The aim of this work is to study and to optimize the diffraction efficiency in sillenite BTO crystals. A theoretical analysis, where the effects of coupling beams are significant,is carried on. In addition, non-mobile transmission gratings are considered.  The diffraction efficiency is obtained by using the coupled wave equations in on-Bragg regime using Method Runge-Kutta of order 4 in Matlab. Also, the non-uniformity of the gratings in side the crystal sample, the optical activity of the material and the polarization of the recording beams are taken into account.Fil: Villamizar Amado, Astrid Lorena. Universidad de Pamplona; Colombia. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Arias Hernández, Néstor Alonso. Universidad de Pamplona; ColombiaFil: Molina Prado, Martha Lucia. Universidad de Pamplona; Colombia. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Tebaldi, Myriam Cristina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata. Centro de Investigaciones Ópticas. Provincia de Buenos Aires. Gobernación. Comisión de Investigaciones Científicas. Centro de Investigaciones Ópticas. Universidad Nacional de La Plata. Centro de Investigaciones Ópticas; Argentin

Arias Hernández, Néstor Alonso

Molina Prado, Martha Lucia

Tebaldi, Myriam Cristina

Villamizar Amado, Astrid Lorena

Optica Pura y Aplicada

English

The aim of this work is to study and to optimize the diffraction efficiency in sillenite BTO crystals. A theoretical analysis, where the effects of coupling beams are significant, is carried on. In addition, non-mobile transmission gratings are considered. The diffraction efficiency is obtained by using the coupled wave equations in on-Bragg regime using Method Runge-Kutta of order 4 in Matlab. Also, the non-uniformity of the gratings inside the crystal sample, the optical activity of the material and the polarization of the recording beams are taken into account.En este trabajo se estudia la eficiencia de difracción en cristales silenitas BTO. Se presenta un análisis teórico en donde los efectos de acoplamiento son tenidos en cuenta. Además, se consideran redes nomoviles de transmisión. La eficiencia de difracción es obtenida a través de la solución de las ecuaciones acopladas en el regimen on-Bragg usando el método de Runge-Kutta de orden 4 implementado en Matlab. La no uniformidad de las redes dentro del cristal, la actividad óptica del material y la polarización de los haces son tenidos en cuenta.Unidad de Investigación y Desarrollo OptimoCentro de Investigaciones Óptica

Molina Prado, Martha Lucía

Tebaldi, Myrian Cristina

El Servicio de Difusión de la Creación Intelectual

ÓPTICA PURA Y APLICADA. www.sedoptica.es 
 
Opt. Pura Apl. 50 (2) 181-186 (2017)  © Sociedad Española de Óptica 
 
181 
Type:	Introduction	to	Research		Section:	Quantum	and	Non-Linear	Optics	
 
Optimization	of	the	diffraction	efficiency	in	photorefractive	
crystals	
	
Optimización	de	la	eficiencia	de	difracción	en	cristales	
fotorrefractivos	
	Astrid	Lorena	Villamizar	Amado1,	Néstor	Alonso	Arias	Hernández2,	Martha	Lucía	Molina	Prado2	,	Myrian	Tebaldi3	
1.	Estudiante	de	Física,	Semillero	COUP	–	Grupo	de	Óptica	Moderna,	Departamento	de	Física	y	Geología,	
Universidad	de	Pamplona	
2.	Grupo	de	Óptica	Moderna,	Departamento	de	Física	y	Geología,	Universidad	de	Pamplona	
3.	Centro	de	Investigaciones	Ópticas	(CONICET	La	Plata,	CIC)	and	UIDET	OPTIMO,	Facultad	Ingeniería,	
Universidad	Nacional	de	La	Plata,	Argentina.	
	(*)	E-mail:	molinaprado@gmail.com			Received:	02/12/2016	 Accepted:	05/05/2017	DOI:	10.7149/OPA.50.2.49044	
ABSTRACT:	The	aim	of	this	work	is	to	study	and	to	optimize	the	diffraction	efficiency	in	sillenite	BTO	crystals.	A	theoretical	 analysis,	where	 the	 effects	 of	 coupling	 beams	 are	 significant,	 is	 carried	 on.	 In	 addition,	non-mobile	transmission	gratings	are	considered.	The	diffraction	efficiency	is	obtained	by	using	the	coupled	wave	equations	in	on-Bragg	regime	using	Method	Runge-Kutta	of	order	4	in	Matlab.	Also,	the	non-uniformity	of	the	gratings	inside	the	crystal	sample,	the	optical	activity	of	the	material	and	the	polarization	of	the	recording	beams	are	taken	into	account.	
Key	words:	Photorefractive	crystals,	diffraction	efficiency.	
RESUMEN:	En	este	trabajo	se	estudia	la	eficiencia	de	difracción	en	cristales	silenitas	BTO.	Se	presenta	un	análisis	teórico	en	donde	los	efectos	de	acoplamiento	son	tenidos	en	cuenta.	Además,	se	consideran	redes	no-moviles	 de	 transmisión.	 La	 eficiencia	 de	 difracción	 es	 obtenida	 a	 través	 de	 la	 solución	 de	 las	ecuaciones	 acopladas	 en	 el	 regimen	 on-Bragg	 usando	 el	 método	 de	 Runge-Kutta	 de	 orden	 4	implementado	en	Matlab.	La	no	uniformidad	de	 las	 redes	dentro	del	 cristal,	 la	actividad	óptica	del	material	y	la	polarización	de	los	haces	son	tenidos	en	cuenta.	
Palabras	clave:	Cristales	fotorrefractivos,	eficiencia	de	difracción.	
	
