A one-step introduction of functional defects into a photonic crystal is demonstrated. By using a multi-beam phase-controlled holographic lithography, line-defects in a Bragg structure and embedded waveguides in a two-dimensional photonic crystal are fabricated. Intrinsic defect introduction into a 3-dimensional photonic crystal is also proposed. This technique gives rise to a substantial reduction of the fabrication complexity and a significant improvement on the accuracy of the functional defects in photonic crystals.
© 2008 Optical Society of America
Photonic crystals (PhCs), proposed in 1987 [1, 2], have undergone rapid development in recent years. The generation of broadband short light pulses in hollow fibers , light localization and controlled emissions in PhCs , and integrated nano-photonics  are among important applications of PhCs. These and many other applications require accurate introduction of defects into the PhCs materials, much in the same way as defects are introduced into the semiconductors. There has been a paramount need to develop means to incorporate functional defects directly into PhCs, both effectively and accurately .
Multiple beam holographic lithography  has been widely employed to fabricate large size, defect-free PhC templates . Various techniques exist for introducing functional defects into PhCs generated by holographic lithography. One method is to introduce functional defects via laser writing, after the crystal has already been formed [9, 10]. However, this technique degrades the resolution and accuracy of the defect, and introduces a costly and complicated second step to the manufacturing process .
Special PhC features can also be introduced in a single step, using multi-beam phase controlled interference holography. With this method, each of the interfering beams, which is large enough in transverse dimension to neglect the propagation diffraction, is controlled using variable phase retardation to determine the interference pattern. This is the space domain analogy of time domain pulse shaping, which was proposed to control physical and chemical processes sometime ago , and has given rise to many significant innovations and developments in different branches of science and technology . In previously reported works, both simulated numerically  and demonstrated experimentally [14, 15], phase control has been used to vary the space group of the generated intensity pattern.
Besides the multi-beam approach, an advanced phase-control holographic lithography has also been used to introduce defects into a PhC in a single step [16, 17]. This technique is based on the multi-beam diffraction in a phase mask, and it is sometimes called diffraction optical element (DOE). As a result, each pixel must be very small to produce effective diffraction, requiring thousands, even millions of pixels for incorporation defects into a PhC structure via holographic lithography. In this approach, the phase mask needs to have very fine scale in order to produce multiple higher-order diffraction beams. Furthermore, this approach is limited because a particular phase mask is suitable only for one particular structure; and the fabrication of the mask itself may be an extra burden compared with coherent multi-beam interference [16, 17]. Very recently, a tunable liquid-crystal spatial light modulator (LC-SLM) has been used to circumvent this problem in a similar experiment, which used holographic lithography to manufacture quasicrystals .
In this communication, we report the demonstration of a novel and simple method of one-step introduction of a functional defect into a PhC using multi-beam phase-controlled holographic lithography. A tunable LC-SLM is introduced to control the phases of the multiple beamlets, making the light intensity distribution dynamically reconfigurable; and a genetic algorithm (GA)  is used to find an amplitude and a phase pattern that will achieve the desired optical intensity distribution with a relatively low number of beamlets. This method has the following merits compared to previously employed techniques used to manufacture functional defects in PhCs: (I) far fewer beamlets are employed to insert defects (tens, as opposed to thousands), (II) the LC-SLM means that different PhCs can be quickly manufactured without having to go through an etching process to build a new phase mask.
We fabricated a line-defect in a one-dimensional (1-D) Bragg structure and an embedded waveguide in a two-dimensional (2-D) PhC structure. We also show that more complicated defects in three-dimensional (3-D) PhCs can be achieved. With this multi-beam phase-controlled technique, the defects are intrinsically and conveniently introduced into a PhC.
2. Design and experiment
Holographic lithography is based on the recording of an intensity pattern produced by multi-beam interference. The interference intensity pattern is given by 
with i, j denoting different light beams. E i and φi are the electric field and the initial phase of the beams respectively, and k i is the wave vector. The laser interference pattern can be controlled by changing the polarization and amplitude of E i, the wave vector ki, as well as the phase φi for each of the input light beams. In order to simulate the intensity pattern recorded in the photoresist material-SU8, the effects of the refraction and 40% film shrinkage in axial direction of SU8 are considered in the following simulating . The GA is used to find the set of E i and φi values that generates the desired multi-beam interference intensity pattern.
The experimental setup is shown in Fig. 1. The output beam from a continuous-wave Nd:YVO4 laser at 532 nm and with power of 120 mW was divided with a mask to multiple beamlets. The multiple beamlets then passed through a LC-SLM  with each unit pixel area of 2 mm×2 mm. The LC-SLM provides a pixel-dependent arbitrarily phase variable for each beamlets, which was controlled through a multi-channel voltage control unit. The multiple parallel beams were focused with a lens to generate phase-dependent wave-vectors of different directions, and then interfered in the focus region to provide an intensity pattern for holographic lithography. A microscope and a CCD detector were used to monitor the intensity distribution, thereby examing the formation of hologram. The samples used to record the periodic structures were glass substrates coated with photoresist, which contained the resin Epon-SU8 (from Shelld). The exposure time is about 30 s and the processing procedures can be found in Ref. 8.
A GA is employed to obtain the desired interference fringes for holographic lithography. The “chromosome” is an encoded binary string which includes the values of both the amplitudes E i and the phase φi, where E i is 0 or 1 and φi is arbitrarily changed from 0 to 2π. The polarization of Ei is chosen prior to the simulation (Fig. 1(b)). GAs are good at finding approximations in a wide variety of complex optimization problems and have already been successfully implemented in holographic lithography  and waveguide design . As a notable example, we demonstrate the periodical removal of a line in a 1-D Bragg structure, using an interference pattern discovered by our GA. The target structure is set to produce a line defect in every 4 periods of a Bragg structure. The basic Bragg structure can be achieved by the interference of beam 2 and beam 7 in the mask (Fig. 1(b)). The periodic defect structure can be produced by changing the amplitudes and phases of beams 1–8 via GA. The lattice constant of the defect periods is determined by the smallest wave-vector difference (e.g. the wave-vector difference between beam 2 and beam 3) in the transverse direction.
