This is a short tutorial about why a transmission hologram recorded in a wet emulsion might not be viewable when dry. It is a response to a question by Justin W on the Holography Forum.

Kaveh Bazargan
kaveh@holographer.org

About the author
Kaveh’s interest began with seeing the Royal Academy of Arts exhibition of holograms in London in 1976. He was studying physics at Imperial College London, and continued to complete a PhD in ‘display holography’. His passion remains the quest to achieve ultimate realism with holography.

Over the years I have often wanted to be able to simulate the recording and reconstruction process in holograms. I am particularly interested in display holograms, as opposed to HOEs. During my studies I did write little programs for specific cases, but not a general purpose program. There are two parts to such a program: first to calculate the reconstructed image, having fed in the the recording data, and second, to present this in a nice graphical form, preferably three-dimensional.

Recently I discovered POV-ray (http://www.povray.org), a 3D image rendering engine, and realised it was the ideal tool for the job. Please see the the amazing images people have created with it. POV-ray is free, platform-independent (Linux, Unix, Macintosh, Windows), and produces images equal to renderers at any price. The beauty of using POV-ray is that not only is it a superb 3D rendering engine, but it is also an excellent programming language. Never calling myself a real programmer, I slowly put some equations in, got results, and soon got carried away! Slowly I built in more capability. The more I work on it the more possibilities I see to simulate the holographic process. I can now see that we can simulate full color holograms, ‘pseudocolour’ holograms, and rainbow holograms. At the moment it is limited to transmission holograms, but I hope to extend it to reflection holograms too.

The basic equations used are those of Champagne (J. Opt. Soc. Amer. 57 (1967) p. 51). In this document I’ll not go into the equations, but just the functionality of the program. This version is really still being worked on, and there will be bugs, so I do appreciate feedback.

License

HoloPov is released under the GNU LGPL license (see http://www.fsf.org/licenses/licenses.html#LGPL). The code is therefore ‘open’ and you can modify it and use it free of charge, and to redistribute it, within the restrictions of the license. In the spirit of Free Software, I encourage people to test and modify the code to improve and debug it, and help serve the holographic community.

Requirements

You should be running POV-ray on your computer. I have used version 3.5. The files should run with any installation of POV-ray above 3.5. POV-ray runs on most platforms and is device independent.

Examples

To see some examples of the kind of output produced, please see my other article, “White light transmission holograms”

Availability

HoloPov is available for download here.

File structure

There are three files in the package, holopov.pov and holopov.inc, and optics.inc. You will also need to get CIE.inc from http://www.ignorancia.org/zips/lightsys4.zip. holopov.pov is the file that is run through POV-ray. It will #include holopov.inc which will in turn #include several other files. Most of these are standard files distributed with POV-ray, but optics.inc is a library file of mine, and CIE.inc is a third party file which, amongst other things, works out the approximate color for a given wavelength.

The coordinate system

We use the coordinate system of POV-ray, i.e. a left-handed cartesian system. The hologram is always in the x–y plane, centered at the origin. If you imagine your computer screen as being the hologram, with the x-axis pointing to the right, and y pointing up, then the z-axis will be pointing into the screen. The center of the screen is <0,0,0>

I have chosen to have the observer in the z < 0 region, and other points in the z > 0. But I think that any point can have any coordinates. If the formulae have been coded properly, then this should be the case I think. I have not dared test this yet!

Overall concept

The basic idea is that you dial in the recording and the reconstruction parameters, and the program will work out the position of the image. By ‘parameters’, I mean the following:

• position of the object
• position of the recording source
• position of the reconstruction source
• position of the observer
• recording wavelength
• reconstruction wavelength

All these values are entered into the main file, i.e. holopov.pov.

The object

These are the parameters that define the object:

• obj_dist
• obj_angle
• obj_side_angle
• obj_theta
• object_size
• grid_sep

After a lot of thought, I decided that spherical polar coordinates were the best way to enter the data for the object, at least for me.

obj_dist is the distance of the center of the object from the origin, i.e. from the center of the plate. This value is always positive.

obj_angle is the angle of the object in the y–z plane, i.e. above or below the horizon.

obj_side_angle is the angle of the object in the x–z plane, i.e. the angle if we were looking directly from above.

obj_theta is the angle around the y-axis. This the angle one would see if one were looking down the y-axis from above.