REFERENCES	AND	LINKS	/	REFERENCIAS	Y	ENLACES	[1]	 Yeh,	“Introduction	to	Photorefractive	Nonlinear	Optics,”	Wiley,	New	York,	(1993)	[2]	 L.F.	 Magaña,	 I.	 Casar,	 J.G.	 Murillo,	 “Beam	 energy	 exchange	 in	 sillenite	 crystals	 (Bi12SiO20	 and	Bi12TiO20),	 considering	 the	 variation	 of	 light	modulation	 along	 sample	 thickness	 in	 a	 strong	non-linear	regime,”	Opt.	Mater.	30,	979–986	(2008).	 	https://doi.org/10.1016/j.optmat.2007.05.026	[3]	 Marrachi,	A.,	 Johnson,	R.V,	Tanguay,	 Jr.A.R	 “Polarization	properties	of	photorefractive	diffraction	 in	electrooptic	and	optically	active	sillenite	crystals	(Bragg	Regime),	J.	Opt.	Som.	A.	B,	(1986)	
ÓPTICA PURA Y APLICADA. www.sedoptica.es 
 
Opt. Pura Apl. 50 (2) 181-186 (2017)  © Sociedad Española de Óptica 
 
182 
[4]	 Gonzales,	G.,	Zuñiga,	A.	“Optimization	of	the	diffraction	efficiency	in	nonuniform	gratings	in	sillenite	crystals	considering	the	variation	of	fringe	period,	optical	activity	and	polarization	angles	in	a	strong	non-linear	regime,”	Revista	Mexicana	de	Física,	(2009)	[5]	Molina	Prado	M.L.,	et	al.	“Medida	de	la	eficiencia	de	difracción	fotorrefractiva	como	función	del	vector	de	onda	de	la	red	en	un	cristal	BTO,”	Bistua,	9	(2),	43-47	(2011).		[6]	Molina	M.L.	 et	 al,	 “Eficiencia	 de	 difracción	del	 registro	 de	 speckle	modulados	 generados	 a	 partir	 de	superficies	reflecto-difusoras,”	Bistua,	10	(2),	2012.		[7].	 J.G.	 Murillo,	 L.F.	 Magaña,	 M.	 Carrascosa	 and	 F.	 Agulló-López,	 “Temporal	 evolution	 of	 the	 physical	response	 during	 photorefractive	 grating	 formation	 and	 erasure	 for	 BSO”.	 J.	 Appl.	 Phys.	 78.	 5686	(1995)	 	https://doi.org/10.1063/1.359626	[8]	 C.L.	Woods,	C.L.	Matson	and	M.M.	Salour,	“Crystallographic	fringe	orientation	diffraction	efficiency	in	bismuth	silicon	oxide”.	J.	Appl.	Phys.	A	40,	177-182	(1986)	 	https://doi.org/10.1007/BF00617401	[9].	 J.P.	 Herriau,	 D.	 Rojas,	 and	 J.P.	 Huignard,	 “Highly	 efficient	 diffraction	 in	 photorefractive	 BSO-BGO	crystals	at	large	applied	fields”.	Ferroelectrics	75,	271-279	(1987)		 	https://doi.org/10.1080/00150198708008227		
1.	Introduction	The	 photorefractive	 effects	 in	 electrooptical	 crystals	 have	 been	widely	 studied	 for	 applications	 to	 real-time	holography,	optical	data	storage,	and	more	recently	for	phase-conjugate	wavefront	generation	[1-6].	The	high	recording	sensitivities	combined	with	crystal	quality	and	availability	has	been	found	in	a	cubic,	paraelectric	and	electrooptic	Bi12TiO20.	The	 nonuniform	 pattern	 received	 by	 the	 crystal	 results	 in	 a	 refractive	 index	 change	 due	 to	 the	photorefractive	and	linear	electro-optic	effect.	The	optimum	diffraction	efficiency	has	been	observed	for	two	different	orientations	of	the	crystallographic	axes	with	respect	to	the	space	charge	field	Esc	[3].	The	sign	of	the	charge	carriers,	which	cause	the	space-charge	field,	can	be	determined	by	the	energy	transfer	direction	and	the	electro-optic	coefficient	measurements.	In	 this	 work	 we	 considered	 non-moving	 transmission	 gratings	 under	 an	 applied	 field,	 the	 variation	 of	fringe	 period	 and	 we	 included	 optical	 activity	 and	 birefringence	 in	 the	 Bragg-regime.	 One	 crystal	orientation	is	considered,	with	the	grating	vector	is	parallel	to	the	face	[001].	
	 Fig.1:	Longitudinal	holographic	configuration	𝐾"	‖ < 001 >	.			
2.	Set	Coupled	Wave	Equations	The	BTO	is	a	material	with	electrooptic	coupling	and	optical	activity,	the	wave	equation	that	governs	light	propagation	is:	
ÓPTICA PURA Y APLICADA. www.sedoptica.es 
 
