Advances in Lasers and Electro Optics_3 pot

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 crysta

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

acrylate in grating formation is to make the LC phase-separate by the formation of linked polymer matrix

cross-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 (MM-TMS), methacryloxymethyltrimethoxysilane (MM

3-N-(2-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

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 CH 2

O O

H 2 C O

O O

CH3

CH3

CH3

CH3O O

H2C O

O Si Si

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

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

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

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

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 %]

3.4 Effectiveness of M M -TMOS on formation of holographic gratings

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

ω-Figure 11 shows the real-time diffraction efficiency of holographic gratings formed with various ω-methacryloxyalkyltrialkoxysilanes capable of radical photo-polymerization and hydrolysis- condensation

When spacer was changed from methylene to propylene (MP-TMOS), the diffraction efficiency was dropped to 72% with longer induction period (576 sec) This seems to be because of the higher hydrophobicity of the spacer group compared with MM-TMOS The rate of the hydrolysis-condensation of trialkoxysilyl functions seems very important

By changing the trialkoxysilyl functional group from trimethoxy to triethoxy (MP-TMOS to

MP-TEOS) with the same propylene spacer, not only the diffraction efficiency was decreased

to 13%, but the induction period was also elongated to 693 sec, which again strongly suggested that the hydrolysis-condensation process of trialkoxysilane function is playing an essential role in grating formation

In grating formation, induction period basically depends on the time of the formation of cross-linked polymer matrix In classical grating formation by radical polymerization of multi-functional acrylates, induction period is observed because polymerization does not start until the complete consumption of oxygen present in the system as an inhibitor In the present system, the induction period depends on the actual gelation time of recording solution assisted by hydrolysis-condensation of trialkoxysilyl functions The induction period varies by the physical property of ω-methacryloxyalkyltrialkoxysilane derivatives The rate of the hydrolysis-condensation of trialkoxysilyl functions by moisture strongly depends on the hydrophobicity of the methacrylate monomer Polymerization of recording solution leads to changes in the chemical potential of the system, and increases the miscibility gap between LC and polymerized matrix

Fig 11 Real-time diffraction efficiency of the gratings formed with various

ω-methacryloxyalkyltrialkoxysilanes in the recording solution with 65 wt% matrix compounds

of TMPTA : ω-methacryloxyalkyltrialkoxysilane : NVP = 10 : 80 : 10 wt% and 35 wt% TL203, and KC-DPI (0.2 wt% - 2 wt% to matrix compounds) with one beam intensity of 20 mW/cm2

High Performance Holographic Polymer Dispersed Liquid Crystal Systems Formed

To investigate the effects of chemical structures of spacer between trialkoxysilylalkyl group and methacrylate group, hydrophilic urethane and hydroxylpropylene groups were introduced in the spacer of the monomer structure The highest diffraction efficiency of 75% and remarkably shorter induction period of 75 sec were obtained for grating formed with

MU-TEOS having urethane linkage in spacer group In addition, gratings formed with MHTEOS having hydroxylpropylene group in the spacer showed the shortest induction period

-of 18 sec, although the diffraction efficiency was considerably low (20%)

To summarize the results, we may consider that the radically mono-functionally polymerizable 3-methacryloxypropyltrialkoxysilane became apparently multi-functional cross-linkable monomer by hydrolysis and condensation of trialkoxysilyl group as shown in Scheme 1, which induced the high concentration of cross-linking with moderate rate by the hydrolysis

Scheme 1 proposed matrix formation processes: 1) radical cross-linking by TMPTA, 2) simultaneous radical cross-linking of TMPTA and small amounts of multi-functional methacrylate formed via hydrolysis-condensation of trialkoxysilyl group, followed by further cross-linking by hydrolysis, 3) competing rapid cross-linking of (meth)acrylate functions and sol-gel process of trialkoxysilane function, followed by further cross-linking

by radical polymerization and sol-gel process

Si(CH 3 ) 3

TMPTA

Low cross-linking density

Partial cross-linking and conversion to multi-functional methacrylate by hydrolysis-condensation