To have an object in the observer space, i.e. z < 0, we can use values of obj_theta, obj_angle > 90^∘ or < −90^∘.

object_size needs some explanation. I have defined an object to be not just a point, but a 3-dimensional grid of points. The three coordinates in object_size define the extent of the image in each axis. For example <2,2,2> means a cube, with dimension 2 on each side. grid_sep defines the distance between successive grid points. The smaller the value of grid_sep, the more grid points will be present. If object_size is set to <2,0,2>, the grid will effectively be a sheet, with zero height.

Please note that values corresponding to object distance and angle refer to the average object position, i.e. the center of the object.

Recording geometry

Here are the values to set:

• ref_dist
• ref_angle
• ref_side_angle
• ref_theta
• lambda_r

Again we use spherical polar coordinates. lambda_r is the wavelength used for recording the hologram. Again we should be able to use angles > 90^∘ and < −90^∘ to have, for example, a converging beam. I need to do a bit more thinking on this. For example, how to distinguish between a converging beam and a diverging beam, both referring to the same point in space. I hope that we can have a sign system that caters for all combinations.

Reconstruction geometry

Here are the corresponding values in reconstruction:

• rec_dist
• rec_angle
• rec_side_angle
• rec_theta
• lambda_c

Observer

These values define the observer:

• obs_dist
• obs_angle
• obs_side_angle

I have been using angles of zero, and a negative observer distance to signify an observer on the z-axis, but to be consistent with above.

Camera settings

These are the more or less standard settings used in POV-ray for the camera:

• camera_loc – The camera position. Presently in cartesian coordinated, but might be better to change it to polar in future releases.
• camera_look – Center of the rendered image.
• camera_angle – As in the normal POV-ray definition. To zoom in, we decrease this value.

H1 and H2

I have managed to simulate the effect of having a master hologram, or as it is normally referred to, the H1. The hologram doing the final imaging, I will always refer to as the H2. The dimensions of the two plates are set by:

• h1_width
• h1_height
• h2_width
• h2_height

The main reason for setting the dimensions of the two plates is to take into account the image cut-off, or vignetting, that occurs as the viewer moves around. We also need a value to define the position of the H1 relative to H2. We could of course give the distance between the two, but I have opted to define h1_sep, which is the distance from the center of the object to that of the H1. So as we change the position of the object, all other values being equal, we are also implicitly changing that of the H1.

Options

I have put in a lot of options which have a bearing on how the final image is drawn. These are set as parameters that are tested later in holopov.inc. A value of 0 implies the value is false, any other value sets the value to true. Here are the list of options now available:

• object_points_true – Draws a sphere at each object grid point. The color is a color approximating to the recording wavelength. This is achieved by the Wavelength() macro in CIE.inc (see the file for details). The radius of the sphere is determined by object_sphere_rad in holopov.inc.
• object_grid_true – Joins up the object grid points with cylinders, color as above. The radius of the cylinder is set by grid_cylinder_rad in holopov.inc.
• image_points_true – As with object_points_true, but for image points. Color corresponds to the reconstruction wavelength. Radius is set by image_sphere_rad. I have set the object and image radii separately.
• image_grid_true – As with object_grid_true, but for the image. The radius of the cylinder is set by grid_cylinder_rad.
• ref_beam_true – Draws four lines from the reference point to the four corners of the recording plate. The ‘lines’ are actually cones which taper to zero at the reference point. The maximum radius of the cone is equal to ray_radius. Color approximates to the reference wavelength.
• rec_beam_true – As with ref_beam_true, but for the reconstruction beam. Color as expected.
• obj_beam_true – Draws a cylinder from the center of the plate to the position of the observer. Color that of the recording wavelength, radius set by ray_radius.
• image_beam_true – Joins the center of the image to the observer position. The section from the observer to the plate is a solid cylinder, radius determined by ray_radius, and the section from the plate to the image is drawn as a dotted line. Paramters for this are dotted_space and dotted_ray_rad.
• eye_true – Draws a pair of eyes, a bit evil looking at present, at the observer position, and looking at the center of the image.
• plate_true – Draws the recording plate, the dimensions of which are determined by Plate, in holopov.inc. I have left this in the .inc file, as I have not had to change it much, but it can be moved to the main .pov file.
• vignette_true – This option will draw only those parts of the image which would be visible with the geometry described, i.e. the image will be cut off at the edges of the plate.
• disp_comp_true – Uses ‘dipersion compensation’. This means it disregards the value set for rec_angle, and adjusts it so that the image is reconstructed as close to the object position as possible. In other words, it compensates for lateral chromatic dispersion.

