1. Introduction

Surface relief gratings (SRGs) are periodic undulations on the surface of an optical material leading to an alternating refractive index at the nanoscale. Light incident onto a SRG is directed into diffraction orders that are reflected and transmitted at angles related to the surface characteristics. By varying the grating’s pitch, depth and vector orientation, the diffraction orders can be manipulated and optimized to suit a particular application [1]. To date, the large majority of fabricated SRGs consist of constant-pitch sinusoidal undulations with a single common grating vector orientation over the entire grating. Such gratings have been used in the fields of acousto-optics [2], optical data processing [3], optical coherence tomography [4] and most recently, plasmonic solar cells [5]. SRGs have also been applied to designs such as rotating disk scanners [6], and are primary components to many waveguides and fiber optic cables [7–9]. Several theoretical models have been previously developed to explain and predict the interaction of SRGs with incident light [10,11]. Also, designs and patents have been developed to optimize diffraction from linear SRGs with the aim of reducing polarization loss while improving diffraction efficiency at higher orders [12], or producing more robust grating surfaces able to endure adverse operating conditions, while remaining efficient, for applications requiring lengthy laser exposures [13].

Several methods exist for fabricating SRGs. For instance, mechanical methods such as crystal diamond turning [14] have been used to create and finely control the characteristics of relief gratings, with a pitch accuracy of 10 nm. Also, chemical processes have been used to create grating patterns. One example of such grating formation is through a buckling effect when a polydimethylsiloxane film undergoes oxidation under applied stress [15]. In this case, the resulting surface deformation produces a chirped diffraction grating and the pitch is determined by controlling the time duration in which the sample undergoes oxidation. The most common production techniques for SRGs are ion beam micromachining and lithographic inscription. Ion beam micromachining [16] uses a laser to cut the grating pattern one line at a time into a material’s surface, whereas lithographic inscription, which can be accomplished using a laser [17] or an electron beam [18], directs a laser beam through a dielectric mask applied to the material’s surface with the grating pattern cut out. Even though the lithographic inscription technique requires no subsequent processing, a dielectric mask still needs to be fabricated beforehand and a different mask is required each time the grating parameters need to be changed. Regardless, all of the above-mentioned fabrication techniques employ multiple stages throughout their development process or, in some cases, are limited by the physical constraints of the inscription tools. An alternative laser lithography technique, taking advantage of the photomechanical effect in azobenzene derivatives, can be used to produce SRGs both quickly and cost effectively. Furthermore, this method allows for the inscription of complicated grating structures simply by tuning the parameters of the inscribing laser pattern.

Azobenzene thin films are known to undergo repetitive cis to trans photoisomerization when exposed to a laser beam at or near their absorption wavelength [19]. By applying a periodic interference pattern from the recombination of laser beams, the pattern can be photomechanically inscribed onto the azobenzene film’s surface without any masks and in a single fabrication step usually in under a minute. For linear SRGs, the pitch and depth are controlled with the laser incidence angle and exposure time respectively using a Lloyd’s mirror interferometer. This technique has proven to produce stable relief gratings that has been used in various applications including resonant bandgap filters [20], as well as permitted the creation of superimposed crossed gratings [21] and circular gratings with both constant pitch and chirped pitch [20, 21]. Since the inscription of gratings using this technique is only dependent on the laser interference pattern, complicated and non-uniform grating structures can also be inscribed in a single-step process. It has already been shown that modifying the optical set-up of this SRG inscription technique can produce superimposed holographic non-uniform gratings that are able to focalize light along their diffracted orders [24].

In this paper, this SRG fabrication technique is used while altering half of the collimated laser beam with a spherical lens in order to produce large-scale (~3 cm²) two-dimensional non-uniform SRGs on azobenzene molecular glass thin films. The localized physical properties of the resulting non-uniform gratings, such as the depth, pitch and grating vector orientation are measured experimentally and compared to a theoretical model. This was done to enable the efficient and cost-effective production of non-uniform diffraction gratings where the localized grating pitch, chirping rate, and grating vector orientation can all be easily modified and set during the fabrication process.

