SUPPORTING INFORMATION
Mechanism of multiple grating formation in high-energy recording of
holographic sensors
Ali K. Yetisen,1,a) Yunuen Montelongo,2 Nicholas M. Farandos,3 Izabela
Naydenova,4 Christopher R. Lowe,5 and Seok Hyun Yun1
1Harvard
Medical School and Wellman Center for Photomedicine, Massachusetts General Hospital, Boston,
Massachusetts, 02114, USA
2Electrical
Engineering Division, Department of Engineering, University of Cambridge, Cambridge, CB3
0FA, UK
3Department
of Chemical Engineering, Imperial College London, South Kensington Campus, London, SW7
2AZ, UK
4Centre
for Industrial and Engineering Optics, School of Physics, College of Sciences and Health, Dublin
Institute of Technology, Dublin 8, Ireland
5Department
of Chemical Engineering and Biotechnology, University of Cambridge, Tennis Court Road,
Cambridge CB2 1QT, UK
_____________________________
a)
Author to whom correspondence should be addressed. Electronic mail: ay283@cam.ac.uk.
1
The Model for the Superpositioning of Waves and Thresholding
A plane wave propagating in an arbitrary direction can be expressed as follows:
2𝜋(𝑛+𝑖𝑘)
)(𝑘𝑥 𝑥+𝑘𝑦 𝑦+𝑘𝑧 𝑧)−𝜔𝑡}
𝜆
𝑬 = |𝑬|(𝐸𝑥 𝒊 + 𝐸𝑦 𝒋 + 𝐸𝑧 𝒌)𝑒 𝑖{(
= |𝑬|(𝐸𝑥 𝒊 + 𝐸𝑦 𝒋 +
2𝜋𝑘
)(𝑘𝑥 𝑥+𝑘𝑦 𝑦+𝑘𝑧 𝑧) 𝑖{(2𝜋𝑛/𝜆)(𝑘𝑥 𝑥+𝑘𝑦 𝑦+𝑘𝑧 𝑧)−𝜔𝑡} 𝑖𝜙
𝜆
𝐸𝑧 𝒌)𝑒 −(
𝑒
(1)
𝑒
where Ex, Ey and Ez represent the state of polarization, and satisfy Ex2+Ey2+Ez2=1; |E| is the magnitude of the electric field
wave; kx, ky, and kz are unit vector components and determine the direction of the wave vector, where kx+ky+kz=1; the
complex refractive index is n+ik (k is not the wave vector but the imaginary part of the refractive index). The first exponential
represents the optical attenuation in the medium. The second exponential describes the phase of propagation. The third
exponential contains a phase shift φ acquired upon reflection at an interface.
The average intensity I at each point in space can be determined by describing the three individual beams illustrated in
Fig. 1 and calculating the complete interference pattern. Computed variables include phases φ i=0 (incident), φm=arbitrary
(object), φr=φi+π=π (reflected from hydrogel/water or air interface); c=speed of light in vacuum; indexes of refraction na
(air), nw (water); nh (wet hydrogel≈1.372); np (substrate≈1.491).1 The parameter 𝛾 is the exponential decay in a complex
form, which was estimated by fitting experimental data to the following equation:
𝐼𝑖
𝐼𝑓
=
1
(2)
𝑒 −𝛾𝑥
where Ii , If are the initial and final wave intensities for hydrogel+Ag0 NP layer, respectively. The imaginary part of the index
of refraction can be estimated from 𝛾≈183655 m−1. This represents the amount of energy that is absorbed by the medium. The
index of refraction may also be expressed as a complex number:
𝑛ℎ = 𝑛ℎ + 𝑖 𝛾
𝜆
(3)
4𝜋
When the wave travels through the interface of two materials, the intensity is partially reflected from the interface, while
