Both reversible volume phase diffraction gratings [1
1. P. Rochon, J. Gosselin, A. Natansohn, and S. Xie, “Optically induced and erased birefringence and dichroism in azoaromatic polymers,” Appl. Phys. Lett. 60(1), 4–5 (1992). [CrossRef]
] and high-efficiency surface-relief gratings [2
2. P. Rochon, E. Batalla, and A. Natansohn, “Optically induced surface gratings on azoaromatic polymer films,” Appl. Phys. Lett. 66(2), 136–138 (1995). [CrossRef]
] can be inscribed onto cast azopolymer films by exposure to an interfering laser pattern at an absorbing wavelength. The primary mechanism invoked in the creation of a phase grating throughout the volume of the film is the induction of local birefringence. This process does not involve a change in the surface profile of the film but rather a change in the local refractive index. As for the relief gratings, a high degree of localized mass transport of the polymer chains is responsible for the creation of a surface profile to depths nearing that of the original film thickness. This phenomenon involves pressure gradients as a driving force, present due to different photochemical behaviors of the azo chromophores at different regions of the interference pattern. Several models have been proposed to explain the creation of surface-relief gratings [3
3. C. J. Barrett, A. L. Natansohn, and P. L. Rochon, “Mechanism of optically inscribed high-efficiency diffraction gratings in azo polymer films,” J. Phys. Chem. 100(21), 8836–8842 (1996). [CrossRef]
Since birefringence volume gratings can be recorded by a laser on azo films in a matter of seconds, they become very sensitive to operating light conditions and can be erased by exposure to randomly polarized light since the molecular alignment can be easily perturbed. On the other hand, relief gratings take several minutes to form, and once the final shape has been achieved, these gratings are very stable. Nonetheless, both volume phase and surface relief gratings can be erased by heating the film above its glass transition temperature (~130°C). A schematic of the experimental set-up used for creating these gratings is illustrated in Fig. 1
Fig. 1 Experimental set-up for inscribing a surface-relief grating.
Half of a collimated laser beam is reflected by a mirror oriented at 90° with respect to the azo film, thus resulting in a sinusoidal interference pattern on the film’s surface. The depth and spacing of the gratings can be controlled with the exposure time and laser incidence angle respectively. This experimental set-up allows the simultaneous inscription of both birefringence volume phase and surface-relief grating structures, but the polarized diffraction from the birefringence gratings is largely overshadowed by the higher efficiency diffraction from the surface-relief gratings. In addition, unlike volume gratings, the polarization of the diffracted light is unaffected by the surface grating.
In the last decade, polymers containing azobenzene or its derivatives have been intensively investigated for their photoresponsive properties and potential applications in diffractive optical elements, information storage, optical switching, nonlinear optics, sensors, and actuators. For instance, surface-relief gratings can be used to couple light either in the azo film itself or in another adjacent medium, yielding narrowband resonant waveguide filters [4
4. R. J. Stockermans and P. L. Rochon, “Narrow-band resonant grating waveguide filters constructed with azobenzene polymers,” Appl. Opt. 38(17), 3714–3719 (1999). [CrossRef] [PubMed]
]. More recently, single-layered graphene grafted with azo polymer brushes has been used to significantly enhance the diffraction efficiency of the photoinduced surface-relief gratings on azo molecular glass films [5
5. D. Wang, G. Ye, X. Wang, and X. Wang, “Graphene functionalized with azo polymer brushes: Surface-initiated polymerization and photoresponsive properties,” Adv. Mater. 23(9), 1122–1125 (2011). [CrossRef] [PubMed]
]. The azo compound has been proven to allow the inscription of multiple superimposed relief gratings and the interactions between these gratings has been studied [6
6. L. Lévesque and P. Rochon, “Surface plasmon photonic bandgap in azopolymer gratings sputtered with gold,” J. Opt. Soc. Am. A 22(11), 2564–2568 (2005). [CrossRef] [PubMed]
Furthermore, diffraction gratings can be used to project other optical element holograms by recording and then later retrieving the phase and amplitude of light [7