H1 and H2 options

We’ll list the H2 options first, as the the H2 is always present, but H1 is optional:

• h2_visible – This just means that the H2 plate is drawn in the final rendering. Sometimes it might be best not to draw it.
• h2_vignette_true – This will simulate the vignetting effect, or the cut-off, of the h2. In other words, an image point is drawn only when the straight line from that point to the eye intersects with the H2. This works for real and virtual images.
• h2_vignette_visible – This draws a faint volume in space, rather like a distorted pyramid, showing how the vignetting works. It is included for educational purposes.
• h1_visible – As above, but shows the H1. I have only tried this in viewer space, i.e. a real image of the H1 projected, but it should work in the case of a virtual image too.
• h1_vignette_true – As with H2, but this is far more important for holgraphers, as it determines the viewing angle for the image.
• h1_vignette_visible – As with H2.

Camera options

• camera_observer_true – The camera is at the observer position (between the two eyes!). I have added 0.1*y to its value later, so that the camera is not blocked by the image or object beams cylinders, if these are present.
• look_at_image_true – Always keeps the center of the image in the center of the picture.
• orthographic_true – Uses orthographic viewing. Useful if looking down at one of the axes. Actually, we are using a ‘cheat’ orthographic view, because true orthographic is not compatible with screen.inc which we need in order to put the data in the corner of the image. In our case we set a high value for the distance of the camera, and adjust the camera angle to a very small one. If you are interested, see #if (orthographic_true)… in holpov.inc.

Multiple images

Quite often we want to superimpose in the same picture, the result of varying one of the values, for example, observer position, reconstruction wavelength, etc. I have defined a variable called repeat_image. This can be set to a value which determines which parameter is to be varied. For example, we can say #declare repeat_image = Lambda_c to vary the reconstruction wavelength. Note the upper case first letter to distinguish this ‘choice’ from the actual value, which is lambda_c. (I have previously set Lambda_c to a number, so this is just a trick to make it easier to remember what is to be varied.)

In holopov.pov I have inserted and commented out all the possible variables, so these just need to be uncommented. If nothing is to be varied, then we choose None.

The value repeat_amplitude determines the ‘amplitude’ of the variation around the central value set previously. So, for example, if lambda_c has been set to 550, and repeat_amplitude is set to 150, then the range for repetition will be from 400 to 700nm. The number of steps is determined by number_of_steps. A value of 2 will result in two variations, one for 400 and one for 700nm.

Animation

In addition to multiple images in the same output, we can animate the scene by varying any of these parameters. Using multiple images and animations are independent, so it is possible to have a multiple image that is animated too. For instance, we can show the effect of using different wavelengths to reconstruct in the same scene, using multiple images, then animate that whole scene by moving the observer in an arc, by varying obs_theta.

The equivalent of repeat_amplitude in animation is the animate variable, and it is set in the same way as in multiple images.

There is one more parameter that determines how the frames are distributed through time: if sinusoidal_true is set to zero, then the frames are evenly distributed through time. Otherwise, the distribution is sinusoidal. In other words, the variation is fast near the central value, and slows down towards the extreme values. When the final animation is ‘looped back and forth’, this gives a more natural, smooth motion.