the remaining intensity is transmitted through the interface (Figure 1). The intensity (𝐼𝑟1 and 𝐼𝑟2 ) of the reflection at the
hydrogel/water interface is found by applying the exponential decay and the Fresnel equations for both transversal
polarizations:
Ir1= 𝐼𝑖 e-γx (
Ir2= 𝐼𝑖 e-γx (
1−𝛼1 𝛽1 2
1+𝛼1 𝛽1
)
𝛼1 −𝛽1 2
𝛼1 +𝛽1
)
(4)
(5)
where 𝐼𝑖 is the incident intensity (normalized as 1), and 𝛼1 and 𝛽1 are defined as:
𝛼1 =
𝑐𝑜𝑠 𝜃𝑡
𝑐𝑜𝑠 𝜃𝑖
(6)
2
𝑛𝑎
𝛽1 =
(7)
𝑛ℎ
where 𝜃𝑖 and 𝜃𝑡 are the tilt and transmission angles extracted from Snell’s law. The average intensity of the two polarizations
is given by:
𝐼𝑟 =
𝐼𝑟1 +𝐼𝑟2
(8)
2
The intensity of the two transversal polarizations reflected from the mirror 𝐼𝑚1 and 𝐼𝑚2 are estimated as:
Im1 = (1 − 𝐼𝑟1 ) (
Im2 = (1 − 𝐼𝑟2 ) (
1−𝛼2 𝛽2 2
1+𝛼2 𝛽2
)
𝛼2 −𝛽2 2
𝛼2 +𝛽2
)
(9)
(10)
where 𝛼2 and 𝛽2 are constants defined as:
𝛼2 =
𝛽2 =
1
(11)
𝛼1
1
(12)
𝛽1
𝐼𝑚 takes into account the lost intensity when the ray is reflected from mirror and enters the hydrogel:
𝐼𝑚 = 𝐼𝑖 𝑒 −𝛾𝑥 − 𝐼ℎ𝑟 −
𝐼𝑚1 +𝐼𝑚2
2
(13)
The three interference waves: 𝐸𝑖 incident, 𝐸𝑚 reflected back from the mirror, and 𝐸𝑟 reflected from hydrogel/water inner
interface can be expressed as:
Ei=|Ei| 𝑒 𝑖(𝒌𝒊 ·𝒓−𝜔𝑡−𝜑𝑖)
(14)
Em=|Em| 𝑒 𝑖(𝒌𝒎 ·𝒓−𝜔𝑡−𝜑𝑚 )
(15)
Er=|Er| 𝑒 𝑖(𝒌𝒓 ·𝒓−𝜔𝑡−𝜑𝑟 )
(16)
where k represents the respective wave vectors with phase 𝜑, and 𝒓 is the position vector. The solution is derived from the
addition of the three waves. The intensity is extracted with the square of the absolute value of the field. The sample tilt angle
during fabrication was varied to understand the effect on the formation of a Bragg grating.
Synthesis of the Recording Medium
A poly(methyl methacrylate) (PMMA) substrate (t = 0.5 mm) was O 2 plasma treated under a vacuum of 1 torr for 3 min to
render the surface hydrophilic. The monomer mixture prepared for the recording medium consisted of 2-hydroxyethyl
methacrylate (HEMA, 91.5 mol%), ethylene dimethacrylate (EDMA, 2.5 mol%) and methacrylic acid (MAA, 6 mol%). 2 This
solution (100 µl) was mixed (1:1, v/v) with 2-dimethoxy-2-phenylacetophenone (DMPA) in propan-2-ol (2%, w/v) and
copolymerized on the PMMA substrate using UV light. The resulting pHEMA matrix (~10 µm) was doped with Ag + ions
3
(0.1 M) by depositing AgNO3 (200 µl) as on elongated blob on a glass surface and placing the polymer matrix on the blob for
1 min. The matrix was dried under a tepid air current for 30 s. The pHEMA matrix was immersed in a photographic
developer (pH ~13.0) that consisted of 4-methylaminophenol sulfate (0.3% w/v), ascorbic acid (2% w/v), Na 2CO3 (5% w/v)
and NaOH (1.5% w/v) for 30 s, followed by immersion in an acetic acid bath (5% v/v). 3 This process converts Ag+ ions in
the pHEMA to Ag0 NPs with an average diameter (Ø) from ~10-50 nm.1 In contrast to traditional photography, the developer
works as a reducing agent independently of any exposure to light.