The results obtained in this work can be applied to study and enhance the propagation of surface plasmon resonance (SPR) in non-uniform gratings. Since SPR can only be excited when the incident light polarization is along the grating vector, a non-uniform grating should be able to excite SPR in several possible light polarizations. A recent study from our group on the excitation of SPR using chirped one-dimensional linear gratings has shown an increase in the bandwidth of the SPR excitation wavelengths [25], but only for a single light polarization. The study conducted herein will enable a better understanding of SPR excitation in non-uniform gratings since SPRs are strongly correlated to the grating’s pitch, depth and vector distributions [26].

2. Procedure

2.1 Sample preparation

Soda lime microscope slides, having dimensions of 35 mm by 35 mm and a thickness of 1 mm, were thoroughly washed and dried in an oven at 95°C for 25 minutes in preparation for the deposition of azobenzene thin films. Disperse Red 1 molecular glass (gDR1) [27] was dissolved in dichloromethane at a weight ratio of 3% through mechanical shaking for 1 hour. The solution was then filtered using a 0.45-microns filter to remove any remaining solids.

Approximately 1 mL of gDR1 solution was deposited onto a cleaned soda lime substrate using a Headway Research spin coater. The sample was spun at 1070 RPM for 25 seconds to evenly coat the substrate with a thin film having a nominal thickness of 240 nm, as measured using a Dektak II profilometer. The finished samples were then baked in an oven at 95°C for 30 minutes to evaporate any residual solvent from the spin coating process.

2.2 SRG writing process

A Coherent Verdi V5 diode-pumped CW laser (λ = 532 nm) was used for the SRG inscription process. As depicted in Fig. 1, this laser went through a spatial filter, a collimating lens, a quarter wave plate, and an adjustable iris before being incident on a Lloyd’s mirror interferometer. The quarter wave plate was used to circularly polarize the laser beam, since this polarization has been found to be the most efficient for writing SRGs [28]. The spatial filter consisted of a microscope objective focusing the laser beam onto a 25-microns aperture. The collimating lens was placed at exactly one focal distance away from the aperture. A variable iris was then used to set the beam diameter to 14 mm for all samples. The laser power was adjusted to obtain an irradiance of 428 mW/cm² after collimation.

 

Fig. 1 Top view of the laser and Lloyd’s mirror interferometer used for inscribing non-uniform SRGs. S: Sample. M: Mirror. L and LH: Lens and Lens Holder. VI: Variable Iris. QWP: Quarter Wave Plate. CL: Collimating Lens. SF: Spatial Filter.

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The Lloyd’s mirror interferometer consisted of a mount that held an azobenzene thin-film sample firmly at a 90° angle to a mirror. This set-up was positioned on a computer-controlled rotating stage to allow angular adjustments with respect to the incident laser beam, dictating the grating’s central pitch. This design resulted in half of the laser beam being incident on the mirror while the other half was incident on the sample’s surface. The reflected light from the mirror perfectly overlaid that on the sample surface resulting in a grating area in the shape of a half circle with width equal to half the diameter of the adjustable iris. This was a typical description of the set-up required to fabricate a constant-pitch SRG with a common grating vector.

For inscribing a non-uniform SRG, a spherical lens, having a 5.7 mm diameter and a 14.3 mm focal length, was positioned at a set distance f away from the sample’s surface. The lens was positioned in the half of the laser beam directly incident on the sample. A lens holder was fabricated using a 3D printer in order to block any direct laser light not passing through the spherical lens, as illustrated in Fig. 1.

