Advances in Lasers and Electro Optics Part 13 docx

choose 2f-2f classical imaging systems with 1/2f + 1/2f = 1/f to image the speckles of the source onto the object plane and the ghost image plane.. For instance, using the Sun as light s

The Physics of Ghost Imaging 583

Fig 20 Schematic illustration of It is clear that the amplitude pairs

j 1 × l 2 with l1 × j 2, where j and l represent all point sub-sources, pair by pair, will

experience equal optical path propagation and superpose constructively when D1 and D2 are

located at  , z1  z2 This interference is similar to symmetrizing the wavefunction of identical particles in quantum mechanics

It is not difficult to see the nonlocal nature of the superposition shown in Eq (59) In Eq

(59), G(2)(r1, t1; r2, t2) is written as a superposition between the paired sub-fields Ej (r1, t1)

E l(r2, t2) and El(r1, t1)Ej(r2, t2) The first term in the superposition corresponds to the

situation in which the field at D1 was generated by the jth sub-source, and the field at D2 was generated by the lth sub-source The second term in the superposition corresponds to a different yet indistinguishable situation in which the field at D1 was generated by the lth sub-source, and the field at D2 was generated by the jth sub-source Therefore, an interference is concealed in the joint measurement of D1 and D2, which physically occurs at

two space-time points (r1, t1) and (r2, t2) The interference corresponds to |Ej1E l2 + El1E j2| 2 It

is easy to see from Fig 20, the amplitude pairs j 1 × l 2 with l 1 × j 2, j ‘1 × l ‘2 with l ‘1 × j‘2,

j 1 × l ‘2 with l ‘1 × j 2, and j ‘1 × l 2 with l 1 × j‘2, etc., pair by pair, experience equal total optical

path propagation, which involves two arms of D1 and D2, and thus superpose constructively when D1 and D2 are placed in the neighborhood of = , z1 = z2 Consequently, the

summation of these individual constructive interference terms will yield a maximum value

When ≠ , z1 = z2, however, each pair of the amplitudes may achieve different relative phase

and contribute a different value to the summation, resulting in an averaged constant value

It does not seem to make sense to claim a nonlocal interference between [(E j goes to D1) × (E l goes to D2)] and [(E l goes to D1) × (E j goes to D2)] in the framework of Maxwell’s

electromagnetic wave theory of light This statement is more likely adapted from particle physics, similar to symmetrizing the wavefunction of identical particles, and is more

suitable to describe the interference between quantum amplitudes: [(particle-j goes to D1) × (particle-l goes to D2)] and [(particle-l goes to D1) × (particle-j goes to D2)], rather than

waves Classical waves do not behave in such a manner In fact, in this model each source corresponds to an independent spontaneous atomic transition in nature, and consequently corresponds to the creation of a photon Therefore, the above superposition corresponds to the superposition between two indistinguishable two-photon amplitudes,

sub-and is thus called two-photon interference [9] In Dirac’s theory, this interference is the result

of a measured pair of photons interfering with itself

In the following we attempt a near-field calculation to derive the point-to-point correlation of

G(2)( , z1; , z2) We start from Eq (59) and concentrate to the transverse spatial correlation

In the near-field we apply the Fresnel approximation as usual to propagate the field from

each subsource to the photodetectors G(2)( , z1; , z2) can be formally written in terms of the Green’s function,

(61)

In Eq (61) we have formally written G(2) in terms of the first-order correlation functions G(1),

but keep in mind that the first-order correlation function G(1) and the second-order

correlation function G(2) represent different physics based on different measurements Substituting the Green’s function derived in the Appendix for free propagation

into Eq (61), we obtain G(1)( , z1)G (1)( , z2) ~constant and

Assuming a2( ) ~constant, and taking z1 = z2 = d, we obtain

(62)

where we have assumed a disk-like light source with a finite radius of R The transverse spatial correlation function G(2)( ; ) is thus

(63)Consequently, the degree of the second-order spatial coherence is

(64)

The Physics of Ghost Imaging 585

For a large value of 2R/d ~ Δθ, where Δθ is the angular size of the radiation source viewed

at the photodetectors, the point-spread somb-function can be approximated as a δ-function

of | ư | We effectively have a “point-to-point” correlation between the transverse

planes of z1 = d and z2 = d In 1-D Eqs (63) and (64) become

(65)and

(66)which has been experimentally demonstrated and reported in Fig 18

We have thus derived the same second-order correlation and coherence functions as that of the quantum theory The non-factorizable point-to-point correlation is expected at any intensity The only requirement is a large number of point sub-sources with random relative phases participating to the measurement, such as trillions of independent atomic transitions There is no surprise to derive the same result as that of the quantum theory from this simple model Although the fields are not quantized and no quantum formula was used in the above calculation, this model has implied the same nonlocal two-photon interference mechanism as that of the quantum theory Different from the phenomenological theory of intensity fluctuations, this semiclassical model explores the physical cause of the phenomenon

5 Classical simulation

There have been quite a few classical approaches to simulate type-one and type-two ghost imaging Different from the natural non-factorizable type-one and type-two point-to-point imaging-forming correlations, classically simulated correlation functions are all factorizable

We briefly discuss two of these man-made factoriable classical correlations in the following (I) Correlated laser beams

In 2002, Bennink et al simulated ghost imaging by two correlated laser beams [26] In this

experiment, the authors intended to show that two correlated rotating laser beams can simulate the same physical effects as entangled states Figure 21 is a schematic picture of the

experiment of Bennink et al Different from type-one and type-two ghost imaging, here the

point-to-point correspondence between the object plane and the “image plane” is made artificially by two co-rotating laser beams “shot by shot” The laser beams propagated in opposite directions and focused on the object and image planes, respectively If laser beam-1

is blocked by the object mask there would be no joint-detection between D1 and D2 for that