7. C. Falldorf, C. Dankwart, R. Gläbe, B. Lünemann, C. Kopylow, and R. B. Bergmann, “Holographic projection based on diamond-turned diffractive optical elements,” Appl. Opt. 48(30), 5782–5785 (2009). [CrossRef] [PubMed]
8. R. Shi, J. Liu, J. Xu, D. Liu, Y. Pan, J. Xie, and Y. Wang, “Designing and fabricating diffractive optical elements with a complex profile by interference,” Opt. Lett. 36(20), 4053–4055 (2011). [CrossRef] [PubMed]
]. Sub-wavelength dielectric focusing grating structures with non-periodic patterning have been fabricated from different nano-engineered materials by chemical vapor deposition and dry etching, and their shape provided the ability to control the focused light’s profile [9
9. D. Fattal, J. Li, Z. Peng, M. Fiorentino, and R. G. Beausoleil, “Flat dielectric grating reflectors with focusing abilities,” Nat. Photonics 4(7), 466–470 (2010). [CrossRef]
10. L. Chrostowski, “Optical gratings: Nano-engineered lenses,” Nat. Photonics 4(7), 413–415 (2010). [CrossRef]
]. The main advantage of using azo-polymer for such nanostructures is that the reading and inscription of a hologram each consist of a single fabrication step with no post-exposure adjusting or chemical processing. Holographic volume diffracting lenses were previously recorded in azo polymer [11
11. G. Martinez-Ponce, T. Petrova, N. Tomova, V. Dragostinova, T. Todorov, and L. Nikolova, “Bifocal-polarization holographic lens,” Opt. Lett. 29(9), 1001–1003 (2004). [CrossRef] [PubMed]
] and azo liquid crystal films [12
12. J. Zhang, H. Ming, P. Wang, L. Tang, J. Xie, Q. Zhang, and J. Liu, “Holographic lens in azobenzene liquid crystal polymer films,” in SPIE Proceedings 5281, C. F. Lam, C. Fan, N. Hanik, and K. Oguchi, eds., 614–618 (2004). [CrossRef]
] using a different experimental set-up than the one presented here, however, the major set-back of these lenses is that they were polarization dependent since they rely on birefringence phase gratings. The available literature on the inscription and study of chirped relief gratings in azo polymers is very limited and only includes single grating structures. There are no readily available published studies on the optical interactions between superimposed linear and chirped surface-relief grating structures.
The azo-polymer compound was diluted in dichloromethane with a mix ratio of 3% weight/weight, and thoroughly mixed. Several test samples were prepared by spin-casting this solution on a glass slide. The experimental set-up in Fig. 1
was modified as illustrated in Fig. 2
Fig. 2 Modified experimental set-up for chirped grating inscription.
Light from a Verdi-V5 diode-pumped laser with a wavelength of 532 nm and output power of 2 W was passed through a spatial filter and collimated to ~4.8 cm in diameter, yielding a plane wave writing beam with an irradiance of ~27.6 mW.cm−2. A λ∕4 plate was subsequently used to create circularly polarized light, with the size of the beam being regulated to a diameter of ~1 cm using a variable iris. A small converging lens was placed along the half-circle of the laser beam that was directly incident on the azo-polymer film, hence, creating a curved wavefront which interfered with the plane wavefront from the other laser half-circle incident on the mirror. The laser was incident at an angle which would create a linear surface-relief diffraction grating with a spacing of 800 nm, however, any other spacing could have equally been chosen. The laser exposure time was set to 5 minutes per grating.
This arrangement produced a chirped holographic lens grating with an off-angle focal point that depended only upon the location of the focal point of the physical mini-lens during the inscription process. This mini-lens was positioned on a linear track which enabled it to be placed at various distances away from the sample. In order to make a dual-focus chirped holographic lens grating, an initial exposure lens hologram was inscribed for 5 minutes, then, the mini-lens was moved a fixed distance forward or backward, followed by a second laser inscription of a superimposed lens grating.
Subsequently, a He-Ne laser, having a power of 2 mW, was used to measure the focusing ability of the gratings, as depicted in Fig. 3
Fig. 3 Experimental set-up for measuring the lens grating focusing.
. A high dynamic range CCD camera was positioned on a linear track and pictures of the laser beam were taken as a function of distance travelled along the diffracted orders.