The grating pitch that would be produced in the absence of the spherical lens and lens holder in Fig. 1 will be thereafter referred to as the central pitch of the non-uniform SRG. Four different non-uniform SRGs were produced with central pitches of 500 nm, 1000 nm, 1500 nm and 2000 nm at a fixed distance f of 35 mm. Also, two other non-uniform SRGs, having a central pitch of 2000 nm, were fabricated with the lens and lens holder positioned at distances f of 45 mm and 55 mm. The combination of each lens distance f and grating central pitch measurement underwent an optimization process through which the laser exposure time was varied between 95 and 400 seconds in order to standardize the grating depth to approximately 100 nm at the center of each grating. This was done to ensure that the depth distribution could be fairly compared between the various non-uniform gratings inscribed and to minimize inconsistencies in the grating analysis process.

2.3 Grating analysis

A Bruker Dimension Edge Atomic Force Microscope (AFM) was used to measure the profile and modulation depth of each SRG. The AFM recorded three-dimensional information from the grating surface with scan dimensions of 5-µm squares for the 500 nm and 1000 nm central pitch gratings, and 10-µm squares for the 1500 nm and 2000 nm central pitch gratings. These scan dimensions provided the data set necessary to measure the grating depth, pitch, and vector orientation at various positions throughout the SRG. Each scan produced a 256 by 256 data points matrix, with the grating height value at each point. Twenty-five scans were conducted on the surface of each grating, forming a square matrix around the grating’s center. The movement of the AFM tip from one scan point to the next was done using the computerized AFM stage. The collected data was then exported as a text file for analysis. A silver coated non-uniform SRG produced using this method is pictured with a reference scale in Fig. 2(a). In Figs. 2(b) and 2(c), a representative three-dimensional AFM image and a profile view, showing a sinusoidal modulation depth of approximately 130 nm, are shown respectively. The AFM image and profile were taken at a random position on a grating.

 

Fig. 2 a) Example of a non-uniform SRG at 1000 nm central pitch and 55 mm lens distance with reference scale, b) three-dimensional AFM image, and c) cross-sectional profile view of the same SRG.

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The analysis of the gratings under the various test conditions demonstrated that the grating central pitch and lens distance were the key parameters for defining the SRG’s chirping rate and grating vector orientation. In order to characterize the influence of the grating central pitch and lens distance on the fabricated SRG pattern, the laser light interference properties were analytically modeled and are explained in Section 3. A comparison between the theoretical model and the experimental results is given in Section 4.

3. Theory

A Cartesian coordinate system was used for the derivation of an analytical model of the non-uniform SRG inscription. As illustrated in Fig. 1, the X-Y reference plane was chosen to be on the sample’s surface, with the X-axis being horizontal and the Y-axis being vertical with respect to the optical table. The Z-axis was taken to be perpendicular to the sample’s surface.

For a constant-pitch grating having a common grating vector orientation, the pitch equation can be derived from the phase difference between two consecutive cycles in the irradiance interference equation on the azobenzene film’s surface, according to Eq. (1) [24]:

For non-uniform gratings, the directly incident laser beam on the azobenzene film’s surface is diverging due to the spherical lens and an additional interference phase angle is imposed to account for the time delay for the beam to reach various areas on the sample’s surface. An expression for this additional phase angle has been previously obtained when using a cylindrical lens along the direct laser beam path [22,23]. However, when using a spherical lens, the geometrical analysis of the phase difference between the direct and reflected beams is different since it must account for distance variations both along the X-axis and Y-axis. This phase difference between the direct and reflected beams is given by:

 

Fig. 3 The approximate overlay on the azobenzene sample surface upon recombination of the two interfering laser beams for the inscription of non-uniform SRGs at a distance of a) 35 mm, b) 45 mm, and c) 55 mm.