Running the animation

Please note that a command for running the animation must be given, otherwise the animation settings will be ignored. The precise method for running the animation depends on your installation. You should either choose animation through the user interface, or use a command line if you are using Linux or the command line version on Mac OS X (which is what I use). In any case, the most important point is that the clock variable must go from -1 to 1. Other options, such as total number of frames, resolution, etc, are set according to your preference.

Of course POV-ray can only produce the individual frames for an animation. You will need to use a utility to convert these to a true movie file. I use QuickTime Pro, and it works pretty well. The best way to look at the animation is to loop the animation back and forth continuously. In QT Pro, choose ‘loop back and forth’ from the Movie menu.

Other comments

Data output

The reason for including screen.inc is to allow data to be output to the file. This works very nicely. Presently only a few values are output, and there is no choice as to what to print. I need to spruce this up a bit.

The maths

This was a bit more complex than I thought. The basic equations are those of Champagne of course, but there is a basic complication in display holography which is not normally mentioned. In a conventional optical system, the ‘pupil’ is normally given, and the principal ray is that which goes through the center of the pupil. So we can apply the equations and get the answer. In a display hologram, we don’t know where the pupil will be. In other words, the precise spot on the hologram that the observer looks through is not fixed. If the wavelength is changed, the image will shift, and the observer will be looking through a different part of the hologram in order to see the same image point. The same happens of course if the observer moves. That point is the center of our optical system.

So before using the elegant equations of Champagne, I had to use an iterative method (Newton’s approximation) to find the principal ray. Only then could I apply the equations. This process has to be repeated for each point in a grid.

I will detail the mathematical background in another article soon.

Divisions by zero

I have caught a few cases where division by zero occurs, e.g. when obj_dist = 0. In this case I use the simple trick of giving it a very small distance. (I think I am showing that I am not a real programmer!) I appreciate any more cases of such errors, as well as suggestions of better ways to avoid them.

Lighting

This is set as a simple set of two lights, which can be changed.

Things to do

Short term

• Check angles for points in −z region
• Group items better as macros in holopov.inc
• Improve data display
• Provide option for a solid looking cube, rather like ball and sticks, as now.
• Provide a user-friendly GUI. It is important that this is portable, and
  more or less device independent. I am presently trying out Revolution
  (http://www.runrev.com)

Longer term

• Show a ‘real’ object and its image, not just grid points or cubes
• Incorporate conventional optics (like a field lens) to modify image

Kaveh Bazargan
kaveh@holographer.org

About the author
Kaveh’s interest began with seeing the Royal Academy of Arts exhibition of holograms in London in 1976. He was studying physics at Imperial College London, and continued to complete a PhD in ‘display holography’. His passion remains the quest to achieve ultimate realism with holography.

I am going to look at the techniques available for making white light viewable transmission holograms. I will go step by step, starting with a straightforward laser transmission hologram, and look at the causes of image blur when we view this in white light, and from there go on to ways of fixing this blur. I will then look at image plane (full aperture) and rainbow holography, and finally examine the little used method of dispersion compensation. There is, of course, reflection holography for white light viewing, but I will not discuss that here.

I find that simulated animations convey the best intuitive picture of a physical process. So I have used animations throughout this article.

Laser transmission holograms

Let’s start by looking at the simplest off-axis hologram. We’ll use a collimated reference beam, and a wavelength of 550 nm. This is a convenient wavelength to choose, as it is roughly in the centre of the visible spectrum, i.e. midway between 400 and 700 nm. We’ll bring in the reference beam from an angle of 45^∘ to the normal to the plate. The object is centred on the plate, i.e. object angle is 0^∘. We have chosen an object made up of a series of small spheres on a 3D grid. This helps us later in looking at distortions and aberrations in the image.

The ‘perfect’ image

Let’s reconstruct the image of the hologram we have just made, using a reconstruction beam identical in geometry and wavelength to the original reference beam. As we would expect, we get an image that is geometrically identical to the object, i.e. it is not distorted. Moreover, no matter what angle we look at it from, the position of the image remains identical to that of the original object. In other words, there are no aberrations in the image. The undistorted laser-reconstructed image, almost indistinguishable from the original object, is technically the most impressive image. We can call this the perfect holographic image.

Figure 1: Reconstucting the ‘perfect’ image.