Opt. Pura Apl. 50 (2) 181-186 (2017)  © Sociedad Española de Óptica 
 
183 
	 𝛻C𝐸 + FGHIG 𝐷 = 0;	 (1)	where	E	is	the	total	light	field,	kO = 2π λ	is	the	magnitude	of	the	optical	wave	vector,	λ	is	the	wavelength	and	ϵO	is	the	free	space	permittivity.	For	a	birefringence	medium	with	optical	activity	and	electro-optic	coupling,	this	constitutive	relation	can	be	expressed	in	the	form:		 DZ = ϵO ϵZ[ + GZ[ + ∆ϵZ[ E[;	 (2)	where	GZ[	is	the	optical	activity	antisymetric	tensor,	∆ϵZ[	is	the	change	in	the	permittivity	tensor	induced	by	the	electrooptic	effect	and	ϵZ[	is	the	symmetric	optical	permittivity	tensor.		For	the	K_	‖ < 001 >	orientation,	the	x	axis	is	parallel	to	the	<001>	crystallographic	axis,	as	show	in	Fig.	1.	 The	 transformation	 matrix	 that	 converts	 from	 the	 crystallographic	 system	 to	 the	 light	 propagation	system	is:	
	 T = 0 0 1−1 2 −1 2 01 2 −1 2 0 	 (3)	For	this	orientation,	the	permittivity	tensor,	exclusive	of	optical	activity	components,	is:	
	 ϵZ[ = ϵ 0 00 ϵ + ∆ϵ 00 0 ϵ − ∆ϵ 	 (4)	If	∆ϵ = nObrcdEe,	where	nO	 is	 the	 index	of	 refraction,	 rcd	 is	electrooptic	 coefficient	and	Ee	 is	 the	electric	field.	This	field	consists	of	two	components:		 Ee = EO + Efgcos	(K_. r + ϕ)	 (5)	EO	 is	 the	 applied	 electric	 field	 and	 Efg	 is	 the	 space	 charge	 field	 which	 depend	 of	 the	 diffusion	 and	saturation	fields:		 Efg = mEk lGHmlnHlGHm lnmlo H d C	 (6)	In	Eq.	(6)	EO	Is	the	applied	electric	field	of	eq.	(5),	Ed	is	the	diffusion	field	and	Eq	is	an	upper	limit	on	the	space	charge	field	corresponding	in	a	single	trap	model	to	saturation	of	available	acceptor	site	(assuming	monopolar	charge	transport	in	which	electrons	are	the	only	mobile	carriers).	The	electric	field	of	the	two	beams	can	be	written	as:		 E x, y, z = R(z)eZrs.t + S(z)eZrv.t	 (7)	The	polarization	states	of	the	incident	probe	beam	R	and	of	the	diffracted	signal	field	S	are	two	orthogonal	linear	polarization	states	which	components	is	given	by:		 R = yRl + (xcosθx − zsenθx)Ry	;	 (8)		 S = ySl + (xcosθz − zsenθz)𝑆y	 (9)	The	set	of	coupled	wave	equations	can	be	derived	now	that	structure	of	the	light	field	has	been	defined.	The	optical	activity	tensor	can	be	added	to	the	permittivity	tensor,	the	constitutive	relation	for	the	K_	‖ <001 >	orientation	is:		 D|Dl = ϵO ϵ ia−ia ϵ + ∆ϵ EyEl ;	 (10)	where	a	is	a	parameter	defined	by	2nOρ kO.	Using	the	slowly	varying	envelope	approximation	derives	the	coupled	wave	equations.	The	resulting	set	of	coupled	wave	equation	is:	
ÓPTICA PURA Y APLICADA. www.sedoptica.es 
 