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Finally, an expression for the irradiance distribution on the sample’s surface can be obtained to analytically model the modulation depth distribution of the gratings. It was assumed that the irradiance would be linearly proportional to the modulation depth of the SRGs. The equation for the total irradiance of the recombined beams can be derived from Maxwell’s equations and is shown as follows:

4. Results

4.1 Experimental results

In Fig. 4, the grating pitch, modulation depth and vector orientation obtained from each AFM scan are represented for four different non-uniform SRGs inscribed with central pitches of 500 nm, 1000 nm, 1500 nm and 2000 nm at a fixed f = 35 mm. In this figure, each arrow corresponds to data collected from a single AFM scan. The arrow’s length is directly proportional to the grating’s localized pitch, with a reference scale bar indicated on this figure, while the arrow’s orientation is representing the localized grating vector. The color map is illustrating the grating modulation depth profile normalized to the deepest value on each grating, as indicated in the title of each part of Fig. 4. The center of each arrow in the X-Y plane represents the actual coordinates on the sample’s surface where the AFM scan was taken.

 

Fig. 4 SRG pitch and vector measurements overlaid on a normalized grating modulation depth color map for a non-uniform SRG produced experimentally at a central pitch of a) 500 nm, b) 1000 nm, c) 1500 nm, and d) 2000 nm.

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As seen in Fig. 4, as the central grating pitch is increased, the grating pitch rate of change along the X-axis is also significantly increased. From Eq. (1), it is known that the central grating pitch is directly related to $\theta$, the angle between the Lloyd mirror and the incident laser beam, meaning that larger central pitch gratings are inscribed at smaller laser incidence angles. In Eq. (3), only the x position, and not the y position, is directly influenced by $\theta$, meaning that for small values of $\theta$, the pitch rate of change will only occur along the X-axis and that it will be constant along the Y-axis.

Also, from this figure, the central arrow length and orientation in each plot was verified to match the value of the pitch and grating vector orientation that would be inscribed if the lens was removed. This occurs because a light beam going through the center of the lens will continue undisturbed.

In addition, it can be seen that the grating vector orientation rate of change along the Y-axis is more pronounced for larger central pitch gratings. Each grating’s pitch rate of change is a direct consequence of the additional distance that the laser beam must travel after passing through the spherical lens before it interferes with the reflected laser beam off the Lloyd mirror. It was observed that the SRG pitch is lower than the central pitch at all positions left of the grating center, and the SRG pitch is greater than the central pitch right of the grating center. The path length of the laser light reflected off the mirror is equidistant along the Y-axis, however the spherical lens causes a two-dimensional divergence of the laser beam, attributing to a propagation distance change along the Y-axis. Upon recombination, the effect of this path length change is a clockwise rotation of the grating vector when moving in the positive Y direction, as seen in Fig. 4. The SRG vector is also affected by the localized pitch, which results in a clockwise rotation of the grating vector when moving in the positive X direction.

4.2 Theoretical results

The simulation results from the analytical model shown in Section 3 are presented in Fig. 5.

 

Fig. 5 Simulated SRG pitch and vector representations overlaid on a normalized irradiance distribution for a non-uniform SRG produced from Eqs. (2) and (4) for central pitch values of a) 500 nm, b) 1000nm, c) 1500 nm, and d) 2000nm.

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In this figure, the gratings’ pitch and vector orientation distributions were calculated analytically, using Eq. (2), to produce hypothetical gratings that simulate the 4 non-uniform gratings described in the experimental section. Furthermore, the theoretical laser irradiance distribution, ignoring reflection effects, was generated analytically using Eq. (4).

In the theoretical model, the interference phase $\delta$ between the direct and reflected laser beams, represented by Eq. (2), was calculated for a hypothetical grating at 25 x-y coordinates identical to those used in the AFM scans. Thus, theoretical values for the grating pitch and vector orientation were obtained and illustrated by arrows in Fig. 5. Similarly to Fig. 4, the arrow’s length is directly proportional to the grating pitch, with the same reference scale bar indicated on the figure, while the arrow orientation is representating the grating vector. For each arrow, a Python computer code was developed in order to generate a 200 x 200 data points matrix that represents a single AFM scan with dimensions of 5-$\mu$m squares for the 500 nm and 1000 nm central pitch SRGs, and 10-$\mu$m squares for the 1500 nm and 2000 nm central pitch SRGs. The computer program didn’t allow the export of theoretical matrices larger than 200 x 200 data points into the desired file format. A representative figure of an analytical SRG, obtained by executing the Python code is presented in Fig. 6.