Si OR O OH Si RO

No hydrolysis-condensation

C2H5

Si O O O Si HO OH

Si OH OR

Si OH OR

Si O Si O O O O Si O RO Further cross-linking by sol-gel process

O Si O Si O O O O Si O O Si

Si O O Si HO HO

Si OH RO Si OH OR

O Si O Si O O O O Si O O Si

Si O O O Si HO O

Si OH RO Si O O Si

O O

=Acrylate functional group

by radical polymerization and sol-gel process

In case of methacryloxymethyltrimethylsilane, cross-linking density is not high enough to form grating This process corresponds to type 1) in Scheme 1 In TMOS or TEOS system, the hydrolysis of trialkoxysilyl group is relatively slow compared with the fast radical polymerization of TMPTA Thus, grating formation is not rapid, but following cross-linking

by hydrolysis assisted the formation of polymer matrix and further diffusion of LC to form gratings with high diffraction efficiency By the introduction of urethane function in the spacer, the hydrophilic nature of the spacer increases the hydrolysis of triethoxysilyl group

by moisture, and converts mono-functional methacrylate to apparently multi-fumctional methacrylate, and assisted the formation of polymer matrix by radical polymerization together with cross-linking by hydrolysis condensation This process corresponds to type 2)

in Scheme 1 In case of the introduction of hydroxypropylene spacer, too much hydrophilic nature of the spacer strongly enhanced the hydrolysis of the trialkoxysilyl group, and created the situation where apparently high concentration of multi-functional (meth)acrylates in the initial polymerization solution, and resulted in rapid formation of grating by radical cross-linking, but low diffraction efficiency This process corresponds to type 3) in Scheme 1

The spectral responses of the demultiplexer are measured by use of a broadband doped fiber amplifier (EDFA) source and are monitored by means of an optical spectrum analyzer for each channel Figure 12 shows the spectra of the two output channels with the uniform gratings formed with MU-TEOS at the ratio of TMPTA: MU-TEOS: PI solution = 40: 50: 10 wt% (35 wt% of TL203 to monomer solution) All the channels had a 3-dB bandwidth

erbium-of 0.13 nm and a channel spacing erbium-of 0.4 nm, and the interchannel cross-talk level, defined as the difference between the maximum power of a channel at the band edge and its power at the adjacent signal’s band edge, is reduced by ~18dB

1546 1547 1548 1549 1550 1551 1552 1553 1554 -25

-20 -15 -10 -5 0

Fig 12 Spectra of two output channels with 0.4 channel spacing for uniform gratings

formed with MU-TEOS at the ratio of TMPTA: MU-TEOS: PI solution = 40: 50: 10 wt% (35 wt% of TL203 to monomer solution)

High Performance Holographic Polymer Dispersed Liquid Crystal Systems Formed

3.5 Effects of triethoxysilyl, hydroxyl, and methacrylate functional groups based on PPG derivatives on performance of holographic gratings

Figure 13 shows the AFM surface topology of the gratings formed with the formulation with

65 wt% polymer matrix compound of the ratio 20: 10: 50: 20 in TMPTA: NVP: Mu-TEOS: PPG-DTEOS and 35wt% of E7 Very regular and well-defined gratings were fabricated as shown in Figure 13(a) scanned in 10 μm length The grating spacing was approximately 839.8 nm as shown in Figure 13(b) scanned in 3 μm length, which was in good agreement with the calculated spacing value of 965 nm by Bragg’s equation (grating spacing Λ = λ / 2sinθ, λ is 532 nm wavelength of laser light and θ is 16 degree of incident external half angle

in this experiment) Polymer matrix layers are shown as the sinusoidal pattern in profile of AFM topology since the LC layers were washed out by methanol from the positions of the valley parts in sinusoidal pattern, which maybe concluded that polymer matrix with PPG-DTEOS was exactly formed by photo-reaction in high intensity regions and E7 was phase-separated in low intensity regions of interference pattern of two laser beams

(a) (b)

Fig 13 AFM surface topologies of the gratings formed with the formulation with 65 wt% polymer matrix compound of the ratio 20: 10: 50: 20 in TMPTA: NVP: Mu-TEOS: PPG-DTEOS and 35wt% of E7 in (a) 10 μm and (b) 3 μm scanning lengths

The effectiveness of functional groups of PPG derivatives were observed on real-time diffraction efficiency as shown in Figure 14 Photo-reactive solutions (matrix components) were prepared from the ratios of 20: 10: 50: 20 wt% in TMPTA: NVP: Mu-TEOS: PPG

derivatives Holographic recording solutions with E7 were ready to make holographic gratings in the ratio of 65 wt% and 35wt% as photo-reactive solutions and E7, respectively

