s the grating. The surface on the grating is smooth, but the refractive index inside the grating varies sinusoidally. To model a VHG, algorithms such as efficient Kogelnik’s theory or robust Rigorous Coupled-Wave Analysis (RCWA) are required.
  • The SRG, shown in Figure 1 (a), can be fabricated by several methods, such like e-beam writing, lithography, nanoimprinting, or diamond turning. Unlike the VHG, the SRG doesn’t have a spatially varying refractive index. Instead, the surface of the grating is made of periodic microstructures. To model a SRG, an algorithm like the Fourier Modal Method, also called RCWA, is required.

This article will present the tools for VHG.

See the Knowledgebase article, Simulating diffraction efficiency of surface-relief grating using the RCWA method, for tools for SRG.

Two coupled wave analysis

Let’s review the two coupled wave theory, which is mainly used in the Volume Holographic Grating model.

Consider a simple case that a hologram plane, with normal vector n, is illuminated by two plane waves of the same wavelength, propagated along the directions of wave vectors 1 and k2. The plane waves are first refracted by Snell’s law when crossing the hologram and have new wave vectors k1 and k2 inside the hologram (Figure 2 (a)). The grating vector can then be defined by the following equation:

After development, when the hologram is illuminated by a playback plane kp, the diffracted wave kd, can be determined by solving the equation:

where the kp and kd are the wave vectors of playback and diffracted waves inside of the hologram emulsion (Figure 2 (b)). Note the grating vector K can be chosen from two reversed directions. The sign convention for the equation (2) assumes the direction of K is chosen to satisfy K.k>0.

Figure 2. (a) Two construction beams refract into the hologram material (b) The playback ray refracts into the volume hologram

Now, we consider the fringes in the hologram are represented by sinusoidally varying refractive index n and α as in equation (3).

Where n0 is the average index, n1 is the amplitude modulation of the refractive index and K is the grating vector.

The TE (transverse electric) polarized electric field of diffracted and direct wave for transmission and reflection holograms can be calculated with the following 4 equations. Note, in this theory, it is assumed the energy only exchange between the incident, zero order (direct wave) and first order diffraction waves. To remove this limit, the theory of rigorous coupled wave analysis is required.

where

For TM (Transverse Magnetic) polarization, we can simply replace the κ by κTM as follow and still use the previous equations.

In case of conical diffraction, where the incident ray is not perpendicular to the grating, the eigen polarization are defined as follows:

Figure 3. In Kogelnik’s coupled wave theory, the hologram is consider to be thick enough that each incident ray either directly passes at the 0th order or is diffracted as the 1st order, for both reflection and transmission holograms.

Assumptions and limitations

Kogelnik’s Coupled Wave theory has the advantage over other theories that can predict accurately the response of the efficiency of the zero and first orders for volume phase gratings. However, this accuracy may decrease when either the thickness is low or when over-modulated patterns (high refractive index modulations) are recorded. It is therefore necessary to discuss the limits of applicability of Kogelnik’s theory for user’s reference.

  • Index modulation: The index modulation cannot be too large compared to average index. In other words, n1/n << 1. This is normally true for most of practical cases. A rule of thumb is the ratio of n1/n should not be larger than 0.16 for reflection holograms and 0.06 for transmission holograms. [2]
  • Thickness: The hologram is assumed to be thick. A rule of thumb is to ensure that:

  • Multi-order diffraction: This is same limitation as the thickness. For thick holograms, the energy of input ray will be only transferred to either direct 0th or diffracted +1st order waves. For thin holograms, efficiency for other diffraction orders, such like -1 -2 +2 -3 +3 …, may not be zero.
  • Multiplexing: There can only be one set of fringes existing in the hologram at once. Multiplexing of more than one fringe set is not allowed with Kogelnik’s method.
  • Birefringent material: The hologram material is assumed to be isotropic. Birefringent material is not allowed.

Note these limitations can be removed by using different algorithm. Contact Zemax support for more information if your hologram could break these conditions.