Recording the Holograms in Denisyuk Reflection Mode
The pHEMA-Ag0 NP sample was aligned with a tilt angle (𝜃𝑖 ) with respect to the plane of a mirror. A 6 ns Nd:YAG pulsed
laser beam (532 nm, 350 mJ, Q-switch delay=400 µs) was deviated by dielectric mirrors, expanded, collimated to obtain a
beam (Ø =1 cm) and directed onto the sample. When the laser light is absorbed by the Ag 0 NPs at the antinode regions, the
laser light might selectively heat, attenuate or displace Ag0 NPs.1, 4
Hologram probing
The hologram was probed by a custom made reflection spectrophotometer with a resolution of 0.5 nm (Fig. 4b). The sample
was illuminated perpendicularly with a white light source and the diffracted light was collected at angles away from the
reflected light. The wavelength of the diffracted light is governed by Bragg’s law:
𝜆𝑝𝑒𝑎𝑘 = 2𝑛𝑑 𝑠𝑖𝑛 𝜃𝐵
(17)
where λpeak is the wavelength of the first order diffracted light at the maximum intensity in vacuo, n is the effective index of
refraction of the recording medium, d is the spacing between the two consecutive recorded NP based layers (constant
parameter), and 𝜃𝐵 is the Bragg angle determined by the recording geometry.
Simulations of the Grating
The tilt angle during fabrication was varied in order to understand the effect on the formation of a multilayer and the blazed
grating. Figure S1 shows the interference pattern of the three distinct plane waves and the tilt angle of the hologram is varied
from 0° to 90°. The incident angle is propagated from the top in Denisyuk reflection mode, as evident from the intensity of
the light, which exponentially decays as it propagates through the recording media. In addition to the multilayer structure, a
larger periodic wave corresponding to a transmission grating appears in the vertical direction. Although this image shows the
intensity distribution, the photochemical patterning occurs at a specific threshold. Therefore, the waves with above certain
4
intensities allow patterning. Figure S2 shows the interference pattern arising from the superposition of the reflected waves as
the tilt angle from the normal is varied from 0°-90° with 5° increments assuming that transmission (20%), thickness (10 μm)
and intensity (0.5). The periodicity of the transmission grating diminishes as the tilt angle approaches 0°. Figure S3 illustrates
the interference pattern arising from the superposition of the reflected waves as the transmission is varied from 0-100% with
10% increments the tilt angle (5°), thickness (10 μm), and intensity (0.5). As the transmission increases, the amount of
intensity required increases. This represents a compromise between the absorption and the intensity. Figure S4 shows the
formed pattern based on thresholding the intensity assuming that the transmission is 20% and thickness is 10 μm. In the
simulated pattern, black regions correspond to the non-patterned material, while white regions represent photochemically
patterned material. Along with the vertical standing wave (~193 nm), a larger period wave (~3 μm) is in the horizontal
direction. As the angle from the normal increases, the multilayer pattern diminishes. Figure S5 shows the interference pattern
arising from the superposition of the reflected waves as the thickness of the polymer matrix is varied from 0.5-10.0 µm with
0.5 µm increments assuming that transmission (20%), angle (5°) and intensity (0.5). As the thickness of the hydrogel
increases, the number of layers in the multilayer structure also increases. At least 5 µm thick hydrogel is required to obtain
~10 stacks (Fig. S5). Thus, using simulation, we can predict the optical characteristics such as the periodicity of the
transmission grating, regions that will be patterned at a given transmission or absorption values and the number of multilayer
gratings that will be formed during fabrication.
5
FIG. S1. The interference pattern arising from the superposition of the reflected waves as the tilt angle is varied 0-90° from the normal in
Denisyuk reflection mode. The incident angle is propagated from the top. Intensity field distribution I = |E| 2 produced by laser light
interference inside the hydrogel, created by three beams: (1) incident beam, (2) beam reflected from the mirror and (3) beam
reflected internally at the polymer-water interface. Scale bars=3 µm.
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FIG. S2. The interference pattern arising from the superposition of the reflected waves as the tilt angle from the normal is varied from 0°90° with 5° increments in Denisyuk reflection mode. The incident angle is propagated from the top. Scale bars = 3 µm.
7
FIG. S3. The interference pattern arising from the superposition of the reflected waves as the transmission is varied from 0-1 with 0.1
increments in Denisyuk reflection mode. The incident angle is propagated from the top. Scale bars = 3 µm.
FIG. S4. Threshold of intensity of the cross section as the intensity is varied from 0-1 in Denisyuk reflection mode. The incident angle is
propagated from the top. The period of the surface grating is 3.01 μm at 5°. Scale bars = 3 µm.
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FIG. S5. The interference pattern arising from the superposition of the reflected waves as the polymer matrix thickness is varied from 0.510.0 µm with 0.5 µm increments in Denisyuk reflection mode. The incident angle is propagated from the top. Scale bars = 3 µm.
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