By changing the functional groups of PPG derivatives as triethoxysilyl, hydroxyl, and methacrylate groups, remarkable differences were observed on diffraction efficiency even though the induction periods for grating formation were similar with 33 second The highest final diffraction efficiency of 93% was observed in holographic gratings formed from PPG-DTEOS with difunctional triethoxysilyl groups at 240 second of irradiation of light In the case of PPG-HTEOS together with hydroxyl and triethoxysilyl groups, maximum diffraction efficiency was approximately 78% When the PPG-MTEOS together with methacrylate and triethoxysilyl groups was used, maximum diffraction efficiency reached at 96% and gradually decreased to 60% These phenomena may be considered that the functional groups of PPG derivatives affected strongly on the diffraction efficiency attributed to the difference of reaction kinetics and extend of LC phase separation In the case of PPG-DTEOS and PPG-HTEOS, phase separation of LC should not be so fast compared with the case of PPG-MTEOS, and further cross-linking by the formation of siloxane network helped the LC

to gradually phase-separate into low intensity regions of interference patterns, and maximum diffraction efficiency was reached at slower exposure time than PPG-MTEOS

PPG-DTEOS PPG-HTEOS PPG-MTEOS

Fig 14 Real-time diffraction efficiency of the gratings formed with DTEOS, HTEOS, and PPG-MTEOS in the recording solution with 65 wt% polymer matrix

PPG-compounds of the ratios 20 : 10 : 50 : 20 wt% in TMPTA : NVP: Mu-TEOS: PPG derivatives, and 35 wt% E7

High Performance Holographic Polymer Dispersed Liquid Crystal Systems Formed

1,3-bis(3-Stable transmission holographic polymer dispersed liquid crystal gratings were efficiently prepared via network formation by radical polymerization of tri-functional acrylate assisted

by hydrolysis-condensation reaction or trialkoxysilane functional group of methacryloxyalkyltrialkoxysilane, induced by radical and proton species produced in the photo-decomposition of initiating system composed of 3, 3’-carbonylbis[7’-diethylaminocoumarine] as a photo-sensitizer and diphenyliodonium hexafluorophosphate

ω-as a photo-initiator

The longest grating spacing of 0.9 μm indicated the least volume shrinkage

At higher concentration of methacrylate, gratings formed with trimethoxysilylmethyl methacrylate capable of siloxane network formation showed remarkably higher diffraction efficiency than that formed with trimethylsilylmethyl methacrylate, which does not have functional groups to be cross-linked by hydrolysis

High diffraction efficiency of 72% was obtained in gratings formed with

trimethoxysilylpropyl acrylate and E7 (35wt%) with 0.2 wt% 3,

3’-carbonylbis(7-diethylaminocoumarin), and 1 wt% diphenyliodonium hexafluorophosphate In SEM morphology, very regular and well-defined gratings were observed for the gratings formed with trimethoxysilylpropyl acrylate Although gratings formed with high concentration of trimethoxysilylpropyl acrylate had some cracks in polymer matrix, the largest grating spacing was observed indicating the lowest volume shrinkage

We developed a very useful holographic recording materials based on polypropylene glycol (PPG) derivatives functionalized with triethoxysilyl, hydroxyl, and methacrylate groups by taking into account the reaction rates and extent of phase separation of E7

of nematic LC in transmission holographic polymer dispersed liquid crystal (HPDLC) systems

Holographic gratings were clearly formed from the radical polymerization and condensation reaction of recording solution with PPG derivatives, which was demonstrated

hydrolysis-by AFM topology with very regular and well-defined morphology having the grating spacing of approximately 839.8 nm

The highest final diffraction efficiency of 93% was observed in holographic gratings formed from PPG-DTEOS with difunctional triethoxysilyl groups at 240 second of irradiation of light In the case of PPG-HTEOS together with hydroxyl and triethoxysilyl groups, maximum diffraction efficiency was approximately 78% When the PPG-MTEOS together with methacrylate and triethoxysilyl groups was used, maximum diffraction efficiency reached at 96% and gradually decreased to 60% These phenomena may be considered that

the functional groups of PPG derivatives affected strongly on the diffraction efficiency attributed to the difference of reaction kinetics and extend of LC phase separation

Consequently, if I 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 I believe that these novel recording materials controlled in nanometer scale can be significantly contributed to the development and progress in the optics, electronics, photo-induced patterning, microsystem, and nanotechnololgy Especially, electrically switchable holographic gratings are very promising for actual application such

as 3-D image storage, full color LC display, dense wavelength division multiplexing (DWDM), and polarization-selective element Moreover, this research will contribute to the establishment of new-generation display or device technology and will encourage the activity of industries

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Guymon, C A (2006) Polymer, 47, 2289