Opt. Pura Apl. 50 (2) 181-186 (2017)  © Sociedad Española de Óptica 
 
184 
	 ~x = − gf Rl	dRldz = ρcosθ Ry + i Ccosθ Ry + i Cfgcosθ SleZ	dSydz = − ρcosθ Sl		 z = gf Sy + i gf Sl + i gf RleZ;	 (11		where	ρ	is	the	optical	rotatory	power	and:	 C = πnObλ rcdEO	Cfg = πnObλ rcdEfg 12		For	the	longitudinal	configuration	we	obtained	the	diffraction	efficiency	𝜂,	defined	by:	η = I(z)IZ(0)	where	I(z)	is	the	intensity	of	the	diffraction	light,	IZ(z)	is	the	intensity	of	the	incident	light.	For	theoretical	analysis	was	used	the	Runge-Kutta	method	and	its	algorithm	in	Matlab.	TABLE	1.	Parameter	for	BTO	[3,7-9]	taken	for	our	calculations		 BTO	𝜖	𝐷𝑖𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐	𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡	 47	𝑛		𝑟𝑒𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛	𝑖𝑛𝑑𝑒𝑥	 2.58	𝑟cd	𝑒𝑙𝑒𝑐𝑡𝑟𝑜	𝑜𝑝𝑡𝑖𝑐	𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡	(𝑚𝑉d)	𝑁¢	𝐴𝑐𝑐𝑒𝑝𝑡𝑜𝑟	𝑑𝑒𝑛𝑠𝑖𝑡𝑦	(𝑚b)	𝜌	𝑂𝑝𝑡𝑖𝑐𝑎𝑙	𝐴𝑐𝑡𝑖𝑣𝑖𝑡𝑦	º𝑐𝑚d	(𝜆 = 632𝑛𝑚)	 5.1	x	10-12	4	x	10-22	65		 	
3.	Results	and	Discussion	In	Fig.	2	we	show	the	result	 for	the	dependence	of	 the	diffraction	efficiency	on	the	sample	thickness	 for	different	 applied	 field	 for	 a	BTO	 grating	 using	 red	 light	 (632	nm)	 for	 reading.	 For	 this	 case,	 KG∥<001>;	m=0.9	and	L	=	1µm;	10	µm;	the	largest	value	of	the	diffraction	efficiency	occurs	for	an	applied	field	of	10	kV/cm.	We	can	see	that	the	smallest	of	diffraction	efficiency	occurs	for	the	smallest	value	of	the	applied	field:	1	kV/cm.			
	Fig.	2.	BTO	crystal	diffraction	efficiency	in	terms	of	the	crystal	thicknesses	using	red	read-out	light	and	different	applied	electric	fields	Ea=	1kV/cm,	5kV/cm,	7kV/cm	y	10kV/cm,	ϵ = 47,	m=0.9,	at	various	values	(a)	Λ=1µm,	(b)	Λ=10µm.	
 In	Fig.	3	we	show	the	result	for	the	dependence	of	the	diffraction	efficiency	on	the	sample	thickness	for	a	BTO	 grating	 using	 red	 light	 (632	nm)	 for	 reading	 for	 different	 grating.	 For	 this	 case,	KG∥<001>;	 Ea=	 10	
ÓPTICA PURA Y APLICADA. www.sedoptica.es 
 
Opt. Pura Apl. 50 (2) 181-186 (2017)  © Sociedad Española de Óptica 
 
185 
Kv/cm;	m	=0.9	and	m=0.3;	the	largest	value	of	the	diffraction	efficiency	occurs	for	a	grating	period	of	10	microns	 and.	We	 can	 see	 that	 for	 all	 values	 of	 the	 grating	 period	 considered,	 the	 diffraction	 efficiency	reaches	a	maximum	value	and	the	smallest	of	these	occurs	for	the	smallest	value	of	the	grating	period:	1	micron.			
	