 

Fig. 6 Simulated SRG pattern generated by the phase analysis of $\delta$ at a random position on the surface of a hypothetical grating having a central pitch $\Lambda$ = 1500 nm.

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Additionally, the irradiance distribution generated using Eq. (4) enabled modeling the expected grating modulation depth distribution of a non-uniform SRG produced experimentally. This was done assuming a linear dependence of the irradiance on the grating’s depth. The average irradiance was calculated at each point across the hypothetical grating by averaging the irradiance values generated in a 101 by 101 data points matrix. The 25 irradiance values were then normalized and used to generate the 4 color contour plots in Fig. 5.

It can be observed that the hypothetical gratings’ pitch and vector orientation very closely match those same parameters measured experimentally, with only very slight discrepancies, which may be attributed to the misalignment of the X-Y coordinate system origin used during the AFM scans. However, some discrepancies can be identified in the gratings’ modulation depth distribution between simulated and experimental SRGs. In Fig. 4, the difference between the deepest and shallowest sections of the SRG is drastically more pronounced than those in Fig. 5. Some factors that could contribute to these discrepancies include the Gaussian nature of the inscribing laser beam, the azobenzene thin-film quality and localized thickness, imperfections on the Lloyd mirror surface, or most importantly, the non-uniform dependence of the grating depth on the laser irradiance [25]. Since the grating modulation depth is directly related to the diffraction efficiency, it is expected that the hypothetical gratings would have a stronger diffraction signal than the experimental gratings.

4.3 Variable lens position

Finally, the spherical lens, used in the experimental set-up shown in Fig. 1, which was held at a constant f = 35 mm thus far, was moved away from the sample to 45 and 55 mm, yielding 2 other non-uniform SRGs having a central pitch of 2000 nm. These results were compared to those in Fig. 4(d) and are shown in Fig. 7.

 

Fig. 7 SRG pitch and vector orientation measurements overlaid on a normalized grating modulation depth color map for a non-uniform SRG produced experimentally at a lens distance of a) 35 mm, b) 45 mm, and c) 55 mm.

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As the lens distance f is increased, both the SRG vector orientation and chirping rates of change is decreased. This result is intuitive since the diverging laser beam from the spherical lens is spread over a greater area. As f goes to infinity, the non-uniform SRG will become linear. There are only minor differences between the SRG modulation depth profiles of the 3 graphs presented in Fig. 7.

5. Conclusion

Azobenzene molecular glass thin films were spin coated on soda lime microscope slides and non-uniform surface relief diffraction gratings were inscribed on the resulting thin films upon exposure to the interference pattern from the recombination of collimated and spherically divergent laser beams. A spherical lens was positioned at a distance of 35 mm from the sample surface yielding grating pitch and vector orientation variations along the X-axis and the Y-axis. Subsequently, the localized non-uniform gratings’ pitch and vector orientation, as well as the modulation depth, were measured experimentally using an atomic force microscope. A theoretical model was developed to simulate the experimental SRGs by analyzing the phase of the recombined laser light as well as the irradiance profile on hypothetical gratings. The analytical results for the gratings’ pitch and vector orientation distributions were successfully compared to those obtained experimentally. The effect of the spherical lens position was also studied and it showed a significant influence on the non-uniform gratings’ pitch and vector orientation. This technique of SRG production provides an inexpensive and time efficient method for the fabrication of non-uniform gratings that offers easy control over the grating’s pitch, vector orientation and modulation depth. Also, it would be possible to substitute the spherical lens along the direct beam path in the experimental set-up with other optical elements, such as a spatial light modulator, to create other grating distributions. In the future, these non-uniform SRGs can be used in various photonics applications, including plasmonics.