“shot”, while if laser beam-1 is unblocked, a coincidence count will be recorded against that angular position of the co-rotating laser beams A shadow of the object mask is then reconstructed in coincidences by the blockingưunblocking of laser beam-1

A man-made factorizable correlation of laser beam is not only different from the factorizable correlations in type-one and type-two ghost imaging, but also different from the standard statistical correlation of intensity fluctuations Although the experiment of Bennink

non-et al obtained a ghost shadow, which may be useful for certain purposes, it is clear that the

Fig 21 A ghost shadow can be made in coincidences by “blocking-unblocking” of the correlated laser beams, or simply by “blocking-unblocking” two correlated gun shots The man-made trivial “correlation” of either laser beams or gun shots are deterministic, i.e., the laser beams or the bullets know where to go in each shot, which are fundamentally different from the quantum mechanical nontrivial nondeterministic multi-particle correlation

physics shown in their experiment is fundamentally different from that of ghost imaging In fact, this experiment can be considered as a good example to distinguish a man-made trivial deterministic classical intensity-intensity correlation from quantum entanglement and from

a natural nonlocal nondeterministic multi-particle correlation

(II) Correlated speckles

Following a similar philosophy, Gatti et al proposed a factorizable “speckle-speckle”

classical correlation between two distant planes, and , by imaging the speckles of the common light source onto the distant planes of and , [13]

(67)where is the transverse coordinate in the plane of the light source.9

The schematic setup of the classical simulation of Gatti et al is depicted in Fig 22 [13] Their

experiment used either entangled photon pairs of spontaneous parametric down-conversion (SPDC) or chaotic light for obtaining ghost shadows in coincidences To distinguish from

9 The original publications of Gatti et al choose 2f-2f classical imaging systems with 1/2f + 1/2f = 1/f to image the speckles of the source onto the object plane and the ghost image plane The man-mde speckle-speckle image-forming correlation of Gatti et al shown

in Eq (67) is factorizeable, which is fundamentally different from the natural factorizable image-formimg correlations in type-one and type-two ghost imaging In fact, it

non-is very easy to dnon-istingunon-ish a classical simulation from ghost imaging by examining its experimental setup and operation The man-made speckle-speckle correlation needs to have two sets of identical speckles observable (by the detectors or CCDs) on the object and the image planes In thermal light ghost imaging, when using pseudo-thermal light source, the classical simulation requires a slow rotating ground grass in order to image the speckles of the source onto the object and image planes (typically, sub-Hertz to a few Hertz) However,

to achieve a natural HBT nonfactorizable correlation of chaotic light for type-two ghost imaging, we need to rotate the ground grass as fast as possible (typically, a few thousand Hertz, the higher the batter)

The Physics of Ghost Imaging 587

Fig 22 A ghost “imager” is made by blocking-unblocking the correlated speckles The two identical sets of speckles on the object plane and the image plane, respectively, are the

classical images of the speckles of the source plane The lens, which may be part of a CCD camera used for the joint measurement, reconstructs classical images of the speckles of the

source onto the object plane and the image plane, respectively s o and s i satisfy the Gaussian

thin lens equation 1/s o + 1/s i = 1/f

ghost imaging, Gatti et al named their work “ghost imager” The “ghost imager” comes

from a man-made classical speckle-speckle correlation The speckles observed on the object and image planes are the classical images of the speckles of the radiation source, reconstructed by the imaging lenses shown in the figure (the imaging lens may be part of a

CCD camera used for the joint measurement) Each speckle on the source, such as the jth

speckle near the top of the source, has two identical images on the object plane and on the image plane Different from the non-factorizeable nonlocal image-forming correlation in type-one and type-two ghost imaging, mathematically, the speckle-speckle correlation is factorizeable into a product of two classical images of speckles If two point photodetectors

D1 and D2 are scanned on the object plane and the image plane, respectively, D1 and D2 will have more “coincidences” when they are in the position within the two identical speckles,

such as the two jth speckles near the bottom of the object plane and the image plane The

blocking-unblocking of the speckles on the object plane by a mask will project a ghost

shadow of the mask in the coincidences of D1 and D2 It is easy to see that the size of the

identical speckles determines the spatial resolution of the ghost shadow This observation has been confirmed by quite a few experimental demonstrations There is no surprise that

Gatti et al consider ghost imaging classical [27] Their speckle-speckle correlation is a

man-made classical correlation and their ghost imager is indeed classical The classical simulation

of Gatti et al might be useful for certain applications, however, to claim the nature of ghost

imaging in general as classical, perhaps, is too far [27] The man-made factorizable

speckle-speckle correlation of Gatti et al is a classical simulation of the natural nonlocal

point-to-point image-forming correlation of ghost imaging, despite the use of either entangled photon source or classical light

6 Local? Nonlocal?

We have discussed the physics of both type-one and type-two ghost imaging Although different radiation sources are used for different cases, these two types of experiments demonstrated a similar non-factorizable point-to-point image-forming correlation:

In type-one ghost imaging, the δ-function in Eq (68) means a typical EPR position-position

correlation of an entangled photon pair In EPR’s language: when the pair is generated at the source the momentum and position of neither photon is determined, and neither photon-one nor photon-two “knows” where to go However, if one of them is observed at a point at the object plane the other one must be found at a unique point in the image plane In type-two ghost imaging, although the position-position determination in Eq (69) is only partial, it generates more surprises because of the chaotic nature of the radiation source Photon-one and photon-two, emitted from a thermal source, are completely random and independent, i.e., both propagate freely to any direction and may arrive at any position in the object and image planes Analogous to EPR’s language: when the measured two photons were emitted from the thermal source, neither the momentum nor the position of any photon is determined However, if one of them is observed at a point on the object plane the other one must have twice large probability to be found at a unique point in the image plane Where does this partial correlation come from? If one insists on the view point of intensity fluctuation correlation, then, it is reasonable to ask why the intensities of the two light beams exhibit fluctuation correlations at = only? Recall that in the experiment of

Sarcelli et al the ghost image is measured in the near-field Regardless of position, D1 and D2

receive light from all (a large number) point sources of the thermal source, and all

sub-sources fluctuate randomly and independently If ΔI1ΔI2 = 0 for ≠ , what is the physics

to cause ΔI1ΔI2 ≠ 0 at = ?