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630

26 Multicolor Stationary Light

Yi Chen1, Serguei Andreevich Moiseev2 and Byoung Seung Ham1

1Center for Photon Information Processing, and the Graduate School of IT, Inha University

2Kazan Physical-Technical Institute of Russian Academy of Sciences

It is well established that light is the fastest information carrier in nature However, controlling light for localized application is very difficult Thus, manipulation of light velocity becomes a crucial task in optical and quantum information processing (Nielsen & Chuang, 2000) Recently light localization using EIT has been demonstrated for stationary light (Bajcsy et al., 2003) Stationary light gives novel effects to nonlinear quantum optics in the context of lengthening light-matter interaction time Compared with ultraslow light, where the medium’s length is a limiting factor, stationary light is free from spatial constraint For example, the interaction time using ultraslow light in a semiconductor quantum dot, whose spatial dimension is less than a few tens of nanometers, is much less than nanosecond By using a stationary light technique, however, we can enormously increase the interaction time of the light with such a nano optical medium In this chapter,

we discuss stationary light based on the EIT-induced ultraslow light phenomenon We theoretically investigate how to dynamically manipulate multicolor (MC) stationary light in the multi double lambda-type system by simply changing the parameters of control fields, and demonstrate ultralong trapping of light, which is different from the conventional quantum mapping phenomenon Quantum coherent control of the stationary light has potential applications to various quantum optical processing such as quantum nondemolition measurement and quantum wavelength conversion

This chapter is organized as follows In section 2, quantum coherent control of two-color stationary light is described In section 3, quantum manipulation of MC stationary light is presented In section 4, we give the results of MC stationary light, with discussions Finally, section 5 offers conclusions

2 Quantum coherent control of light

2.1 Model and theory

In this section, we present dynamic control of two-color stationary light in a double-lambda type four-level system using EIT Fig 1 shows the energy level diagram We assume that

initially one weak probe quantum field E+ which couples the transition |1> − |3> enters the medium in a forward propagation direction k+, where all the atoms are in the ground state

|1> Under the influence of classical control field Ω+, which is resonant with the transition

|2> − |3> and propagates in the same direction κ+, the probe field propagates at an ultraslow group velocity After the probe pulse enters the medium completely, we adiabatically switch on the second control field Ω−, which is resonant with the transition |2>

− |4> and propagates in the backward direction κ− The probe pulse will be almost stationary if the stationary light condition is satisfied The propagation direction k− of the

field E- is determined by phase matching with the Bragg condition: k−= k+−κ+ +κ− Here

Ω+ and Ω− are the Rabi frequencies of two control fields Below we will study the evolution

of the stationary light when (1) Ω− is switched off after a certain time while keeping the control filed Ω+ on, and (2) Ω+ is switched off after a certain time while keeping the control filed Ω− on (see Fig 2)

Fig 1 Energy level diagram for two-color stationary light

For theoretical analytical purposes, we introduce a quantum field

0

field, respectively, which is traveling in a ±z direction Aσ is the slowly varying field operator, is the Planck’s constant, and ε0 is the electric permittivity We assume that the quantization volume V is 1 Under the rotating-wave approximation, we obtain the following Hamiltonian for the quantum fields and atoms in the interaction picture:

Multicolor Stationary Light 619

where P nm j =(P mn j )+are the atomic operators, gσ =℘σ ωσ /(2εo V)is the coupling

constant of photons with atoms, ℘ is the dipole moment for each transition, and ϕσ ± are

the phases of the control fields, ω =+ ω31and ω−=ω41 Using Hamiltonian (1) and adding the

decay constantsγ3, γ4, and γ2for the atomic transitions |1> − |3>, |1> − |4> and |1> −

|2>, we derive equations for the field Aσ and atomic operators j

mn

P Under the typical adiabatic condition for the slow light propagations (Fleischhauer & Lukin,

2000; Zibrov et al., 2002), we introduce new field operators

slow-light propagation v g≅ Ωc 2+/Ng2+<< , we ignore the atomic population on the c

excited levels |3> and |4> and assume slowly varying amplitudes of the laser fields Then

we can obtain the following coupled wave equations for the new field operators:

21

' 2

Fig 2 Temporal scheme of the switching sequence of the two control fields (a) Ω− is turned

off at t=t2, (b) Ω+ is turned off at t=t2.

