## Abstract

We report on the design and fabrication of novel diffractive phase elements that reconstruct distinct intensity patterns in the far-field on illumination with two specific wavelengths. The elements contain deep surface-relief structures that represent phase-delays of greater than 2π radians. The design process incorporates a modified version of the iterative Fourier transform algorithm. A 16 phase-level element for dual wavelength (blue and red) operation, with high diffraction efficiency, is demonstrated experimentally.

© Optical Society of America

## 1. Introduction

Diffractive phase elements (DPEs) are high-efficiency optical devices that, among other applications, are widely used to manipulate the far-field intensity of a laser beam.[1,2,3] DPEs are usually composed of multilevel, micron-scale structures etched into a transmitting substrate, e.g., fused silica for UV to near-infrared wavelengths. Here, we exclusively consider a specific class of beam-shaping DPE: the pattern formation element (PFE). These are periodic elements that generate intensity patterns in the focal plane formed from many, closely packed focal spots, i.e., diffraction orders. The large number of diffraction orders that can be accurately manipulated by the PFE enables the formation of complicated patterns. These patterns can be imaged so that individual spots are not resolved by the eye, resulting in an apparently continuous image. Applications of PFEs include laser material processing, marking and laser display.

One of the disadvantages usually associated with DPEs is their single wavelength operation characteristic. Illumination with a different wavelength from that specified during design will normally result in severe far-field distortion and reduction in the diffraction efficiency. In this paper, we examine the design and fabrication of a pattern formation element that operates optimally under illumination of two specified wavelengths. One of the main applications of this new class of element may be in medical applications where the control of two wavelengths is required, e.g., λ_{1} for alignment and λ_{2} for incision. Other applications include laser displays, since images consisting of more than one colour are more impressive than their monochrome equivalents, and in spectroscopy, where the manipulation of multiple wavelengths corresponding to spectral emissions is required.

The storage of multiple spot array generators has been shown theoretically for different angles of incident illumination for a continuous phase kinoform by Turunen *et al*.[4] They suggested that multiple patterns could be similarly stored in a DPE, each being reconstructed using illumination of different wavelengths. However, the simple fan-outs that they produced required a very complex DPE profile containing substantially more phase freedom than is usually necessary for this task. This requirement means that more complex output patterns cannot be easily produced using their method.

In section 2, we show that surface-relief profile acts as a constraint between the effective phase-profiles that are experienced by two different wavelengths. We also demonstrate that this surface-relief constraint can be overcome by the use of deep surface-relief structures in the DPE. The resulting requirement for the addition of many phase levels in the element and the associated limits of the fabrication process are discussed. Also, the impact of the choice of input colours on the maximum obtainable efficiency is calculated using an iterative method. A design process is presented for a two-colour PFE based on the iterative Fourier transform algorithm (IFTA). The design process reduces the quantisation error from the use of a limited number of discrete phase levels in the element. In section 3, experimental results are given for a fabricated PFE, operating under coincident red and blue colour illumination.

## 2. Design

The phase-delay, φ, from a DPE surface-relief structure of height h is found from

The chromatic variation of the refractive index of fused-silica can be found using an empirical formula given by Malitson.[6]

Altering the input wavelength to a new value will alter the associated phase-delay of this structure by an amount proportional to the ratio of the two wavelength values. Since the resulting change is linear, a surface-relief profile cannot be built up that produces a desired phase profile for one wavelength and a different phase profile for a new wavelength. Therefore, the phase profiles for the two wavelengths are constrained to each other by the physical dimensions of the surface-relief profile.

#### 2.1 Deep surface-relief structures

Noach *et al*. demonstrated the use of deep surface-relief structures as a solution to this problem.[5] Deep surface-relief features represent phase-delays of greater than 2π radians for the chosen wavelength. The advantage of such features comes from their resulting phase-delay for a different illumination wavelength. Increasing the height of a surface-relief structure by an amount corresponding to 2π radians phase-delay for a certain wavelength (λ_{1}) has no effect on the effective total phase-delay. However, the increase in the height alters the effective phase-delay for a source of different wavelength (λ_{2}), as the phase difference does not represent 2π radians. This means that deep structures do not constrain the phase profile for a new wavelength as thin structures do.

Noach *et al*. used the repeated addition of 2π radians blocks to individual structures of the phase profile for one wavelength, until the effective phase profile for a different wavelength was close to that desired for the second wavelength. This process requires the addition of many ‘2π blocks’ to form a good likeness of the required phase profile, as the generated phase levels are not evenly spaced. This is demonstrated in figure 1, where the effective phase-delays for red light are shown, under the addition of 2πL radians to a π/2 radians phase-delay for blue light (L is an integer).

#### 2.2 Fabrication considerations

Current fabrication technology employed by Heriot-Watt University is a multi-step, mask and etch, photolithographic process.[7] This process cannot usually support a surface-relief profile containing many surface-relief levels (>16) due to the increasing strain that is placed on fabrication tolerances. For multilevel elements, mask misalignment is known to have dominant effect on performance loss over other possible errors incurred during the fabrication process.[8] The effects of mask misalignment errors are cumulative, causing further degradation of the elements performance with each additional fabrication step. Fabrication of deep multilevel structures is also problematic when using a photolithographic process. The resolution that can be achieved when the masking structure is transferred into photoresist is seriously reduced for a deep multilevel structure due to depth variations in the substrate that are present from the previous fabrication levels. This results in a maximum number, and a maximum depth of phase levels, that can be effectively fabricated in a DPE. To satisfy the fabrication process that is used in this paper, we choose a 16-level element, which corresponds to a 4-level element for both the primary and secondary wavelengths.