Upon exposure to the inscribing laser beam, the interference term of the resulting irradiance on the azo film’s surface can be written as:
is the phase difference between the direct and reflected beams, which can be expressed as:
are respectively the direct and reflected wave vectors,
is the position vector and
is a constant phase difference between the two beams.
As illustrated in Fig. 4
Fig. 4 Laser beam geometry.
, for a linear surface-relief grating to form at point A, both direct and reflected wave fronts must be linear. Therefore, we have that:
is the free-space writing laser wavelength. Ignoring the constant phase,
The interference term of the irradiance is now given by:
It can be seen that when Eq. (5)
is plotted as a function of x
, the period gives the grating spacing
, such as:
Now, assume a cylindrical lens is placed along the direct half laser beam with its axis along the vertical with respect to the optical bench, as in Fig. 2
. The laser beam focal point will be at a distance f
from the sample. The direct laser wavefront now becomes curved and the light is delayed before reaching the mirror by a distance
, as illustrated at point B in Fig. 4
. The further away point B is located from either sides of point A on the sample’s surface, the more the light is delayed. In order to calculate this delay, we can obtain from Fig. 4
Using the binomial approximation, we obtain:
, we now have:
This phase delay must be added to
in order to obtain:
Therefore, in the case of a surface-relief holographic grating of a cylindrical lens, the interference term of the irradiance becomes:
This indicates that the grating spacing changes symmetrically with respect to Point A in Fig. 4
. It is those pitch variations that lead to light focalization from the holographic lens grating upon exposure at a probing He-Ne beam. One can see that as
, Eq. (11)
becomes Eq. (5)
. A similar development could be done for a spherical lens, but the analysis becomes more complex since the irradiance now varies along both the
Fig. 5 The theoretical relative irradiance as a function distance along the sample’s surface.
illustrates numerical examples of how the grating spacing is affected by Eqs. (5)
. It can be seen that when
, both equations give the same grating spacing, however, when
, the spacing given by Eq. (5)
stays the same, but that given by Eq. (11)
For superimposed linear gratings at normal incidence, light will be diffracted with wave vectors satisfying the grating equation
are integers representing the diffraction orders, and
are the grating spacings. This equation is also valid for light diffraction from superimposed chirped gratings, as the surface undulations from the first gratings are not erased, but rather added to the second grating inscribed. Figures 6(a)
Fig. 6 Light focalization representation in transmission from superimposed (a) chirped lens gratings with the same linear spacing and different focal points (b) chirped lens gratings with different linear spacing and same focal points (c) chirped lens grating and a linear grating with the same spacing.
gives a graphical representation of the excepted focalization of various combinations of superimposed chirped holographic lens gratings.
4. Results and discussion
First, the circularly polarized He-Ne laser light, depicted in Fig. 3
, was incident on a chirped diffraction grating acting as a single cylindrical holographic lens. The CCD camera was approximately placed in the focal point of the holographic lens and the computerized linear track was zeroed. Then, various pictures were taken along the 1st diffracted order at 6.7 mm intervals while scanning from −46.7 to 46.7 mm, as illustrated in Fig. 7
Fig. 7 Pictures taken as a function of distance travelled along the 1st backward diffracted order of a chirped grating of a cylindrical holographic lens.
. The focal point of this chirped grating was located approximately 10 cm away from the sample’s surface.
Maximum focalization occurs at 13.3 mm, as seen in Fig. 7
. Since the resulting undulations on the azo surface are the recording light’s intensity and phase, a probing laser would reproduce the same light profile at the same geometry, hence, resulting in focalization occurring only along a single diffracted order. A similar focalization was also apparent in reflection of the same diffracted order. The polarization independence of the incident light off-angle focalization was verified with a polarizer, hence, confirming that the focalization is occurring due to the chirped surface-relief grating rather than any volume birefringence grating produced. The grating’s varying pitch yields slightly different diffraction angles, which results in light focalization at their intersection in the 1st backward diffracted order. The focus resolution of this chirped lens grating is dictated only by that of the physical mini-lens used during the inscription process illustrated in Fig. 2
Then, a spherical mini-lens was substituted in the experimental set-up in Fig. 2
, and a chirped holographic grating acting as a spherical lens was inscribed. Various pictures were taken along the 1st backward diffracted order at 7 mm intervals while scanning from −4.7 to 4 cm, as illustrated in Fig. 8
Fig. 8 Pictures taken as a function of distance travelled along the 1st backward diffracted order of a chirped holographic lens grating.
