Ersoy O.K. Diffraction, Fourier optics, and imaging (Wiley, 2006)(ISBN 0471238163)(427s) PEo

Reference wave

Object wave

Recording medium

Figure 13.1. Geometry 1 for recording a hologram.

13.2COHERENT WAVE FRONT RECORDING

Suppose that an object (desired) wave Uðx; yÞ is expressed as

Another reference wave Rrðx; yÞ is expressed as

The two waves will be incident on a recording medium that is sensitive to intensity as shown in Figure 13.1 or Figure 13.2.

It is important that the waves are propagating at an angle with each other as shown. The intensity resulting from the sum of the two waves is given by

Iðx; yÞ ¼ jAðx; yÞj2þjBðx; yÞj2þ2Aðx; yÞBðx; yÞ cosðcðx; yÞ fðx; yÞÞ ð13:2-3Þ

where the last term equals AB þ A B and includes both Aðx; yÞ and fðx; yÞ.

Object wave

2q

Reference wave

Recording medium

Figure 13.2. Geometry 2 for recording a hologram.

The transmission function of optical recording devices including photographic film is sensitive to intensity. We will assume that the sensitivity is linear in intensity. Bðx; yÞ will be assumed to be constant, equal to B, say, a plane wave incident perpendicular to the hologram as shown in Figure 13.2 and approximately a spherical wave far away from its origin. The transmission function of such a device can be written as

where C and b are constants. tðx; yÞ represents stored information. Now suppose that the generated hologram is illuminated by another reference wave R. The wave emanating from the hologram can be written as

Note that U3 is the same as U except for a multiplicative term. Thus, U3 appears coming from the original object and is called the virtual image. U4 is the same as U except for a multiplicative term. U4 converges toward an image usually located at the opposite side of the hologram with respect to U3. This is illustrated in Figure 13.3. U3 and U4 are also called twin images. Usually U3 forms an image to the left of the hologram, and U4 forms an image to the right of the hologram. Then, these images are called virtual image and real image, respectively. However, which image is virtual and which image is real actually depend on the properties of the reference waves used during recording and reconstruction. These issues are further discussed in Section 13.6.

Suppose that Rr and R are the same, and they are constant, as in a plane wave perpendicular to the direction of propagation. Then, U3 is proportional to U, and U4 is proportional to U . In this case, U and U would be overlapping in space, and a

Figure 13.3. Formation of virtual and real images.

viewer would not see the original image due to U only. This was the case when Gabor first discussed his holography method.

13.2.1Leith–Upatnieks Hologram

Instead of a simple plane perpendicular plane reference wave, R can be chosen as

The term e j4pay causes U4 to propagate in a direction that does not overlap with the direction in which U1, U2, and U3 propagate if a is chosen large enough.

Alternatively, the geometry shown in Figure 13.2 can be used. In this case, the reference wave illumination is normally incident on the hologram, but the wave coming from the object is at an average angle 2y with the z-axis. The reference wave equals D, a constant, whereas the object wave can be written as

where Aðx; yÞ is now complex and contains the information-bearing part of the phase

of Uðx; yÞ. Further analysis gives the same results as in Eqs. (13.6)–(13.9). We note that U3 is proportional to Aðx; yÞe j2pay whereas U4 is proportional to A ðx; yÞej2pay.

Hence, they propagate in different directions.

EXAMPLE 13.1 Determine the minimum angle 2ymin so that all the wave components are separated from each other at a distance sufficiently far away from the hologram.

Solution: At a sufficient distance from the hologram, for example, in the Fraunhofer region, the wave components will be similar to their Fourier spectra. Denoting the Fourier transform by FT, we have

FT½U1& ¼ C1dðfx; fyÞ

FT½U2& ¼ b1AF AF

FT½U3& ¼ b2AF ðfx; fy aÞ

FT½U4& ¼ b3AF ð fx; fy aÞ

where C1; b1; b2; b3 are constants, indicates autocorrelation, and AF is the FT of

Aðx; yÞ.

Suppose that the bandwidth of Aðx; yÞ equals fmax in cycles/mm. FT½U1& is simply a delta function at the origin. FT½U2& has a bandwidth of 2fmax because it is proportional to the autocorrelation of AF with itself. U3 is centered at a, and U4 is

Object

(a)

Spherical

reference

Hologram plane

Object

(b)

Figure 13.4. Two geometries for lensless Fourier transform holography.

centered at a along the fy direction. These relationships are shown in Figure 13.4. In order to have no overlap between the terms, we must satisfy

a 3fmax

or

2ymin ¼ sin 1ð3lfmaxÞ

13.3TYPES OF HOLOGRAMS

There are many types of holograms. Transmission holograms transmit light such that the information is viewed through the transmitted light. With reflection holograms,

the information is viewed as a result of reflection from the hologram. Most types of holograms discussed below are transmission holograms.

13.3.1Fresnel and Fraunhofer Holograms

Fresnel and Fraunhofer approximations were studied in Chapter 5. A hologram is a Fresnel hologram if the object to be reconstructed is in the Fresnel region with respect to the hologram. A hologram is a Fraunhofer hologram if the object to be reconstructed is in the Fraunhofer region with respect to the hologram.