Swelling / shrinkage

In this section, we will describe how swelling and shrinkage of the hologram is considered.

When processing, the hologram emulsion may change its thickness. To consider the effect of thickness change, we first separate the grating vector into two components, K and K, where K is perpendicular and K is parallel to the surface normal. If thickness changes from t to t', the new grating vector can be calculated by modifying K as in equation (4).

Figure 4. When the hologram emulsion shrinks, its thickness decreases from t to t’.

How to set up in Sequential Mode

In this section, we explain how to set up the hologram in Sequential Mode. Matters needing attention are also discussed. To add a volume hologram in Sequential Mode we can use any of the 4 hologram surfaces: Hologram 1, Hologram 2, Toroidal Hologram, and Optically Fabricated Hologram. Full details of the volume hologram parameters for each surface are in the OpticStudio Help System. For this example we will use the Hologram 2 surface with parameters as shown below.

Figure 5. Hologram 2 surface parameters in the Lens Data Editor

Among these parameters, Construct X1, Y1, Z1, X2, Y2, Z2, and Construct Wave control the position and wavelength of the construction beams. For more information on these 7 parameters check the Knowledgebase article “How to model holograms in OpticStudio”.

The meanings if the other parameters are as follows.

  • Diffract order: This parameter only works when the surface is a transmission hologram. When it’s 0, we trace the direct 0th order. When it’s 1, we trace the diffracted 1st order. See "How to model holograms in OpticStudio" to know how to distinguish transmission and reflection holograms. Note that there is not this parameter in the diffractive DLL for Non-Sequential Mode because we trace both 0th and 1st order.
  • Volume Hologram?: This defines whether the surface is a volume or surface hologram. It is False when equal to 0, and True if equal to any non-zero integer. Here we set it to 1 as we are modelling a volume hologram.
  • Hologram Thickness: This is the thickness of the hologram emulsion. Note the thickness is virtual and only used for calculating diffraction efficiency. During the ray-tracing process, rays will only see an infinitely thin surface as with other diffractive surfaces.
  • n1 & n2: These two parameters are the refractive indexes of the material where the construction beams are in before they enter the hologram. n1 is for the construction beam 1. n2 is for the construction beam 2. Check the following sections for more information about these two parameters.
  • n: The average refractive index of the hologram emulsion. This is same as “n0” described in the Kogelnik’s theory above.
  • dn: Modulation of the refractive index. This is same as “n1” described in the Kogelnik’s theory above.
  • Shrinkage: The change of hologram thickness after developing. If it’s 0 (default), there is no shrinkage. If it’s not 0, it’s a scale to the thickness. For example, 0.98 means shrinkage of 2%.
  • Index Shift: The change of hologram’s average refractive index after developing.
  • Consider Fresnel?: If set to 1 then Fresnel loss is considered. Set this to 0 to turn off the consideration of Fresnel loss. This is useful if users want to verify the calculation result with their own code.

Note you should add a coating I.0 on the volume hologram surface when the Material is MIRROR and a coating I.99999999 (eight 9s) when the Material is not MIRROR. The hologram surfaces assume no coating on the surface and the effect of Fresnel loss is considered internally in the model.

Figure 6. Material and Coating parameters in the Lens Data Editor.

How to set up in Non-Sequential Mode

In this section, we will explain how to set up the hologram in Non-Sequential Mode. Matters needing attention are also discussed.

To add a volume hologram in Non-Sequential Mode, we have a choice of three objects: Hologram Lens, Hologram Surface, or Toroidal Hologram.

These native hologram models only support simple shapes like circle or rectangle. If the hologram’s shape is neither circular nor rectangular, the object Boolean Native or Boolean CAD combined, plus the object Extruded, can be used to build the arbitrary required shape.

To consider the diffraction efficiency, it is needed to set the Volume Hologram? to 1, and the corresponding parameters will show as in Figure 7.  The meaning for these parameters are same as in Sequential Mode.