But the reconstruction beam is not always identical to the reference beam. It may have a different geometry or wavelength. Now we can take a detailed look at the effects of such changes on the resultant image.

Chromatic blurring

A major cause of image blur is the use of light that is not monochromatic, i.e. is made up of more than one wavelength. The extreme case is, of course, white light, which contains the complete range of visible wavelengths. We can start by using just one wavelength, which is different to that of recording, look at the effect of that, then see what happens with white light.

Changing the wavelength

Figure 2: Reconstructing the image with a red wavelength.

Figure 3: Reconstructing the image with a blue wavelength.

Let’s look at what happens when we change the wavelength of the reconstruction beam, leaving the geometry as before. Suppose we now use a red laser beam, of wavelength 700 nm. This wavelength is around the limit of the eye’s sensitivity, i.e. it is the longest wavelength that the eye can see, so you would not use it in practice, as the image would appear very dim, but it is a nice round figure to work with, and convenient for when we come to look at the dispersion of the spectrum by the hologram. Well, a rule of thumb in (transmission) holography is that red bends more. So, say for the centre of the hologram, whereas with the original green light the light was diffracted through 45^∘, now it bends through a greater angle. You can see that the shape of the object is definitely ‘distorted’. It is somewhat sheared, and ‘squashed’ longitudinally.

Let’s now move our viewing position to the left and right a bit. We notice that the image position depends on where we are looking at it from. This is what is known as ‘aberrations’. It is what causes ‘swing’ in the image in a lot of display holograms. If the image is highly aberrated, i.e. it moves quickly with respect to observer movement, then it can be uncomfortable to view from any position, as each eye will see the image in a slightly different position, and the brain is confused with the signals it is not used to. In severe cases, it can cause a headache, just like when viewing some stereo images.

Going the other way in the spectrum, we can now try illuminating the hologram with a deep blue laser beam, namely 400 nm. Again, this is on the edge of the visible spectrum. As we might guess, the image is distorted and aberrated again. This time the magnification and shift are in the opposite sense to the case of red light. As before, we can see the aberrations in the image by noting that the apparent image position shifts, depending on the observer position.

Depth of the image

Here is a very important point. Looking at each image reconstructed, we can see that the shift in image points is greater for image points located further from the hologram plate. This is a general rule for holographic images: the further the image point from the hologram, the more prone it is to blurring.

Illuminating with white light

Figure 4: Reconstructing in white (or polychromatic) light.

We have seen what happens when we reconstruct the image with a single wavelength that is different to the original. Now lets use white light, which is a continuum of all visible wavelengths. As expected, the deep red wavelengths diffract most, the deep blue least, and all other spectral colours form images in between. What we get is a highly blurred image. If the image size is small, or it is a long way from the hologram, it may be indistinguishable as an image.

Whenever we use broadband (e.g. white light) illumination, we will have some blurring. So we can’t eliminate it entirely, but we can minimize it.

Image plane hologram

Remember that in general, all blurring is proportional to the distance of the image from the hologram. So the first thing we should do is to minimize this distance. But in a transmission hologram, there is a limit to how close the object can be to the hologram plate, as the reference beam must not be impeded. So how can we get the image even closer to the final hologram?

The answer is, of course, to use a two step method: first make a master hologram (conventionally called H1), and use the projected image to make another hologram (H2). Now the iamge can be place anywhere, even straddling the plane of the hologram, partly in front and partly behind it. The average distance of the image is now close to zero, and the image is as close as we can get it to the hologram plane.

Figure 5: An image plane hologram.

Now where the image coincides with the hologram plane, it is perfectly sharp. In practice, display holograms up to around 2cm in depth can be accpetable in such a hologram. This type of hologram is also called an ‘open-aperture’ hologram, to distinguish it from a rainbow hologram, which we’ll look at later.

The open-aperture transmission hologram can have a very realistic look to it. Because the entire visible spectrum is used in image reconstruction, the image viewed is black and white, or achromatic. This seems to look more realistic than, say, a reflection hologram, which has an overall spectral hue. The realism is especially true with objects that we naturally assume to be achromatic, or nearly so, e.g. bone, wood, metal.