The classical superposition is considered “local” The Maxwell electromagnetic field theory requires the superposition of the electromagnetic fields, either or , takes

place at a local space-time point (r, t) However, the superposition shown in Eqs (68) and

(69) happens at two different space-time points (r1, t1) and (r2, t2) and is measured by two

independent photodetectors Experimentally, it is not difficult to make the two detection events space-like separated events Following the definition given by EPR-Bell, we

photo-consider the superposition appearing in Eqs (68) and (69) nonlocal Although the

two-The Physics of Ghost Imaging 589 photon interference of thermal light can be written and calculated in terms of a semiclassical model, the nonlocal superposition appearing in Eq (69) has no counterpart in the classical measurement theory of light, unless one forces a nonlocal classical theory by allowing the superposition to occur at a distance through the measurement of independent photodetectors, as we have done in Eq (59) Perhaps, it would be more difficult to accept a nonlocal classical measurement theory of thermal light rather than to apply a quantum mechanical concept to “classical” thermal radiation

7 Conclusion

In summary, we may conclude that ghost imaging is the result of quantum interference Either type-one or type-two, ghost imaging is characterized by a non-factorizable point-to-point image-forming correlation which is caused by constructive-destructive interferences involving the nonlocal superposition of two-photon amplitudes, a nonclassical entity corresponding to different yet indistinguishable alternative ways of producing a joint photo-detection event The interference happens within a pair of photons and at two spatially separated coordinates The multi-photon interference nature of ghost imaging determines its peculiar features: (1) it is nonlocal; (2) its imaging resolution differs from that of classical; and (3) the type-two ghost image is turbulence-free Taking advantage of its quantum interference nature, a ghost imaging system may turn a local “bucket” sensor into a nonlocal imaging camera with classically unachievable imaging resolution For instance, using the Sun as light source for type-two ghost imaging, we may achieve an imaging spatial resolution equivalent to that of a classical imaging system with a lens of 92-meter diameter when taking pictures at 10 kilometers.10 Furthermore, any phase disturbance in the optical path has no influence on the ghost image To achieve these features the realization of multi-photon interference is necessary

8 Acknowledgment

The author thanks M D’Angelo, G Scarcelli, J.M Wen, T.B Pittman, M.H Rubin, and L.A

Wu for helpful discussions This work is partially supported by AFOSR and ARO-MURI program

Appendix: Fresnel free-propagation

We are interested in knowing how a known field E(r0, t0) on the plane z0 = 0 propagates or

diffracts into E (r, t) on another plane z = constant We assume the field E(r0, t0) is excited by

an arbitrary source, either point-like or spatially extended The observation plane of

z = constant is located at an arbitrary distance from plane z0 = 0, either far-field or near-field Our goal is to find out a general solution E (r, t), or I (r, t), on the observation plane, based

on our knowledge of E(r0, t0) and the laws of the Maxwell electromagnetic wave theory It is

not easy to find such a general solution However, the use of the Green’s function or the

10 The angular size of Sun is about 0.53° To achieve a compatible image spatial resolution, a traditional camera must have a lens of 92-meter diameter when taking pictures at 10 kilometers

field transfer function, which describes the propagation of each mode from the plane of

z0 = 0 to the observation plane of z = constant, makes this goal formally achievable

Unless E(r0, t0) is a non-analytic function in the space-time region of interest, there must

exist a Fourier integral representation for E(r0, t0)

(A-1)

where wk (r0, t0) is a solution of the Helmholtz wave equation under appropriate boundary

conditions The solution of the Maxwell wave equation , namely the Fourier mode, can be a set of plane-waves or spherical-waves depending on the chosen boundary condition In Eq is the complex amplitude of the

Fourier mode k In principle we should be able to find an appropriate Green’s function

which propagates each mode under the Fourier integral point by point from the plane of

z0 = 0 to the plane of observation,

(A-2)

each point on the plane of z0 = 0 are then superposed coherently on each point on the

observation plane with their after-propagation amplitudes and phases It is convenient to

write Eq (A−2) in the following form

the result of a superposition of the spherical secondary wavelets that originated from each

point on the σ0 plane (see Fig A−1),

(A-4)

where we have set z0 = 0 and t0 = 0 at plane σ0, and defined In Eq

(A−4), ( ) is the complex amplitude or relative distribution of the field on the plane of σ0,

which may be written as a simple aperture function in terms of the transverse coordinate , as we have done in the earlier discussions

The Physics of Ghost Imaging 591

Fig A−1 Schematic of free-space Fresnel propagation The complex amplitude ( ) is

composed of a real function A( ) and a phase associated with each of the

transverse wavevectors in the plane of σ0 Notice: only one mode of wavevector k( , ω) is

shown in the figure

In the near-field Fresnel paraxial approximation, when we take the

first-order expansion of r in terms of z and ,

(A-5)

so that E( , z, t) can be approximated as

where is named the Fresnel phase factor

Assuming that the complex amplitude ( ) is composed of a real function A( ) and a phase , associated with the transverse wavevector and the transverse coordinate on

the plane of σ0, as is reasonable for the setup of Fig A−1, we can then write E( , z, t) in the