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Figure 7. Diffraction settings for the User Defined Object

When Volume Hologram? is set to 1, the coating on the diffractive face, which is the Face 1, should always be None.

Figure 8. Coating settings for the hologram objects

The option “Use Polarization” must be turned on so that the diffractive DLL can work. The option "Split NSC Rays" is optional to checked. If the option "Split NSC Rays" is checked, OpticStudio will trace both diffracted and direct transmission rays for any rays incident on the hologram face. If the option "Split NSC Rays" is not checked, OpticStudio will tray only the diffracted ray if the Order parameter of the hologram object is set to 1 and only direct transmission ray if the Order parameter is set to 0.

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Figure 9. “Use Polarization” must be checked in Ray Trace control in order to consider volume hologram

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Figure 10. “Use Polarization” must be checked in Layout in order to consider volume hologram

As described previously, the hologram is always considered as infinitely thin. All the interactions between the light and the hologram only happen and are handled at Face 1, which is the diffractive face.

Figure 11: Hologram as seen in the Layout

 

FAQ

In this section, we discuss some frequently asked questions.

About indexes at two sides of the hologram

The hologram’s behavior is different when the index outside of the hologram changes. For example, in the following picture (Figure 12), we have exactly same hologram sheet on the left and right-hand sides while they diffract rays to different directions. On the left-hand side, the hologram sheet is floating in air, whereas on the right-hand, the hologram sheet is attached to a glass. It can be seen the ray is diffracted into different angle in each case. Therefore, it is important to check if the materials at the two sides of the hologram are set correctly.

 

Figure 12. Hologram sheet in air and glass.

About parameters n1 and n2

The parameters n1 and n2 represent the refractive index of the materials outside of the hologram. n1 is the refractive index seen by construction beam 1, and n2 is the refractive index seen by construction beam 2. If, in construction stage, the two construction beams come from different sides, then n1 and n2 could be different as shown in following picture.

Figure 13. Two construction beams coming from different side

On the other hand, if the two construction beams come from same side during the construction stage, then n1 will be same as n2, as shown below.

Figure 14. Two construction beams coming from same side

See "How to model holograms in OpticStudio" for more information if you are not sure what “construction beams” means.

Note if the n1 and n2 are not correctly set, the hologram’s behavior will be incorrect. This is due to how we construct the hologram in real world. For example, when recording the hologram, we may put a prism at one side of the hologram and remove it while playing back as shown in the following pictures. In this case, n1 and n2 are set to the refractive index of the prism and the waveguide.

 

Figure 15. Construction and playback processes

What if the hologram come from suppliers?

Sometimes we don’t really make the hologram by ourselves but buy it from suppliers, so we may not know all the details about how it is built. In this case, we can build a dummy construction system with given specifications. See the section “Non-sequential example 3” for an example.

About diffraction order

In the examples here, we always have only either 0th or 1st order diffraction.

According to the Kogelnik’s theory, when a wave is incident on the hologram grating, only two significant outgoing light waves are assumed to be present. They are the directly passing wave and diffracted wave. In OpticStudio's native holograms and experimental DLLs, we always consider the directly passing wave as 0th order and the diffracted wave as the 1st order.

Therefore, the parameter Oder for the hologram objects in the non-sequential system settings should always be 0 or 1, when the Volume Hologram? is set to 1. 

Sequential example 1

In this example, we demonstrate how to quickly check diffraction efficiency at different incident angle on the hologram. First, we open the attached native_vhg_kog_seq_example1.zar. In this file, a hologram is designed to diffract and focus a 45 degrees incident collimated beam to a far point.

Figure 16. Sample Hologram in the Layout

The parameters of construction system are set as follows.

Figure 17. Hologram construction parameters in the Lens Data Editor

It means the construction system is as shown in the following diagram. Note how we set Beam 1 source to be at very far point (effectively infinity) to simulate a collimated beam.