Using spatial Fourier transformation σ( , )τ z ∞dke ikz σ( , )τ k

−∞

analytical solutions of Eqs (2) and (3):

),(

~)}

,'('exp{

),

o τωτ τ

τ τ

)

~)(

'/( })(

ck{

),'

1 '

2

αξ

αξαξηηγτ

α α= +− − + α− The functions Ψ+ −, ( , )τo z are determined by the initial conditions for

the field A+ −, which enters the medium (Ω−(t t< 1) 0= ) From Eq (4) we obtain the coupled

fields Ψ+( , ')t z andΨ−( , )t z expressed by nonlocal spatial relations:

and ε is a small value ε→ ) Eq (6) points out that spatial quantum correlations between 0

the fields Ψ+( , )t z and Ψ−( , ')t z spread out within the spatial size 1

ξ− Therefore the spatial correlations of quantum fields E+ and E

-can be determined by Eq (6)

Now we study temporal dynamics of the two-color coupled fields Initially the control field

Ω+ (Ω-=0) is turn on and a probe pulse A+ with a Gaussian shape

,

A t z+ = =A+ −t T , A+,o and T are the amplitude and temporal duration of

the field) enters the medium The solution of Eq (4) describes the propagation of the field

v ≅ Ωc + Ng+<< , and the initial spatial size c l o=v T g with an amplitude decay in

accordance with exp{−ηγ2} Thus, due to the EIT effect, the field A+ is transparent to the

optically dense medium (ξ+ −,l o>> ) At time 1 t t= , we turn on the backward control field 1

−

Ω Using the dispersion relation ( )ω k in Eq (5), we obtain the group velocity for the

interacting fields A+and A−:

Thus the group velocity of the coupled light can be easily controlled by manipulating the

control field Rabi frequencies If the stationary light condition is satisfied, the two coupled

lights can be completely stopped Ignoring small terms proportional to ck3/ξ− 2 in Eq (5),

we find the approximate solution of the integral (Eq (4)) for arbitrarily varying amplitudes

of the control laser fields:

σ σ

σ τ

Multicolor Stationary Light 621 where

(

~{,

)'(')/2()/(

)

' 2 2

2 2

and z o (ξ ξ ) (− 1ξ ξ )

− + + −

= + is a spatial shift of A−with respect to the envelope A+ (z o<< ) l o

2.2 Numerical simulations and discussions

Here we numerically demonstrate quantum manipulation of a traveling light pulse for the two-color stationary light by solving Eqs (2) and (3) For simplicity we ignore the weak decay rate γ2 between the two ground states |1> and |2>, and assume small level splittingω21 to ignore the phase mismatch between the fields Ψ and+ Ψ When the −

backward control field is turned off (Ω = ), Eqs (2) and (3) satisfy slow-light wave − 0equations (Hau et al., 1999; Turukhin et al., 2002) We note that Eqs (2) and (3) coincide with the standing single-frequency light based on a standing wave grating in a three-level system

if g+=g− with Doppler broadening (Bajcsy et al., 2003) It should be noted that Eqs (2) and (3) show general coupled equations of standing lights whether the optical dense medium is Doppler broadened or not At t t= , both control fields 1 Ω and+ Ω are turned on The −

coupled quantum fields Aσare therefore generated and they propagate and evolve together

in the medium If the stationary light condition g+/g−= Ω+/Ω = is satisfied, the coupled − 1

fields Aσ should be standing still with nearly the same amplitude until one control field is turned off This condition corresponds to the appearance of the so-called dark state in the double-Λ system (Korsunsky & Kosachiov, 1999), which can also dramatically change the interaction of the copropagating light fields

Compared with the standing-wave grating case (Bajcsy et al., 2003; Andre et al., 2005), the

1( / )

v cM= ΩΣ Ng+ in Eq (11) can be changed by adjusting the two control fields However, the lowest velocity is realized only if сd∂τ'ηα → and (0 ξ α− +−ξ α+ −)→ 0occur with complete light stoppage (v= ) The fields A0 + and A- are strongly coupled and move or stay together with the following amplitude ratio: ( /g− Ω−) ( , ) ( /A t z− ≅ g+ Ω+) ( , )A t z+ This means the field A- can be generated by the frequency shift ω−=ω41+δω+ if the frequency of the field A+ is tuned to ω+=ω31+δω+ The standing field

At time t t= , we turn off one control laser 2 Ω or + Ω Fig 2 gives the switching sequence of −

two control fields for these two cases: (a) the control field Ω is always on, while the + Ω is −

on only for t1< < ; (b) the control field t t2 Ω is on for + 0 t t< < , while the control field 2 Ω is −

on for t1< t

Fig 3 shows numerical simulation of the two-color stationary light for case (a) mentioned above (also see Fig 2(1)) For simplification, we set γ2= and assume the same Rabi 0

