How to model holograms in OpticStudio

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Holograms are very versatile optical elements that can be used to achieve compact and light systems. They are effectively employed in a wide variety of applications, and in recent years have been especially valuable in the design of Augmented Reality headsets.

This article describes the main characteristics of holograms and gives an overview of the hologram models available in OpticStudio.

Authored By Alessandra Croce, Michael Cheng, Erin Elliott

download example (.zip)

Introduction

What is a hologram?

Holograms are interference patterns recorded on a high-resolution photosensitive emulsion. Two distinctive stages are associated with holograms: the Construction step and the Playback step, pertaining respectively to the hologram creation and its use as an optical element. During the Construction step, two beams of coherent light, called construction beams, interfere with each other, and their interference is recorded on a photosensitive plate. The hologram is the interference pattern once it is developed. It bears the signature of the interfering construction beams and effectively behaves like a diffraction grating when re-illuminated by a readout beam, during the Playback step.

Note that, in this article and in OpticStudio, all holograms are implicitly optically fabricated i.e. generated via interference of two construction beams through the process previously described. This is to distinguish them from Computer-Generated Holograms (CGH), whose interference pattern is digitally generated instead. CGH can be modelled in OpticStudio through any surface allowing a phase profile definition, such as Binary or Grid Phase surfaces.

Hologram ray-tracing equation and main models

OpticStudio treats holograms as infinitely thin surfaces that alter the phase of a ray. The hologram deviates ray paths according to the following equation.

$$\hat{n} \times (\hat{r'_o} - \hat{r'_r}) = (\frac{m\lambda_p}{\lambda_c})\hat{n} \times (\hat{r_o} - \hat{r_r})$$

where is the unit vector normal to the surface of the hologram at ray intersection point; is the unit vector along the first construction beam; is the unit vector along the second construction beam; is the unit vector along the incident readout beam; is the refracted ray; λ_c and λ_p are the construction and playback wavelengths respectively; and m is the diffraction order. A value of m = 0 means the ray is undeviated, while other integer values of m refer to other diffraction orders.

Currently, OpticStudio only models holograms to the extent of deviating ray path. Other properties, such as diffraction efficiency calculations based on diffraction order, wavelength or incident angle, are not considered at present.

There are three main hologram models available in OpticStudio. These are:

• Hologram 1
• Hologram 2
• Optically Fabricated Hologram

With the exception of the Optically Fabricated Hologram, all sequential and non-sequential hologram models are based on Hologram 1 and Hologram 2 sequential surfaces. The main characteristics of all three main hologram models will be described in the following sections.

The Hologram 1 and Hologram 2 surfaces

Both Hologram 1 and Hologram 2 surfaces are based on the assumption that the construction beams come from point sources and have no aberrations. Hologram 1 is to be used when both construction beams diverge or converge, while Hologram 2 is to be used when one construction beam diverges and the other converges. Typical set-up examples for both models is shown in the following image, where blue rays indicate diverging construction beams and green rays indicate converging construction beams.

Both Hologram 1 and Hologram 2 can be used to model transmission or reflection holograms. For transmission holograms, construction beams come from the same side of the hologram surface, as in set-ups (a), (b), (g) and (h). Inversely, for reflection holograms, construction beams come from opposite sides of the hologram surface, as in set-ups (c), (d), (e) and (f).

Note that to use a hologram in reflection, its Material must be set to “Mirror”. This explicitly indicates OpticStudio that rays propagate in the opposite direction after hitting the hologram surface.

The shape of Hologram 1 and Hologram 2 is controlled by Radius and Conic parameters, enabling modelling of plane, spherical or conical holograms. Both surfaces are also characterized by some hologram-specific parameters, defining:

• the XYZ position of construction beam 1, relative to the hologram surface vertex
• the XYZ position of construction beam 2, relative to the hologram surface vertex
• the wavelength of the construction beams
• the diffraction order

Once the characteristics of the hologram are defined through the parameters above, rays landing on the hologram surface (effectively the playback step’s readout beam) will diffract as per the equation previously described. If rays originate from the same position as one of the construction sources, i.e. the readout beam coincides with one of the construction beams, then the hologram diffract rays as if converging to/diverging from the location of the second construction source. An example will clarify this further.

Imagine the following construction setup, representing a transmission hologram with both construction beams diverging. As previously explained, this is a scenario where the Hologram 1 surface should be used.

In the playback step, if the readout beam coincides with construction beam 1, then rays will be diffracted as if coming from construction source 2, as shown in the following image.

The same principle would apply if, for example, construction beam 2 was inversed, and a reflective hologram was considered instead. An example of both scenarios can be found in the article attachments (“Hologram1_transmission.zar” and “Hologram2_reflection.zar” respectively).