(A-7)

Notice that the last equation in Eq (A−7) is the Fourier transform of the function

As we shall see in the following, these properties are very useful in simplifying the

calculations of the Green’s functions g( , ω; , z)

Next, we consider inserting an imaginary plane between σ0 and σ This is equivalent to

having two consecutive Fresnel propagations with a diffraction-free plane of infinity Thus, the calculation of these consecutive Fresnel propagations should yield the same

Green’s function as that of the above direct Fresnel propagation shown in Eq (A−6):

The Physics of Ghost Imaging 593

Therefore, the normalization constant C must take the value of C = −iω/2πc The

normalized Green’s function for free-space Fresnel propagation is thus

[3] A Einstein, B Podolsky, and N Rosen, Phys Rev 35, 777 (1935)

[4] D.V Strekalov, A.V Sergienko, D.N Klyshko and Y.H Shih, Phys Rev Lett 74, 3600

(1995) Due to its nonlocal behavior, this experiment was named “ghost” interference by the physics community

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[6] A Valencia, G Scarcelli, M D’Angelo, and Y.H Shih, Phys Rev Lett 94, 063601

(2005)

[7] G Scarcelli, A Valencia, and Y.H Shih, Europhys Lett 68, 618 (2004)

[8] R Meyers, K.S Deacon, and Y.H Shih, Phys Rev A 77, 041801(2008)

[9] Y.H Shih, IEEE J of Selected Topics in Quantum Electronics, 9, 1455 (2003)

[10] R Hanbury-Brown, and R.Q Twiss, Nature, 177, 27 (1956); 178, 1046, (1956); 178, 1447

(1956)

[11] R Hanbury-Brown, Intensity Interferometer, Taylor and Francis Ltd, London, 1974 [12] M.O Scully and M.S Zubairy, Quantum Optics, Cambridge University Press,

Cambridge, 1997

[13] A Gatti, E Brambilla, M Bache and L.A Lugiato, Phys Rev A 70, 013802, (2004), and

Phys Rev Lett 93, 093602 (2004)

[14] K Wang, D Cao, quant-ph/0404078; D Cao, J Xiong, and K Wang, quant ph/

0407065

[15] Y.J Cai, and S.Y Zhu, quant-ph/0407240, Phys Rev E, 71, 056607 (2005)

[16] B.I Erkmen and J.H Shapiro, Phys Rev A 77, 043809 (2008)

[17] M H Rubin, Phys Rev A 54, 5349 (1996)

[18] J W Goodman, Introduction to Fourier Optics, McGraw-Hill Publishing Company, New

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1988

[20] R.J Glauber, Phys Rev 130, 2529 (1963); Phys Rev 131, 2766 (1963)

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[24] R Meyers, K.S Deacon, and Y.H Shih, to be published

[25] J.B Liu, and Y.H Shih, Phys Rev A, 79, 023818 (2009)

[26] R.S Bennink, S.J Bentley, and R.W Boyd, Phys Rev Lett 89, 113601 (2002); R.S

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25

High Performance Holographic Polymer Dispersed Liquid Crystal Systems Formed with the Siloxane-containing Derivatives and Their

Applications on Electro-optics

Yeonghee Cho and Yusuke Kawakami

Japan Advanced Institute of Science and Technology

Japan

1 Introduction

Holography is a very powerful technology for high density and fast data storage, which have been applied to the systems known as holographic polymer dispersed liquid crystal (HPDLC), in which gratings are formed by anisotropic distribution of polymer and LC-rich layers through photopolymerization of monomers or oligomers and following phase separation of LC in the form of interference patterns of incident two laser beams [1-5] Much attentions have been attracted to HPDLC systems due to their unique switching property in electric field to make them applicable to information displays, optical shutters, and information storage media [6-15]

Many research groups have made efforts to realize useful recording materials for high performance holographic gratings [16-18] Photo-polymerizable materials, typically multi-functional acrylates, epoxy, and thiol-ene derivatives have been mostly studied because of their advantages of optical transparency, large refractive index modulation, low cost, and easy fabrication and modification[19-25] T.J Bunning group has reported investigation that the correlation between polymerization kinetics, LC phase separation, and polymer gel point in examining thiol-ene HPDLC formulations to enable more complete understanding

of the formation of thiol-ene HPDLCs [26] Kim group has developed that the doping of conductive fullerene particles to the formulations based on polyurethane acrylate oligomers

in order to reduce the droplet coalescence of LC and operating voltage [27]

Further extensive research has been devoted to the organic-inorganic hybrid materials having the sensitivity to visible laser beam to resolve the drawbacks of photopolymerizable materials such as volume shrinkage, low reliability, and poor long term stability even high reactivity of them as well waveguide materials, optical coatings, nonlinear optical materials, and photochromic materials [28-29] Blaya et al theoretically and experimentally analyzed the angular selectivity curves of nonuniform gratings recorded in a photopolymerizable silica glass due to its rigidity suppressing the volume shrinkage [30] Ramos et al found that

a chemical modification of the matrix with tetramethylorthosilicate noticeably attenuates the shrinkage, providing a material with improved stability for permanent data storage applications [31]

However, those materials still have significant drawbacks such as volume shrinkage, low reliability, and poor long term stability

Recently, we have focused on the siloxane-containing derivatives by taking advantage of their chemical and physical properties with high thermal stability, high optical clarity, flexibility, and incompatibility[32]