13.3.2Image and Fourier Holograms

In an image hologram, the middle of the object to be reconstructed is brought to focus on the plane of the imaging device. In a Fourier hologram, the plane of the imaging device is the same as the plane at which the Fourier transform of the object transmittance is generated. A Fourier hologram looks like a modulated diffraction grating, and such holograms typically have less need for a high-resolution recording device.

A variant of the Fourier hologram is the lensless Fourier transform hologram. There are two possible geometries for recording such a hologram as shown in Figure 13.4. The main property of such a hologram is that the focus of the spherical reference wave is on the same plane as the object image, either virtual (Figure 13.4(a)) or real (Figure 13.4(b)). Suppose that both the object wave and the reference wave are spherical waves with their sources on the same plane. On the hologram plane, the pattern of intensity has the same type of fringe patterns as in a real Fourier hologram. Hence the name lensless Fourier transform hologram.

As compared with Fourier and lensless Fourier transform holograms, Fresnel holograms have higher density fringe patterns and therefore require higher resolution recording devices. For this reason, Fourier holograms are usually preferred when recording with electronic recording devices.

13.3.3Volume Holograms

Volume holograms are recorded in a thick medium. A major advantage gained is that the hologram is wavelength selective and thereby also works with white light after recording. Such a hologram also has very high diffraction efficiency.

In order to explain how a volume hologram is generated, the simple case of the interference of two plane waves in a thick medium will be discussed. One is the object wave with wave vector k0, and the other one is the reference wave with wave vector kr. They can be written as

The intensity due to the interference of the two waves in the medium can be written as

Equation 13.3 represents a sinusoidal pattern with a period equal to p ¼ 2p=jksj. Consider the case of kr pointing along the z-direction, and k0 being at an

angle y with the z-axis, as shown in Figure 13.5. Then, the period can be written as

The resulting pattern recorded in a thick emulsion represents a volume grating. If the same reference wave is used during reconstruction, it can be shown that the object wave gets reconstructed perfectly provided that the reference wave is incident at the same angle to the hologram as before, and its wavelength is also the same as before [Saleh and Teich, 1991]. If the wavelength of the reference wave changes, the object wave does not get reconstructed.

If white light is used during reconstruction, the component of the white light that is at the same wavelength as the original reference wave provides the reconstruction

Emulsion

Reference wave

q

Volume grating

Object wave

Figure 13.5. Interference pattern of two plane waves generating a volume grating in a thick emulsion.

of the object wave. This is a very useful feature because white light can be used for reconstruction.

13.3.4Embossed Holograms

Embossing is the process used in replicating compact Disks and DVDs with precision of the order of an optical wavelength. The same process is used to replicate a holographic recording in the form of a surface relief hologram. In this way, surface relief holograms are recorded in photoresists or photothermoplastics. Cheap mass reproduction leads to their wide usage in a number of markets, for example, as security features on credit cards or quality merchandise.

The first step in this process is to make a hologram with a photoresist or photothermoplastic. A metal master hologram is next made from the photoresist hologram by electroforming, leading to a thin layer of nickel on top of the hologram. The layer of nickel is next separated from the photoresist master and forms the metal submaster, which can be easily duplicated a number of times. The resulting metal submasters are used in the embossing process to replicate the hologram many times. For this purpose, the metal submaster is heated to a high temperature and stamped into a material such as polyester. The shape is retained when the film is cooled and removed from the press. The embossed pattern is usually metallized with aluminum to create the final reflection hologram.

13.4 COMPUTER SIMULATION OF HOLOGRAPHIC RECONSTRUCTION

The holographic equations can be computed, and the images can be reconstructed from a hologram existing in the computer, for example, by using the NFFA discussed in Chapter 7. Considering the most important terms in Eq. (13.2-3), a hologram can be modeled as

where uðx; yÞ is the object wave, and c is a constant. Equation (13.4-1) includes the twin images and a constant term resulting in a central peak.

Figure 13.6 shows the image of an object whose hologram is to be generated by using the inverse NFFA propagation to yield uðx; yÞ with the following physical parameters:

z ¼ 80 mm, l ¼ 0:6328 m, hologram size: 400 m 400 m, output size: 64:8 mm 64:8 mm

The resulting hologram is shown in Figure 13.7. The reconstruction from the hologram is obtained by using the forward NFFA propagation, and the result is shown in Figure 13.8.

Figure 13.8. Holographic reconstruction of the twin images.

The recording and reconstruction wavelengths are l1 and l2, respectively. l1 may be different from l2.

The final hologram may be of a different size from the initial hologram generated by recording. A point on the initial hologram at z ¼ 0 will be denoted by ½x; y& whereas the corresponding point on the final hologram will be denoted by ½x0; y0&.

The phase due to the object wave at a hologram point ½x; y& can be written relative to the origin as

foðx; yÞ ¼ 2lp ðx x0Þ2 þ ðy y0Þ2 þ ðz z0Þ2&1=2 ½x2o þ y2o þ z2o&1=2g

1

ð13:5-1Þ

Keeping the first two terms of the Taylor series expansion, the last equation can be written for nonconstant terms as