Note that the Hologram Construction Interference user analysis, described in Knowledgebase article “Analyzing hologram construction fringes with a ZOS-API User Analysis”, may be used to visualize the fringes that results from the interference of the two construction beams.

For example, the hologram construction fringes for file “Hologram1_transmission” are shown in the following image.

The Optically Fabricated Hologram surface

The Optically Fabricated Hologram (OFH) is a sequential surface that enables a far more general hologram modeling than Hologram 1 and Hologram 2 surfaces. Indeed, construction beams can be much more complex than just beams converging/diverging to/from a source point: before landing on the hologram surface, they might go through arbitrarily defined optics that may consist of multiple lenses, mirrors, or even other holograms. The aberrations introduced by such construction optics are fully considered and contribute towards the characteristics of the resulting OFH. If required, the construction systems’ parameters can also be set variable and optimized directly from the playback file.

The OFH surface requires use of three ZMX files:

1) the playback file, where the OFH is placed

2) the construction file for beam 1, optionally including beam 1 construction optics

3) the construction file for beam 2, optionally including beam 2 construction optics

The construction files are to be placed in the same folder of the playback file, and their name should be the same but have “_1” and “_2” appended to the end. Both files are then called from the playback file, by specifying the common part of their name (i.e. without the suffix) in the Comment of the OFH surface. For example, if construction files are called “construction_1” and “construction_2”, then in the playback file you should enter the Comment of the OFH as “construction”, as shown in the following image.

The surface at which the two construction beams interfere is the STOP surface in each construction file. Only the ray intercept vectors at the respective STOP surfaces will determine the hologram properties.

In addition to what has been previously explained, the construction files have to fulfil several other requirements. For more information, please refer to help file section The Setup Tab > Editors Group (Setup Tab) > Lens Data Editor > Sequential Surfaces (lens data editor) > Optically Fabricated Hologram.

For example, to use the OFH surface to model the same system as file “Hologram1_transmission.zar” described previously, we would need to define construction_1.zmx as to recreate rays generated by construction source 1 and landing on the STOP.

Similarly, construction_2.zmx should recreate rays generated by construction source 2 and landing on the STOP. You can use a single Coordinate Break surface so that rays are generated from an off-axis source, as shown in the following image.

Once the construction files are created, the OFH surface can be defined in the playback file by referencing the construction files in the Comment parameter, as previously explained. The OFH is also characterized by other surface-specific parameters, defining:

• the substrate shape, that can be:
  • Conical
  • Elliptical, with added polynomial aspheric terms
  • Toroidal
• the parameters relevant to the chosen substrate shape
• the Holo Type flag, corresponding to a choice between Hologram 1 (flag = 1) or Hologram 2 (flag = 2)
• the OPD Mode flag, generally left to 0 (automatic) but overridable if the optical path difference is not correctly calculated for a specific hologram geometry
• the diffraction order

As we are trying to model the same system as file “Hologram1_transmission.zar”, we will use Shape = 0 (hologram substrate is a plane), Holo Type = 1 (both construction sources diverge) and diffraction order -1. The OFH in the playback file will then diffract rays as shown in the following image. As you can see, this is consistent with file “Hologram1_transmission.zar”.

File “OFH.zar” provided in the article attachments contains the playback file as well as the two construction files previously described. As these three files are linked together from the OFH surface, clicking File…Create Archive on the playback file groups them all in the same ZAR.

Check Reference 3 for FAQ.

Other hologram models

OpticStudio includes several other hologram models, both in Sequential and Non-Sequential modes. As previously mentioned, they are all based on the Hologram 1 and Hologram 2 surfaces, but they allow for more flexible hologram shapes. Specifically in Non-Sequential Mode, some parameters control the object outer boundary shape.

All hologram models available in OpticStudio, excluding the previously discussed Hologram 1, Hologram 2 and the OFH, are briefly described below. More information on each surface/object can be found in the relevant Help File section.

• Toroidal Hologram (Sequential). A hologram surface of toroidal shape.
• Hologram Surface (Non-Sequential). A hologram surface of planar, spherical, conical or polynomial aspheric shape, having a circular or user-defined object outer boundary shape.
• Hologram Lens (Non-Sequential). A lens with a hologram surface on the front face. The hologram may be of planar, spherical or conical shape. The lens may have a circular or rectangular outer boundary shape.
• Toroidal Hologram (Non-Sequential). A lens with a toroidal hologram surface on the front face. The lens may have a rectangular, circular or elliptical outer boundary shape.

Summary of OpticStudio hologram surfaces and objects

References

1. W.T. Welford, “A Vector Raytracing Equation for Hologram Lenses of Arbitrary Shape,” in Optics Communications, 14-3, pp. 322-323 (July 1975)
2. OpticStudio Help Files
3. FAQ: https://community.zemax.com/got-a-question-7/faq-of-optically-fabricated-hologram-ofh-how-to-parameters-holo-type-and-diffract-order-390