In this research, first, siloxane-containing epoxides were used to induce the efficient separation of LC from polymerizable monomer and to realize high diffraction efficiency and low volume shrinkage during the formation of gratings since the ring-opening polymerization (ROP) systems with increased excluded free-volume during the polymerization suppress the volume shrinkage [33] Although various epoxide derivatives were used, cyclohexane oxide group should be more suitable to control the volume shrinkage in the polymerization due to their ring structure with more bulky group Actually, we improved the volume shrinkage causing a serious problem during the photopolymerization, by using the ROP system with novel siloxane-containing spiroorthoester and bicyclic epoxides

Generally, the performance of holographic gratings in HPDLC systems strongly depends on the final morphologies, sizes, distribution, and shapes of phase-separated LC domains controlled by adjusting the kinetics of polymerization and phase separation of LC during the polymerization Control of the rate and density of cross-linking in polymer matrix is very important in order to obtain clear phase separation of LC from polymer matrix to homogeneous droplets Too rapid initial cross-linking by multi-functional acrylate makes it difficult to control the diffusion and phase separation of LC At the same time, high ultimate conversion of polymerizable double bond leading to high cross-linking is important for long-term stability These are not easy to achieve at the same time

Till now optimization of cross-linking process has been mainly pursued by controlling the average functionality of multi-functional acrylate by mixing dipentaerythritol pentaacrylate (DPEPA), trimethylolpropane triacrylate (TMPTA) and tri(propyleneglycol) diacrylate, or

by diluting the system with mono-functional vinyl compound like 1-vinyl-2-pyrollidone (NVP) [34-37] In case of TMPTA, considerably high concentration was used Mono-functional NVP adjusts the initial polymerization rate and final conversion of acrylate functional groups by lowering the concentration of cross-linkable double bonds [38] However, the effects were so far limited, and these systems still caused serious volume shrinkage and low final conversion of polymerizable groups Thus, the gratings are not long-term stable, either Moreover, the phase separation of LC component during the matrix formation was governed only by its intrinsic property difference against polymer matrix, accordingly not well-controlled These systems could be called as “passive grating formation” systems

Thus, if we consider the structure and reactivity of siloxane compounds in relation with the property, it will be possible to propose new systems to improve the performance of HPDLC gratings

Second, the objective of this research is to show the effectiveness of the simultaneous siloxane network in formation of polymer matrix by radically polymerizable multi-functional acrylate by using trialkoxysilyl (meth)acrylates, and to characterize the application of dense wavelength division multiplexing (DWDM) systems By loading high concentration of trialkoxysilyl-containing derivatives, volume shrinkage during the formation of polymer matrix should be restrained The principal role of multi-functional

High Performance Holographic Polymer Dispersed Liquid Crystal Systems Formed

with the Siloxane-containing Derivatives and Their Applications on Electro-optics 597 acrylate in grating formation is to make the LC phase-separate by the formation of cross-linked polymer matrix

Our idea is to improve the property of gratings through importing the siloxane network in polymer matrix, by not only lowering the contribution of initial rapid radical cross-linking

of TMPTA and realizing complete conversion of double bonds, but also maintaining the desirable total cross-linking density assisted by hydrolysis-condensation cross-linking of trialkoxysilyl group in the (meth)acrylate component to control the phase separation of LC from polymer matrix [39] Such cross-linking can be promoted by the proton species produced from the initiating system together with radical species by photo-reaction [40-42]

In our system, phase separation of LC is not so fast compared with simple multi-functional acrylate system, and secondary cross-linking by the formation of siloxane network enforce the LC to completely phase-separate to homogeneous droplets, and high diffraction efficiency could be expected We named this process as “proton assisted grating formation” These systems should provide many advantages over traditional systems induced only by radical polymerization by improving: 1) the volume shrinkage by reducing the contribution

of radical initial cross-linking by importing the siloxane network in whole polymer networks, 2) the contrast of siloxane network formed by the hydrolysis of ω-methacryloxyalkyltrialkoxysilane against polymer matrix, and 3) the stability of final gratings via combination of the characteristics of siloxane gel and rather loosely cross-linked radically polymerized system

Finally, poly (propylene glycol) (PPG) derivatives functionalized with triethoxysilyl, hydroxyl, and methacrylate groups were synthesized to control the reaction rate and extent

of phase separation of LC, and their effects were investigated on the performance of holographic gratings The well-constructed morphology of the gratings was evidenced by atomic force microscopy (SEM)

2 Experimental

2.1 Holographic recording materials

Multi-functional acrylates, trimethylolpropane triacrylate (TMPTA) and dipentaerythritol penta-/hexa- acrylate (DPHA), purchased from Aldrich Chemical Co., were used as radically cross-linkable monomers to tune the reaction rate and cross-linking density

Structures of ring-opening cross-linkable monomers used in this study are shown in Figure

1 Bisphenol-A diglycidyl ether (A), neopentyl glycol diglycidyl ether (B),

bis[(1,2-epoxycyclohex-4-yl)methyl] adipate (F) from Aldrich Chemical Co and glycidoxypropyl)-1,1,3,3-tetramethyldisiloxane (C), 1,5-bis(glycidoxypropyl)-3-phenyl- 1,1,3,5,5-pentamethyltrisiloxane (E) from Shin-Etsu Co were used without further purification 1,5-Bis(glycidoxypropyl)-1,1,3,3,5,5-hexamethyltrisiloxane (D), 1,3-bis[2-(1,2-epoxycyclohex-4- yl)ethyl]-1,1,3,3-tetramethyldisiloxane (G), and 1,5-bis[2-(1,2-epoxycyclohex-4-yl)ethyl]- 1,1,3,3,5,5-hexamethyltrisiloxane (H) were synthesized by hydrosilylation of allyl glycidyl ether, or 4-vinyl-1-cyclohexene-1,2-epoxide (Aldrich Chemical Co.) with 1,1,3,3,5,5-hexamethyltrisiloxane, or 1,1,3,3-tetramethyldisiloxane (Silar Laboratories) in toluene at 60~70˚C for 24h in the presence of chlorotris(triphenylphosphine)rhodium(I) [RhCl(PPh3)3] (KANTO chemical co Inc.)