Multicolor Stationary Light 623

frequencies for the control fields Fig 3(c) and Fig 3(d) show the top view of the fields A+

and A- propagation, respectively Fig 3(e) and Fig 3(f) show the temporal evolutions of the field amplitudes A+ and A-, respectively Fig 3(g) and Fig 3(h) show space-time evolution of the fields A+ and A-, respectively These figures show that when the control pulse Ω is −turned off at t t= (2 t2=11), the standing field A- disappears completely, and the original field A+ keeps propagation moving in the same direction at the same group velocity as it had for 0 t t< < (1 t1= ) 4

Fig 4 shows numerical simulation of the two-color stationary light for case (b) (see Fig 2(2)) Similar to Fig 3, Fig 4(c) and Fig 4(d) show the top view of the coupled fields Fig 4(e) and Fig 4(f) show the temporal evolutions of the coupled field amplitudes Fig 4(g) and Fig 4(h) show space-time evolution of the coupled fields As seen in Figs 4(c) and 4(d), when Ω is +turned off at t t= (2 t2=11) while the Ω is kept turned on, the quantum field A− + disappears

at t t= , but the A2 - propagates in the backward direction with new carrier frequency ω− Figs 3 and 4 show that by controlling the coupling fields, the coupled fields E+ and E- can be manipulated for stationary light or frequency conversion, which has potential application for quantum nonlinear optics which needs a longer interaction time Here we note that the maximum trapping time of stationary light is determined by the coherence decay rate γ2between the two ground states |1> and |2> Thus for practical application, we may choose

an optically dense medium with a long spin decay time

In summary, we have demonstrated two-color stationary light and quantum wavelength conversion using a quantum-mechanically reversible process between photons and atomic coherence in a double-Λ four-level system In this scheme, a traveling quantum field can be manipulated by simply adjusting the control fields’ parameters for (1) determination of two-color stationary light, (2) selection of propagation direction, either forward or backward, and (3) dynamic quantum frequency conversion The present quantum manipulation of two-color stationary light can be used for enhanced nonlinear interactions between single photon fields

3 Quantum manipulation of MC stationary light

If we choose a multilevel system, MC stationary light should be possible In this section, we show quantum coherent control of multiple travelling light pulses in an optically dense medium by generalizing the approach we used in the previous section Fig 5 shows an energy level diagram of the present MC stationary light The control fields with Rabi frequencies Ω copropagate forward along with the weak quantum fields (m E m m ∈[1, ,M+]), whereas the other quantum fields (E n n ∈[1, ,M−]) propagate backward along with their control fields with Rabi frequenciesΩ The propagation directions of fields for the effective n

generation and strong interaction between the fields should satisfy the phase matching condition As shown in Fig 5, the quantum fields E and m E are nearly resonant with n

optical transitions 1 − m and 1 − n , respectively The control fields Ω and m Ω have n

frequencies close to the frequencies of transitions 2 − m and 2 − n , respectively (where

Ω = Ω , and φm n( )are the constant phases of the fields)

We assume that initially all the atoms stay in the ground state 1 and only one weak probe quantum fieldE enters into the medium with a forward propagation direction l k l and the

Multicolor Stationary Light 625

12

Fig 5 Energy level diagram for MC stationary light

atoms are driven by one classical control field Ω ≠ with same propagation direction l 0 K l The initial state of light and atoms in the interaction picture is given by (t 0)in l 1atoms

ϕ << = ϕ ( 1atoms=∏j1j and ϕl are the ground state of the atoms and initial state of the probe field, respectively) The quantum field E propagates at a slow group l

velocity due to the control field Ω , and atomic coherent wavel P12 is created between two ground states 1 and 2 After the probe pulse completely enters the medium, we switch

on the additional control fields (Ωm(t t> 1) 0≠ andΩn(t t> 1) 0≠ ) Due to the ultraslow propagation of the probe field, such switching processes can be easily realized The new atomic polarization waves with wave vectors k m n, = −k K l l+K m n, (phase matching condition) which generate the new light field E m n, are created Therefore all the quantum fields become coupled with each other via atomic coherence P12 which has the quantum information of the initial probe fieldE l Below we will study the evolution of MC stationary light in the coherent resonance medium driven by the M++M- control fields