#### 2.3 Performance of an irregularly quantised phase profile

Quantisation of a phase profile results in noise in the output as result of mismatching between the desired phase and allowed phase levels. This quantisation noise has the effect of reducing the efficiency and fidelity of the desired output. It increases severely as the number of phase levels is reduced. An irregularly quantised phase profile will produce more quantisation noise in the output than the equivalent regularly quantised version.

The maximum efficiency that can be obtained from a regularly-quantised DPE is well known.[9] An iterative procedure is applied to investigate the effect of the irregularly-quantised profiles on the maximum obtainable efficiency.[10] We choose a 16-level element as described above. The test profile is an arbitrary binary intensity pattern of size 128 by 128 orders, placed fully, diagonally, off-axis by 4 orders. The chosen illumination sources are He-Cd (λ_{blue} = 442 nm) and He-Ne (λ_{red} = 633 nm) lasers.

For these parameters, the maximum efficiency for the red output profile will be 72%. This is slightly less than the maximum efficiency of 77% that is calculated for the equivalent regular 4-level case. The choice of the illumination sources will have a significant effect on the maximum efficiency that can be obtained in one of the patterns. For blue-green light (λ_{green} = 550 nm), this value, for green, is 73%. For red light, the efficiency is reduced to a maximum of 65% under green-red illumination. The shorter wavelength is chosen as the primary source wavelength in these cases, in order to minimise the physical thickness of the DPEs.

#### 2.4 The iterative Fourier transform algorithm

The iterative Fourier transform algorithm (IFTA) is commonly used in the design of beam-shaping DPEs.[11,12] The IFTA uses the phase freedom in the desired output intensity pattern to produce an input waveform that corresponds to the required DPE. By considering a quantised phase profile within the algorithm, the IFTA minimises the effect of quantisation noise in the output, increasing the fidelity. The sampled phase profile produced by IFTA must be encoded as a realisable surface-relief profile. The angled pixels phase-encoding scheme was applied in this case to achieve high-efficiency off-axis pattern formation.[13]

We apply the IFTA to design a 16-level element with the same 4-level basis as before. The design process has two steps. In the first step, the standard IFTA is used to design a 4-level phase profile using the desired pattern for λ_{1}. Subsequently, a modified IFTA is applied with the desired pattern for λ_{2}. This modified algorithm quantises the phase profile to four uneven levels determined by the chosen wavelengths and the phase profile produced in the first step. The four allowed phase-delays at any point on the element profile are found from the equivalent phase-delay when the wavelength is equal to λ_{2}, after the addition of successive multiples of 2π radians to the phase profile for λ_{1}, using (1). The resulting surface-relief
structure contains a standard 4-level phase profile for λ_{1}, and an irregularly quantised 4-level phase profile for λ_{2}.

#### 2.5 Design results

A DPE was designed to produce simple binary intensity patterns for use with He-Cd and He-Ne lasers (blue-red). The patterns for both colours were positioned off-axis, at the same place in the output plane. Under illumination with blue light, figure 2 was reconstructed with an efficiency (η) of 72% and a signal-to-noise ratio (SNR) of 57. For red light, figure 3 was reconstructed with η = 66%, SNR = 31. SNR measures the image fidelity. It is defined as,

Where P_{mn} is the intensity in the (m,n)^{th} diffraction order, S_{mn} is the desired intensity, and 𝕡 defines the signal pattern window. The proportion of incident power in the pattern is measured by α, such that

The spacing of the diffraction orders that form each pattern will vary in proportion to the wavelength of the source. As a result, a pattern containing mixtures of two colours cannot be directly formed, since concurrent orders for each colour will not overlap. This means that the reconstructed output patterns will have dimensions that are proportional to the wavelength. Manipulating the periodicity of the element individually for each colour could solve this mismatch, however it is not considered here.

## 3. Experimental

The PFE was fabricated in fused silica, with a period of 2mm and illuminated with coincident He-Ne and He-Cd laser beams using a beamsplitter set up. The photograph of the output in figure 4 is slightly over-exposed in an uneven manner for the two colours, so fails to show the sharpness of the images that is observed in the laboratory. The size of the ‘red’ image is approximately 1.5 times that of the ‘blue’ image, as expected from the wavelength disparity. The efficiency of the ‘red’ image is measured as 61%, with the ‘blue’ image being 68%, neglecting Fresnel losses in both cases. This slight variation compared to the theoretical efficiencies can be attributed to fabrication error loss.

## 4. Summary

A design process has been demonstrated for the production of a realisable DPE that exhibits dual wavelength operation. A DPE was fabricated that reconstructed two distinct patterns when illuminated with coincident red and blue wavelength laser.

## Acknowledgements

The authors would like to thank N. Ross and A. Waddie for their contributions to the fabrication process.

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