1,3-bis(3-Methacryloxymethyltrimethylsilane TMS), methacryloxymethyltrimethoxysilane TMOS), 3-methacryloxypropyltrimethoxysilane (MP-TMOS), 3-methacryloxypropyltriethoxysilane (MP-TEOS), 3-N-(2-

(MM-methacryloxyethoxycarbonyl)aminopropyltriethoxysilane (MU-TEOS), and methacryloxy-2-hydroxypropyl)aminopropyltriethoxysilane (MH-TEOS), purchased from

3-N-(3-Gelest, Inc., were used as reactive diluents (Figure 2) Methacrylate with trialkoxysilane are

capable of not only radical polymerization but also hydrolysis-condensation

To investigate the effects of functional groups of photo-reactive PPG derivatives on performance of holographic gratings, three types of PPG derivatives were functionalized

with triethoxysilyl, hydroxyl, and methacrylate groups as shown in Figure 3 PPG

derivative with difunctional triethoxysilyl groups (PPG-DTEOS) and PPG derivative together with hydroxyl and triethoxysilyl groups (PPG-HTEOS) were synthesized by using

1 mol of PPG (Polyol.co Ltd.) with 2 mol and 1 mol of 3-(triethoxysilyl)propyl isocyanate (Aldrich), respectively PPG derivative together with methacrylate and triethoxysilyl groups PPG-MTEOS was synthesized by using 1 mol of PPG-HTEOS with 1 mol of 2-isocyanatoethyl methacrylate (Gelest, Inc.)

1-Vinyl-2-pyrrolidone (NVP) was used as another radically polymerizable reactive diluent Commercial nematic LC, TL203 (TNI=74.6 °C, ne=1.7299, no=1.5286, Δn=0.2013) and E7 (TNI=61 °C, ne=1.7462, no=1.5216, Δn=0.2246) , purchased from Merck & Co Inc., were used without any purification

2.2 Composition of photo-initiator system and recording solution

Photo-sensitizer (PS) and photo-initiator (PI) having sensitivity to visible wavelength of YAG laser (λ= 532 nm) selected for this study are 3, 3’-carbonylbis(7-diethylaminocoumarin) (KC, Kodak) and diphenyliodonium hexafluorophosphate (DPI, AVOCADO research chemicals Ltd.), respectively, which produce both cationic and radical species [43-45] The concentrations of the PS and PI were changed in the range of 0.2-0.4 and 2.0-3.0 wt % to matrix components, respectively

Nd-Recording solution was prepared by mixing the matrix components (65 wt%) and LC (35 wt%), and injected into a glass cell with a gap of 14 μm and 20 μm controlled by bead spacer

2.3 Measurement of photo-DSC and FTIR

The rate of polymerization was estimated from the heat flux monitored by photo-differential scanning calorimeter (photo-DSC) equipped with a dual beam laser light of 532nm wavelength Matrix compounds were placed in uncovered aluminum DSC pans and cured with laser light by keeping the isothermal state of 30 °C at various light intensities

Infrared absorption spectra in the range 4000-400 cm-1 were recorded on polymer matrix compounds by Fourier Transform Infrared Spectroscopy (FTIR) (Perkin-Elmer, Spectrum One)

2.4 Optical setup for transmission holographic gratings

Nd:YAG solid-state continuous wave laser with 532 nm wavelength (Coherent Inc., V2) was used as the irradiation source as shown in Figure 4

Verdi-The beam was expanded and filtered by spatial filters, and collimated by collimator lens Polarized beams were generated and split by controlling the two λ/2 plates and polarizing beam splitter Thus separated two s-polarized beams with equal intensities were reflected by two mirrors and irradiated to recording solution at a pre-determined external beam angle (2θ) which was controlled by rotating the motor-driven two mirrors and moving the rotation stage along the linear stage In this research, the external incident beam angle was fixed at 16° (2θ) against the line perpendicular to the plane of the recording cell

s-High Performance Holographic Polymer Dispersed Liquid Crystal Systems Formed

with the Siloxane-containing Derivatives and Their Applications on Electro-optics 599 Real-time diffraction efficiency was measured by monitoring the intensity of diffracted beam when the shutter was closed at a constant time interval during the hologram recording After the hologram was recorded, diffraction efficiency was measured by rotating the hologram precisely by constant angle by using motor-driven controller, with the shutter closed to cut-off the reference light, to determine the angular selectivity Holographic gratings were fabricated at 20mW/cm2 intensity for one beam, and the optimum condition was established to obtain the high diffraction efficiency, high resolution, and excellent long-term stability after recording Diffraction efficiency is defined as the ratio of diffraction intensity after recording to transmitting beam intensity before recording

H2C O

O O

H 2 C O

O O

CH3

CH3

CH3

CH3O O

H2C O

CH 3

CH 3

CH 3

O O

H 2 C O

O Si

N

O O

High Performance Holographic Polymer Dispersed Liquid Crystal Systems Formed

with the Siloxane-containing Derivatives and Their Applications on Electro-optics 601

Fig 4 Experimental setup for the holographic recording and real-time reading; P: 1/2λ plate, M: mirror, SF: spatial filter, L: collimating lens, PBS: polarizing beam splitter, S: shutter, 2θ: external inter-beam angle, PD: power detector