For a theoretical analysis of MC stationary light based on the multi double lambda-type

where j' '

ll jj

P = l l is an operator of the j-th atom, g l =μl, 1 ωl, 1/(2ε0 S)is a coupling constant

of photons with atoms, μl l', is a dipole moment for the transition between the states

'l → l , v m=ωm1+ Δ , m v n=ωn1+ Δ , andn k0=ω21/c

Using Eq (12) and adding the relaxation constants and Langevin forces associated with the

atomic relaxation processes, we derive the Heisenberg equations for the atomic operators

) 0 (

; t =

F j

)'())(

)'() )2;(0) 1, ) ( 2 12 12

) 0 ( );

(

n m n m j

n m j

n

)'(2)'()2

)'(

) 0 ( );

(

n m j

n m j

n

In Eqs (13)-(15), we ignore the influences of the weak populations of excited states m and

n Thus we have the following equations for the weak field operators A m n, :

0 g c S t z n

i z t A z t

We assume a typical adiabatic condition for slow light propagation γ δm n, t0>> (1 δt0is large

enough temporal duration of the probe pulse) and ignore the fluctuation forces j2 ;(2)

equations for Ψm n, under the slow light propagation condition 2 2

,( ) , ( ) / ,

m n m n m n

v t = Ωc t Ng << : c

Multicolor Stationary Light 627

Eqs (21) and (22) can be solved by using spatial Fourier transformation Assuming that

initial field Ψl( , )t z is determined by the probe pulse field ( , )A t z , after some algebraic l

calculations we finally obtain the following solution:

4 Results and discussion

In this section, we analyze quantum evolution of the MC field Under the slow light

condition and taking into account weak relaxation processes between the two ground levels

1 and 2 , we can get the key information of quantum dynamics control from the

dispersion relation:

'' 2 0

Eg (26) shows that all the laser fields propagate and evolve together with one group

velocity ( )v t in the medium When ( ) 0v t = , that is, when all the laser fields are completely

stopped, we can obtain MC stationary light:

This is the MC stationary light condition Obviously, this relation generalizes the results

obtained for two-color stationary light The amplitudes of these coupled fields can be

manipulated by varying the Rabi frequencies of the corresponding control fields:

( , ) / ( , ) ( ) /( ( ) )

A t z A t z ≅ Ω t g Ω t g Therefore optical control of group velocity, electric

field amplitudes and direction of the MC field can be easily realized by changing the control

The maximum MC field stationary time is determined by relaxation constant γ12and spatial

spreading of the stationary pulse shape which is determined by the dispersion term ''

Multicolor Stationary Light 629

Using the conditions Im(ξm− 1−ξl− 1) 0= andIm(ξn− 1+ξl− 1) 0= , the dispersion can be minimized

From Eqs (34) and (35), we can know that the condition for minimum spreading is

independent of the total number of the control fields M+ and M- Putting Eq (31) into Eq

(27), we find the important relation for the minimum spatial spreading of the MC stationary

After optical trapping of the initial probe pulse by using MC stationary light, it is possible to

generate an arbitrary forward traveling light field A from MC stationary fields by m

controlling the amplitudes of the forward control fields (m∈[1, ,M+],Ωm(t t> 2) 0≠ , and

2

l m≠ t t

Ω > = ) , or to generate an arbitrary backward traveling light field A n by controlling

the backward control fields (n∈[1, ,M−],Ωn(t t> 2) 0≠ , andΩl n≠ (t t> 2) 0= ) Generation of

some number of copropagating quantum fieldsA or m A is also possible n

The electric field amplitude of the m-th component in the MC light at time t out on the

medium output (z=L) is given by

2 0

with the temporal duration δt m travel, =l t m( out) /v travel(t out) Eq (37) shows temporal properties

of the output MC light field We note that this MC wavelength conversion may have

potential applications in optical communications networks

5 Conclusion

We first demonstrated the two-color stationary light and quantum wavelength conversion

using quantum coherence resulting from strongly coupled slow light through EIT in a

double-lambda system Then we generalized the approach to the MC light fields in the multi

double Λ coherent atomic medium driven by the M++M- control intensive laser fields and

showed how to manipulate the MC light field within the adiabatic limit The results show

that the MC light fields can be controlled by simply adjusting the control fields’ parameters

for (1) MC stationary light, (2) selection of propagation direction (forward or backward),

and (3) MC wavelength conversion The maximum stationary time and minimum spatial

spreading of the MC field have also been discussed On-demand quantum manipulation of the MC light field can greatly increase the interaction time of the light and medium, and holds promise for applications in optical buffer, controllable switching, and quantum optical information processing