Figure 18. Initial setup of the system

The Hologram 2 surface assumes one construction beam converges to one construction point (Beam 2) and the other construction beam diverges from the other construction point (Beam 1). However, because of the reciprocity of the construction system, this is identical to the case where Beam 1 is a converging source and Beam 2 is a diverging source. In this case, we could draw it as follows.

Figure 19. Same system built in the opposite direction

The STOP is set before the hologram surface with zero thickness as shown below. This way, we make sure the chief ray will always hit the center of the hologram.

Figure 20. Position of the Stop in the Lens Data Editor

In the sample file, there is an Efficiency vs. Angle graph, where we calculate the diffraction efficiency using Kogelnik’s method with respect to the incident angle of the chief ray. This is found in The Analyze Tab…Polarization and Surface Physics…Diffraction Efficiency…Efficiency vs. Angle. The settings are shown below. 

Figure 21. Efficiency vs. Angle plot settings

Note the incident ray’s polarization state is defined in System Explorer...Polarization. In this file, we set (Jx, Jy) = (1,0) for TE polarization.

Figure 22. Polarization settings under the System Explorer

The following plot shows the results of this analysis:

Figure 23. In the Efficiency vs. Angle graph the y axis represents the diffraction efficiency and the x axis the incident angle of the chief ray. 

Sequential example 2

Let’s take a look on a more complicated design with a waveguide. First, open the attached native_vhg_kog_seq_example2.zar. The way of designing this system is introduced in “Augmented Reality (AR) by hologram”. Here we mainly focus on analyzing the effect of considering diffraction efficiency with the new feature.

In this file, construction Beam 1 is set at a very far point (0, -1E8, -1.35E8). This means it is a collimated beam propagating in direction of vector (0, -0.6, -0.8). Construction Beam 2 is set to (0, 18.66, -45.12), which means it is a converging beam that converges to the point (0, 18.66, -45.12).

Figure 24. Initial set up of the system in the Lens Data Editor and Layout

The two construction beams have a wavelength of 0.55 µm. The material on both sides of the hologram during construction is acrylic, so we set n1 = n2 = 1.493581 (index of acrylic at 0.55 µm).

Figure 25. Parameters of the Hologram in the Lens Data Editor

There are two Image Simulation analyses in this file. They have same settings except one has “Use Polarization” checked and another one unchecked.

Figure 26. Settings in the Image Simulation Analysis

In order to see the effect of considering hologram diffraction efficiency, “Use Polarization” must be checked. We can clearly see the effect in the following images. The simulated image is darker at the top and bottom edge when “Use Polarization” is checked.

Figure 27. Image Simulation analysis

Non-sequential example 1

This example is similar than the sequential example 1 except we analyze the hologram in non-sequential mode. The system is saved in attached vhg_kog_nsc_example1-21.2update.zar. In this file, a hologram is set by a Hologram Lens object. This object allows its face 1 to be a hologram surface. To calculate diffraction efficiency, a Detector Polar is added to receive the diffracted rays.

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Figure 28. Hologram Lens in NSC example 1

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Figure 29. Layout of the whole system in NSC example 1

In the merit function, the diffraction efficiency is calculated and reported with the operand NSDP (Figure 30):

Figure 30. Merit Function Editor

The Universal Plot displays the merit function value as a function of the source ray’s incident angle:

Figure 31. Universal Plot settings

The result of Universal Plot in this example is as follows.

Figure 32. In the Universal Plot, the y axis represents the value of the merit function and the x the source ray incident angle

Non-sequential example 2

In this example, the system is similar than the one analyzed in the sequential example 2. The difference is we rebuild the system in non-sequential mode, the construction system is further optimized for uniform image irradiance, and two more hologram gratings are added to make it supporting color display. The system is attached as vhg_kog_nsc_example2-21_2update.zar.

In following picture, it can be clearly seen how the 3 hologram objects (object 10, 11 and 12) are stacked on the waveguide. These holograms have same parameters except construction beams’ wavelength (parameter Wave) and their corresponding refractive indexes (n1 and n2), This can be checked in the object Properties

Note we have carefully considered the nest rule so that at each boundary between two holograms or between hologram and waveguide, where two faces overlap, rays can correctly see the diffractive face. In this file, the diffractive faces of object 10, 11, and 12 are at the side near to waveguide. In other words, at the -z side.