2.5 Morphology of holographic gratings

Surface morphology of gratings was examined with scanning electron microscope (SEM, HITACHI, S-4100) The samples for measurement were prepared by freeze-fracturing in liquid nitrogen, and washed with methanol for 24h to extract the LC, in case necessary Exposed surface of the samples for SEM was coated with a very thin layer of Pt-Pd to minimize artifacts associated with sample charging (HITACHI, E-1030 ion sputter) Surface topology of transmission holographic grating was examined with atomic force microscopy (AFM, KIYENCE, VN8000) The samples for measurement were prepared by freeze-fracturing in liquid nitrogen, and washed with methanol for 24h to extract the LC AFM having a contact mode cantilever (KIYENCE, OP-75042) was used in tapping mode for image acquisition

3 Results and discussion

3.1 Effects of siloxane-containing bis(glycidyl ether)s and bis(cyclohexene oxide)s on the real-time diffraction efficiency

Real-time diffraction efficiency, saturation time, and stability of holographic gratings according to exposure time were evaluated Figure 5 shows the effects of chemical structures

of bis(glycidyl ether)s (A - E) on real-time diffraction efficiency at constant concentration of E7 (10 wt %) in recording solution [DPHA : NVP : (A - E) = 50: 10: 40 relative wt %]

In general, high diffraction efficiency can be obtained by the formulation of recording solution with large difference in refractive indexes between polymer matrix and LC, and by inducing the good phase separation between polymer rich layer and LC rich layer As expected, gratings formed with C having siloxane component had remarkably higher diffraction efficiency than gratings formed with A and B without siloxane component, which seemed to have resulted from effects of siloxane component to induce good phase separation of E7 from polymer matrix toward low intensity fringes by its incompatible property against E7 Longer induction period for grating formation of C was attributed to lower viscosity of recording solution, and the diffraction efficiency gradually increased and reached to higher value, which resulted from the further phase separation of E7 due to the flexible siloxane chain that helped migration of E7 toward low intensity fringes

A B C D E

Fig 5 Real-time diffraction efficiency of the gratings formed with (A - E) with 10 wt % E7 [DPHA: NVP: (A - E) = 50: 10: 40 relative wt %]

All the gratings formed with (C – E) having siloxane component showed high diffraction efficiencies The highest diffraction efficiency 97% was observed for D with trisiloxane chain, probably due to its incompatible property with E7 However, gratings formed with

E, having phenyl group in the trisiloxane chain, showed the lowest diffraction efficiency Bulky phenyl group attached in the siloxane chain reduced the flexibility of the chain to result in the suppression of phase separation It might have contributed to the increase of the interaction between polymer matrix with E7 having bi-/terphenyl group

Figure 6 shows the real-time diffraction efficiency of the gratings formed with bis(cyclohexene oxide) derivatives (F - H) at constant concentration of E7 (10 wt %) [DPHA: NVP: (F - H) = 50: 10: 40 relative wt %]

Gratings formed with G and H having siloxane component had higher diffraction efficiency than F without it, which seemed to indicate that, as mentioned above, siloxane chain in G and H made the solution less viscous, and incompatible with E7, which helped the easy diffusion and good phase separation between polymer matrix and E7 to result in high refractive index modulation, n Especially, H showed higher diffraction efficiency than E, probably due to flexibility and incompatibility brought about by its longer siloxane chain However, compared with C and D, G and H did not give higher diffraction efficiency, even with longer siloxane chain This may be understood because of the difference in the chemical structure of ring-opening cross-linkable group G and H have bulkier cyclohexene oxide as functional group and have higher viscosity, accordingly its diffusion toward high intensity fringes seems difficult compared with that of C or D

3.2 Volume shrinkage of the gratings depending on the structure of bis(epoxide)

Photo-polymerizable system as holographic recording material usually causes significant volume shrinkage during the formation of gratings, which can distort the recorded fringe pattern and cause angular deviations in the Bragg profile Therefore, it is very important to solve the problem of volume shrinkage in photopolymerization systems

High Performance Holographic Polymer Dispersed Liquid Crystal Systems Formed

with the Siloxane-containing Derivatives and Their Applications on Electro-optics 603

F G H

Fig 6 Real-time diffraction efficiency of the gratings formed with (F – H) and 10 wt % E7 [DPHA: NVP: (F - H) = 50: 10: 40 relative wt %]

For the measurement of volume shrinkage, slanted holographic gratings were fabricated by simply changing the angles of reference (R) and signal (S) beams, as shown in Figure 7 [46]

Fig 7 Fringe-plane rotation model for slanted transmission holographic recording to measure the volume shrinkage

R and S are recording reference (0°) and signal (32°) beams ϕ (16° in this study) is the slanted angle against the line perpendicular to the plane of the recording cell of gratings formed with S and R Solid line in the grating indicates the expected grating d is the sample thickness Actual grating formed by S and R was deviated from the expected grating shown

by dashed line by volume shrinkage of the grating Presumed signal beam (S’), which should have given actual grating was detected by rotating the recorded sample with

reference light R off This rotation of angle was taken as deviation of slanted angle R’ and S’ are presumed compensation recording reference and signal beams ϕ’ is the slanted angle

in presumed recording with S’ and R’, and d’ is the decreased sample thickness caused by volume shrinkage Degree of volume shrinkage can be calculated by following equation;

)d'tan ,d'(tantan'tan1d

d'-1shrinkage volume

50 in relative wt%] was used as the reference

Angular Selectivity (degree)

1.0 DPHA:NVP=50:50 wt%

C G

(a) (b)

Fig 8 Angular deviation from the Bragg profile for the gratings formed with C and G [DPHA: NVP : (C or G) = 50: 10: 40 relative wt %] detected by (a) diffracted S beam, and (b) diffracted R beam