We acknowledge that this work was supported by the CRI program (Center for Photon Information Processing) of the Korean Ministry of Education, Science and Technology via National Research Foundation

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27

Fundamentals and Applications

of Quantum Limited Optical Imaging

Warwick P Bowen1, Magnus T L Hsu1 and Jian Wei Tay2

1School of Mathematics and Physics, University of Queensland, QLD 4072

2Physics Department, University of Otago, Dunedin

an a priori known set, then its unique discrimination, such as in data read-out from a CD or

DVD In general, both types of imaging involve the collection and focusing of light after interaction with the object However, the process of information extraction can be quite different In resolving an unknown structure, a full two dimensional image is usually desired Here, the metric of success is generally the resolution of the final image In most cases diffraction is the key concern, presenting the diffraction limit to the resolution of the final image as approximated by Abbe (Born & Wolf, 1999) There are ways to overcome this

limit, such as by utilising non-linearities (Hell et al., 2009), or using metamaterials (Pendry,

2000) to form so called superlenses, and this is a vibrant and growing area of research The focus of this Chapter, however, is on the second theme of imaging, discrimination between a set of known structures As we will see, this form of imaging is important, not only for read-out of information from data storage devices, but also in other areas such as

microscopy (Fabre et al., 2000; Tay et al., 2009) and satellite navigation (Arnon, 1998; Nikulin

et al., 2001) In structure discrimination, the goal is not to achieve a two dimensional image,

but rather to generate a signal which unambiguously distinguishes each element of the set Hence, the diffraction limit and other constraints on imaging resolution are no longer the primary concern, but rather the signal-to-noise ratio with which the discrimination may be performed To maximise the signal the optical measurement must be matched carefully to the set of structures to be discriminated; whereas the noise typically comes from electronic, environmental, and optical sources Much engineering effort has been applied to minimising the noise sources for important imaging systems; however, fundamentally the quantisation

of light imposes the quantum noise limit (QNL) which is outside of engineering control In this Chapter we consider a general imaging system, and show how the optical mode carrying full signal information may be determined We introduce spatial homodyne

detection (Beck, 2000; Hsu et al., 2004) as a method to optimally extract this signal, showing

how the QNL to measurement sensitivity may be determined and even surpassed using non-classical states of light We illustrate the implications of these techniques for two key

imaging systems, atomic force microscopy (Binning et al., 1985; Fabre et al., 2000) and particle tracking in optical tweezers (Block, 1992; Tay et al., 2009); comparing optimal spatial

homodyne based signal extraction to the standard extraction methods used in such systems today

2 Quantum formalism for optical measurements

The field of optical measurements has progressed significantly, with photo-detection techniques advancing from the use of the photographic plate in the 19th century to the semiconductor-based photodetectors commonly encountered today One is now able to measure with high accuracy and speed, the range of parameters that describe an optical field For example, the amplitude and phase quadratures, the Stokes polarisation parameters, and the transverse spatial profile that are commonly used to parameterise the optical field (Walls & Milburn, 1995) These parameters can be measured and quantified using a range of detection techniques such as interferometry, polarimetry and beam profiling (Saleh & Teich, 1991) However, experimentally measured values for these parameters are estimates due to the presence of classical and quantum noise, and detection inefficiencies

Fig 1 Schematics of (a) a Michelson interferometer with an inset photo of the Laser

Interferometer Gravity-wave Observatory (LIGO), (b) a polarimeter with inset photo of an on-chip polarimeter, and (c) an optical microscope with an inset photo of an optical

microscope M: mirror, BS: beam-splitter and PBS: polarising beam-splitter

Fig 1 shows examples of techniques used for the measurement of (a) amplitude and phase

quadratures (Slusher et al., 1985), (b) polarization (Korolkova & Chirkin, 1996) and (c) spatial

variables (Pawley, 1995) Fig 1 (a) shows a Michelson interferometer whereby an input field

is split using a beam-splitter, followed by propagation of the two output fields through different paths with an effective path difference These two fields are then interfered to produce an output interference signal Depending on the effective path difference, destructive or constructive interference is obtained at the output of the interferometer Variations of this technique include the Mach-Zehnder (Mach, 1892; Zehnder, 1891) and Sagnac (Sagnac, 1913) interferometers A polarimeter is shown in Fig 1 (b), where an input field is phase retarded and the different polarisation components of the input field are separated using a polarization beam-splitter A measurement of the intensity difference