Figure 33. Layout of the system and detail of the three hologram objects

See following two references for more information about nesting rules.

The simulated image is shown below, as well as the effect of index shifts or hologram shrinkages on the image. More discussion of this example can be found in reference [3]

Figure 34. Image simulation of the multi-color design and 3D shaded model of the system.

Non-sequential example 3

In this example, we assume the hologram is not made by ourselves but is from a supplier with following specification.

  • Hologram Type: Transmission
  • Design Wavelength: 632 nm
  • Input Angle 30 degrees
  • Output Angle: 7 degrees
  • Thickness: 7 mm
  • Average refractive index: 1.5
  • Modulation refractive index: 0.00005
  • The element is designed to be used in: AIR

As we don’t know how exactly the hologram is fabricated, we need to build it with a dummy construction system based on the given specs.

To do this, first, we need to convert the input and output angles to vectors as shown in Figure 34, so we can better define the system.

Note the y axis can be replaced by x axis if desired. It depends on which is more convenient to place and rotate the hologram object in the system. However, the z axis cannot be replaced by other axis as the DLL model assumes the hologram surface is at XY plane.

Figure 35. Input and output vectors in the YZ plane.

From the specs and the above picture, we can calculate that vin vector is (0, -sin(7deg), -cos(7deg)) and vout vector is (0, -sin(30deg), -cos(30deg)). Consequently, the construction beams’ parameters can be set as follows:

  • Holotype = 1
  • X1 = 0
  • Y1 = 1E9*(-sin(7 deg)) = -0.121869E9
  • Z1 = 1E9*(-cos(7 deg)) = -0.992546E9
  • X2 = 0
  • Y2 = 1E9*(-sin(30 deg)) = -0.5E9
  • Z2 = 1E9*(-cos(30 deg)) = -0.866025E9

The multiplier 1E9 in above calculations is just a large number to make collimated construction beams.

Note if the Holotype is set to 2, either (X1,Y1,Z1) or (X2,Y2,Z2) need to be multiplied by -1.

For the rest of the parameters, we can directly get them from the data sheet. They are as follows.

  • Wave = 0.633 µm
  • Thickness = 7 mm
  • Index Modulation = 0.00005
  • Hologram Index = 1.5

For the final two parameters, their values depend on how the hologram is designed to be used. In this example, from specs we know the hologram is supposed to be used in AIR, not attached on any substrate. Therefore, we set n1 and n2 as below:

  • n1 = 1.0
  • n2 = 1.0

The final setup and result are shown in the following pictures and can also be checked in the attached file vhg_kog_nsc_example3-21_2update.zar.

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Figure 36. The settings for the Hologram Lens object

Figure 37. In the Universal Plot, the y axis represents the value of the merit function and the x the source ray incident angle

It should be mentioned that in this file, when drawing the Universal Plot, the Splt? of NSTR in merit function is turned off. By doing this, OpticStudio will only trace the ray specified by the parameter Order of the hologram object. This trick helps to see only the diffraction efficiency for a single order in the Universal Plot.

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Figure 38. By turning off "Split Rays" and changing "Diff Order", we can specify which order to trace for the hologram object.

References

  1. Kogelnik, H., "Coupled wave theory for thick hologram gratings, " Bell Syst. Tech. J. 48, 2909-2947 (1969).
  2. Bjelkhagen, H. and Brotherton-Ratcliffe, D., Ultra-realistic imaging: advanced techniques in analogue and digital colour holography. CRC press, 2016.
  3. Han-Hsiang Cheng, Xiaochaoran Tian, "An advanced ray-tracing model for multi-color holographic optical elements," Proc. SPIE 11188, Holography, Diffractive Optics, and Applications IX, 1118817 (18 November 2019); https://doi.org/10.1117/12.2537762

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