As shown in Figure 8, gratings formed with G having bis(cyclohexene oxide) showed smaller deviation from Bragg matching condition than gratings formed with C having bis(glycidyl ether) for both diffracted R and S beams The diffraction efficiency after overnight was only slightly changed, which indicated the volume shrinkage after overnight was negligible

Diffraction efficiency, angular deviation, and volume shrinkage of each system were summarized in Table 1

Gratings formed with only radically polymerizable multifunctional acrylate (DPHA: NVP = 50:50 relative wt %) showed the largest angle deviation, and the largest volume shrinkage of 10.3% as is well known Such volume shrinkage could be reduced by combining the ring-

High Performance Holographic Polymer Dispersed Liquid Crystal Systems Formed

with the Siloxane-containing Derivatives and Their Applications on Electro-optics 605

opening cross-linkable monomers Especially, bis(cyclohexene oxide)s were effective to

reduce the volume shrinkage (5.6 %), probably due to its cyclic structure, although their

diffraction efficiency was lower than those formed with bis(glycidyl ether)s

Angular deviation of diffracted Recording

solution

Diffraction efficiency (%)a S beam

(degree)

R beam (degree)

ϕ’ Degree of volume shrinkage (%)

Table 1 Deviations from Bragg angle of diffracted S and R beams (degree) and degree of

volume shrinkage and diffraction efficiency determined by S beama

The shrinkage effect could be caused by mechanical reduction of the grating pitch and a real

time change in refractive index of the irradiated mixture Which factor is playing a major

role is not clear at present Distinction of these factors will be a future problem

One of the possible reasons for small volume shrinkage is the effective formation of IPN

structure in the grating in the recording system DPHA : NVP : G = 50: 10: 40 relative wt %

The balance between the formation of initial linking of DPHA and following

cross-linking by G might be proper to produce effective IPN structure

Good evidence for these was shown in Figure 9 of SEM morphologies

Figure 9 (a) and (c) show clearly phase-separated polymer layers after the treatment with

methanol, which means almost perfect phase separation between polymer rich layers and E7

rich layers Cross-sectional and surface views of the sample could be observed When 20 wt

% E7 was used, a little incompletely phase separated E7 layers were shown in Figure 9 (d),

although much higher E7 was phase separated than the case of 5 wt % E7 [Figure 9 (b)]

Grating spacing was close to the calculated value from the composition of recording

solution for the grating prepared with 5 wt % E7

3.3 Angular selectivity

When the multiplex hologram recording is required, it is necessary to know the angular

selectivity The smaller the value, the more multiplex data or gratings can be recorded [47-49]

Angular selectivity (Δθang) is defined by Kogelnik’s coupled wave theory as follows [50]:

(a) (b)

(c) (d)

Fig 9 SEM morphologies of gratings formed with H, TMPTA and various concentration of E7 [TMPTA : NVP: H = 50: 10: 40 relative wt %] (a) 5 wt %, (b) 5 wt %, ×60K, (c) 20 wt %, and (d) 20 wt %, ×60K

where n is the average refractive index of recording solution, θ is the internal incident beam angle, T is the thickness of the hologram, λ is the recording wavelength, and n is the modulation of refractive index of the recording solution after recording

Angular selectivity of our samples were similar, irrespective of the structures of epoxides (about 4˚) as typically shown in Figure 10 Solid line represents the simulated theory values according to the Kogelnik’s coupled wave theory

G Montemezzani group reported that the use of Kogelnik’s expression assuming fully symmetric beam geometries in highly birefringent materials such as LC leads to a large error [51] Our experimental data showed only a little deviation from the theoretical values by the Kogelnik’s coupled wave theory This maybe attributed to the slight thickness reduction by small volume shrinkage still existing The role of both factors should be clarified in the future

High Performance Holographic Polymer Dispersed Liquid Crystal Systems Formed

with the Siloxane-containing Derivatives and Their Applications on Electro-optics 607

Angular Selectivity (degree)

1.0

Theory Experiment

Fig 10 Angular selectivity of gratings formed with H , TMPTA, and 5 wt % E7 [TMPTA : NVP : H = 50: 10: 40 relative wt %]

As a preliminary experiment, MM-TMS and MM-TMOS were compared as a diluent for the polymer matrix component (totally 65 wt%, TMPTA : MM-TMS, or MM-TMOS : NVP = 10 :

80 : 10 in wt%, average double bond functionality = 1.1 on mole base), together with 35wt%

LC of TL203 As shown in Figure 11 gratings could not be formed with MM-TMS even with

30 min irradiation of light, because of the low average functionality of the polymerization system G P Crawford reported that HPDLC gratings made with monomer mixtures with average double bond functionality less than 1.3 were mechanically very weak[52] In general, it is difficult to form holographic gratings with low concentration of multi-functional acrylate (average double bond functionality < 1.2) by dilution with mono-functional component in radical polymerization

Dramatic enhancing in the diffraction efficiency to about 86% (induction period of 144 sec) was observed in case of MM-TMOS, even with only 10 wt% TMPTA by using 0.2 wt% KC and 2wt% DPI Only trimethoxysilyl and trimethylsilyl parts are different in these two formulations Hydrolysis of trimethoxysilyl group by moisture and following condensation seems responsible for the increased diffraction efficiency

Effects of Alkyl and Spacer Groups in ω-Methacryloxyalkyltrialkoxysilanes on the Formation and Performance of Gratings

In order to systematically study the influence of alkyl group and spacer group of methacryloxyalkyltrialkoxysilanes on the formation and performance of the formed gratings, their chemical structures were modified as shown in Figure1 The relative concentration was set as TMPTA : ω-methacryloxyalkyltrialkoxysilane : NVP = 10 : 80 : 10 wt% to clearly extract the effects of hydrolysis-condensation of trialkoxysilyl group on the formation of the gratings and the performance of the formed gratings