The GA works as follows. In the first step, chromosomes are randomly generated to form interference intensity patterns by Eq. (1). Then each pattern is compared with the target structure, which is measured with a fitness value defined from the average error between the calculated intensity distribution and the target distribution. The subsequent generations are produced repeatedly by copying, crossing over and mutating the chromosomes until a best fitness value is found . Using the GA simulation, the final interference pattern distribution is searched and the optimal field distribution is automatically obtained. The GA needs 50 generations and apopulation of 100 (5000 iterations) to reach the designated 1-D structure, with a few minutes of computing time by a personal computer. Note that there can be many possible sets of amplitudes and phases for matching the target. One of the possible simple arrangements of pixels is that the amplitudes of beams 4, 5 and 8 are set to be 0, i.e., pixels 4, 5 and 8 are blocked, and the phases of beams 1, 2, 3, 6, 7 are set to be π, 0, 0, π, 0, respectively (Fig. 1(b)). The intensity pattern gives a 97% match to the target, as shown in Fig. 2(a). It clearly shows that defect lines are introduced into a perfect periodical 1-D Bragg structure. The required experimental setup is simple and can be realized in practice.
With the LC-SLM for the phase control, we are able to generate the desired interference pattern, which is recorded with both a CCD detector and a photoresist material-SU8. Figure 2(b) shows the CCD-recorded intensity distribution, and Fig. 2(c) shows the image of scanning electron micrograph (SEM) of the structure produced in SU8.
The above input beams arrangement can be extended to create an embedded waveguide into a triangular 2-D PhC lattice. The 1D Bragg structure is produced by the interference of beams 2 and 7. A 2-D triangle structure is produced with additional beam 9, which interferes with beams 2 and 7 respectively. Following the same idea, we use non-vanishing amplitude beams 1–3, 6, and 7 to produce the line defect in Bragg structure, then add beam 9 to interfere with these beams to produce the line defect in the 2-D triangle PhC. The phases of beams 1–3, 6 and 7 are the same as in the setting for the 1-D defect PhC, and the phase of beam 9 is 0. Note that the intensity distribution is very similar to the structure of a 2-D PhC with a line defect  and that the structure has a photonic band-gap . Figure 3(b) shows the CCD-recorded intensity patterns, which are very similar to the numerically simulated intensity patterns shown in Fig. 3(a). The micro-structure of the exposed material is shown in the SEM in Fig. 3(c) by lens with f=100 mm, showing an accurately and conveniently fabricated PhCs template.
The ratio of the defect period to the unit lattice period of the PhC is determined by the number of the pixels of the SLM, and a larger ratio can be produced with a finer LC-SLM with more variable pixels. As an example, we show a defect mode in every 8 periods of a 2-D square lattice obtained by a similar GA simulation (Fig. 4).
Figure 5 (b) also shows the numerical simulation and the CCD recorded intensity patterns of an embedded plane defect in a 3-D Slanted Pore PhC (a diamond-like structure) . As shown in Fig. 5(a), the beamlets passing through the mask are focused by a focusing lens having numerical aperture NA=0.75. However, the limited aperture of the lens does not support an easy setup with a single focusing lens. Hence, in our experiment, two focusing lenses with f=100 mm were used to focus the beamlets of 1~6 and 7~12 respectively. Then these two light paths were overlapped at the focus region of each focusing lens with an angle equal to 98°.
With more elaborated arrangement of beamlets and the polarization of the light source, more complex structures, such as line defects and point defects in 3-D PhCs with large ratio of the defect period to the base lattice period, can be obtained. In comparison to the defect structure extrinsically introduced into a PhC, the removal of a defect from the PhCs with the one-step holographic lithography will give rise to a much improved accuracy to the defect location as well as negligible disturbance to the PhC structure.
There are also some disadvantages of this experiment setup. First, the overall size of the light field distribution is limited by the focusing area of the lens, which is about 100 µm in diameter. Secondly, the fabricated structure with 5 µm period is still too large to produce a band gap in the telecommunication wavelength. One can use a lens with shorter focal length to obtain smaller period structure at a cost of the overall size of the light field distribution. In our simulation, defect PhCs with period less than 1 µm can be produced by a focusing lens with NA=0.35 (Fig. 4). A 2-D line defect with 3 µm period (not shown here) was also fabricated. We believe that the above limitations can be overcome by using umbrella setups based on the DOE  to fabricate a PhC with a much larger size.
In conclusion, we demonstrate that multi-beam phase-controlled holographic lithography can be used to fabricate functional defect PhCs structures. Examples of templates for line defects in a Bragg structure, embedded waveguides in a 2-D PhC and embedded plane defect in a 3-D Slanted Pore PhC are presented. We believe that the phase-, amplitude- and polarization-controlled multi-beam holographic lithography will have immediate impact on the fabrication of 1-D, 2-D and 3-D PhCs with functional defects.
The authors thank Dr. Y. Xiang for providing the liquid crystal modulator and Y.M. Liang and Y.F. Guan for designing and testing of the power supply driving units. This work is supported by the National Key Basic Research Special Foundation (G2004CB719805), Chinese National Natural Science Foundation (60677051, 10774193). GK and DJ are supported by EC (MIDAS STREP) and DIP. KSW and YZ are partially supported by Research Grants Council of Hong Kong (grant number 603507).
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