We now look at different methods we can use to increase the depth of the white-light reconstructed image.

Rainbow or Benton hologram

Now it’s time to look at the image plane hologram in a bit more detail. Remember that that image plane hologram was made using an intermediate ‘H1’. I won’t go into the details of the two-step process, but the result is that when we look at the final hologram (the H2), we are effectively looking through a ‘porthole’ which is the image of the H1, and which is projected into viewer space. The image is only visible when we look through this porthole.

Figure 6: The H1 ‘porthole’, reconstructing with red...

Figure 7: ...and with blue.

Figure 8: Effect of the H1 when using white light.

Now, just as the image is distorted when the recontstruction wavelength or geometry are different to those of recording, so is the shape of the H1. So we should consider the H1 to be part of the image. The result is that not only is the image distorted and displaced, but for each wavelength it is visible only the ‘porthole’ for that wavelength. For example, for a red image, the porthole is diffracted through a greater angle, and only the top of the image might be visible to the viewer.

When the image is shallow, the full aperture method works fine, and we see a sufficiently sharp achromatic image. But we notice that as we move our head towards the top and bottom extremities of the visible zone, the image loses its ‘achromaticity’ and takes on a color tinge. At the top it looks reddish, and looking from the bottom of the porthole it looks bluish. This is because in those areas we are only picking up the image from a narrow band of wavelengths, not the full spectrum. This last effect is the important one. If only we could make the image as it appears at the very top and bottom, i.e. color-cast, but sharper, we could would be able to view a deep image in white light.

Figure 9: The rainbow hologram.

Well, welcome to rainbow holograms! This is precisely the important step that Stephen Benton introduced. His solution was to reduce the height of the H1. What happens is that the overlap of the H1’s in space is reduced, so at any time the viewer can only see each part of the image in one wavelength or, to be more precise, a small range of wavelengths. Hence the blurring is reduced. But there is a price to pay. As the height of H1 is very limited, there is very little vertical parallax. In other words, we can’t look above and below the image. The image seems to distort and ‘follow us around’ as we move up and down.

Figure 10: Effect of the viewer distance in a rainbow hologram.

Figure 11: Effect of H1 height on image sharpness.

Dispersion compensation

Now we take another approach to reducing chromatic blur. Let’s go back to the case where we changed the wavelength and watched the image distort and be displaced. how can we bring the image back to be aligned with the original position of the object? Well, the answer is to alter the angle of the reconstruction beam. Suppose again we make a hologram with a reference angle of 45^∘ and wavelength 550 nm, and reconstucted with red light at 700 nm. When the reconstuction angle is the same as that of recording, the red light ‘bends more’ and the image seems to be lower than the original object position. As the reconstruction angle is altered to hit the plate at a steeper angle, the image moves towards the object position. At around 64^∘, we find that the image is more or less aligned with the object position. We can call this the ‘dispersion compensated’ angle for that wavelength. For blue light at 400 nm, the angle turns out to be 31^∘.

Figure 12: Effect of replay angle on image location for red light...

Figure 13: ...and blue.

Another thing that we notice is that the distortion in the image is reduced, as is the ‘swing’ as we move around. This means that dispersion compensation not only brings the image back to the original position, but reduces the aberrations too.

Figure 14: Viewing a dispersion compensated image in monochromatic light.

Figure 15: Viewing a dispersion compensated image in white light.

Figure 16: Improving sharpness by using an image plane hologram.

But if we are using white light to view the image, we need each wavelength to hit the plate at a different angle, namely the dispersion compensated angle for that wavelength. There is a very convenient way of achieving this, and that is to use a plain diffraction grating to disperse the beam first, before it hits the hologram. In effect, we compensate for the normal ‘dispersion’ of the white light when it diffracts from the hologram, but ‘predispersing’ it through a diffraction grating first.

As we can see, dispersion compensation goes a long way to eliminating image blur, but there is still residual blur present. Once again, the further the image from the hologram, the greater the blur. By using an image-plane hologram in conjunction with dispersion compensation, we can obtain very sharp images.