. The focal point of this chirped grating was also located approximately 10 cm away from the sample’s surface.
A dual-focus chirped holographic lens grating was inscribed by superimposing two chirped gratings at the same laser inscription angle, but with different positions of the focal point of the physical mini-lens. Similar pictures were taken as a function of the distance travelled along the 1st backward diffracted order. As seen in Fig. 9
Fig. 9 Pictures taken as a function of distance travelled along the 1st backward diffracted order of a dual-focus chirped holographic lens grating.
, the two focal points, which are located approximately 8 cm apart, can be identified around −48 mm and 38.4 mm. The minor angle eccentricity in the two focal points is due to the misalignment of the linear track holding the physical mini-lens during the inscription process. The separation between the focal points of the resulting chirped lens grating is equal to the movement of the physical mini-lens during the first and second exposures.
The images in Figs. 8
were digitally processed, and the light intensity profiles were plotted as a function of the horizontal pixel position and distance travelled, as seen in Figs. 10(a)
Fig. 10 Light intensity profile as a function of horizontal pixel position and distance travelled for (a) single chirped holographic lens grating (b) dual-focus chirped holographic lens grating.
. The focal points are indicated on the figures. The CCD camera had a resolution of 10 microns per pixel, therefore, the focal points outlined in Figs. 10(a)
are approximately 200 microns wide.
In Fig. 11
Fig. 11 Pictures taken as a function of distance travelled along the 1st backward diffracted orders of a dual-focus holographic lens grating with two different spacings.
, a first chirped holographic lens grating with a spacing of 800 nm was inscribed, followed by a second chirped lens grating with a spacing of 700 nm. Both gratings were written with the physical mini-lens, seen in the experimental set-up in Fig. 2
, left at the same location during both exposures. The focal points of the resulting dual-focus lens grating are approximately 4.5 cm apart.
The He-Ne laser source has been used so far to measure the off-angle focusing of these holographic lenses because of the wavelength dependence of the diffracted light. This occurs since
is the light wavelength in the medium and
is the diffraction angle. For a value of
satisfying Eq. (12)
must decrease. Therefore, for a white light source, the focalization of each wavelength will occur at different angles.
For the next part of this experiment, the He-Ne laser, collimating lens and quarter-wave plate were removed from the experimental set-up illustrated in Fig. 3
, and an un-polarized white light source was used to illuminate the test sample through the variable iris. A chirped holographic lens grating was inscribed on an azo film followed by the inscription of a superimposed linear grating having the same spacing. In this case, one of the possible diffraction wave vectors in Eq. (12)
are equal but opposite signs, which gives a diffraction angle of zero. Hence, it was possible to observe light focalization along the 0th order for all light wavelengths, as illustrated in Fig. 12
Fig. 12 Pictures taken as a function of distance travelled along the 0th order for a superimposed lens grating with a linear grating having the same spacing.
Finally, since each focal point in the dual-focus chirped lens grating acts as a point source, an interference pattern can be observed in the far-field. For this part, a different dual-focus holographic lens grating was inscribed with a distance of only 1 cm between the two focal points. Figure 13
Fig. 13 Interference pattern from the dual-focus chirped lens grating at 20 cm away from the sample’s surface.
illustrates the interference pattern from a He-Ne laser source at a distance of 20 cm away from the grating along the 1st backward diffracted order.
A variety of superimposed chirped gratings, acting as holographic lenses, were inscribed on azo-polymer films by interfering plane and converging laser wavefronts. These chirped holographic lens gratings were shown to simultaneously focus incident light, independently of polarization, along various angles, including the 0th order. The focal points of each lens and the off-axis focalization angle were easily controlled during the single step fabrication process. Upon inscribing a dual-focus holographic lens grating with only 1 cm separation between the focal points, a circular interference pattern was observed in the far-field.