	Fig.	3.	BTO	crystal	diffraction	efficiency	in	terms	of	the	crystal	thicknesses	using	red	read-out	light	and	different	grating	period:	Λ=1µm,	2µm,	3µm,	5µm,	10µm,	at	various	values	of	(a)	m=0.9,	Ea=0kV/cm		(b)	m=0.9,	Ea=10	kV/cm	(c)	m=0.3,	Ea=10kV/cm	
 In	Fig.	4	we	show	the	result	for	the	dependence	of	the	diffraction	efficiency	on	sample	thickness	for	a	BTO	grating	using	red-out	light	(632	nm)	for	reading	for	different	modulations.	For	this	case:	KG∥<001>;	Ea=10	Kv/cm;	L	=	1µm;	10	µm.	In	this	case,	the	largest	value	of	the	diffraction	efficiency	occurs	for	a	modulation	of	 0.9.	 For	 all	 the	 range	of	 values	of	 sample	 thickness	 considered,	 the	modulation	 corresponding	 to	0.9	maintains	the	largest	value	for	the	diffraction	efficiency.	
 
 Fig	4.	BTO	crystal	diffraction	efficiency	in	terms	of	the	crystal	thicknesses	using	red	read-out	light	and	different	modulations,	m=0.1,	0.3,	0.6,	0.9	at	various	values	of	(a)	Λ=1µm,	(b)	Λ=10µm.	
ÓPTICA PURA Y APLICADA. www.sedoptica.es 
 
Opt. Pura Apl. 50 (2) 181-186 (2017)  © Sociedad Española de Óptica 
 
186 
4.	Conclusions	We	studied	the	diffraction	efficiency	in	non-moving	transmission	gratings,	in	Bragg	regime	in	longitudinal	configuration	 in	 BTO	 crystal.	 A	 set	 of	 coupled-wave	 equations	 has	 been	 derived	 for	 longitudinal	holographic	orientation	in	crystal	BTO	of	sillenite	family.	These	equations	allow	an	analysis	to	be	made	of	the	polarization	properties	of	light	diffraction	including	the	effects	of	concomitant	optical	rotatory	power.	In	the	date	analytic	we	used	red	light	for	the	reading.	We	considered	the	birefringence,	coupling	electro-optic,	 optical	 activity,	 fringe	 period,	 different	 fields	 applied,	 different	modulations	 and	 the	 polarization	beams.		The	results	show	that	for	the	field	applied	of	1	Kv/cm	corresponded	to	the	small	diffraction	efficiency,	for	the	fringe	period	of	10	𝜇m	the	diffraction	efficiency	is	greater	and	for	a	modulation	of	0.1	the	diffraction	efficiency	is	smallest.		
Acknowledgements	This	research	was	performed	under	the	grants:PR130-00-033	(GA160-BP-II-2013-2.1.2.2.1)	y	PR130-00-001	(GA160-CM-I-2014-2.1.2.2.1)-Unipamplona,	Astrid	Villamizar	acknowledges	 financial	support	of	 the	Vicerrectoría	de	 Investigaciones	of	 the	Universidad	de	Pamplona	(Pamplona-Colombia).	Portions	of	 this	work	were	presented	at	the	IX	RIAO/OPTILAS	in	Pucón,	Chile	2016"		


Optimización de la eficiencia de difracción en cristales fotorrefractivos

LAReferencia - Red Federada de Repositorios Institucionales de Publicaciones Científicas Latinoamericanas

ÓPTICA PURA Y APLICADA. www.sedoptica.es  Opt. Pura Apl. 50 (2) 181-186 (2017)  © Sociedad Española de Óptica  181 Type:	Introduction	to	Research		Section:	Quantum	and	Non-Linear	Optics	 Optimization	of	the	diffraction	efficiency	in	photorefractive	crystals		Optimización	de	la	eficiencia	de	difracción	en	cristales	fotorrefractivos		Astrid	Lorena	Villamizar	Amado1,	Néstor	Alonso	Arias	Hernández2,	Martha	Lucía	Molina	Prado2	,	Myrian	Tebaldi3	1.	Estudiante	de	Física,	Semillero	COUP	–	Grupo	de	Óptica	Moderna,	Departamento	de	Física	y	Geología,	Universidad	de	Pamplona	2.	Grupo	de	Óptica	Moderna,	Departamento	de	Física	y	Geología,	Universidad	de	Pamplona	3.	Centro	de	Investigaciones	Ópticas	(CONICET	La	Plata,	CIC)	and	UIDET	OPTIMO,	Facultad	Ingeniería,	Universidad	Nacional	de	La	Plata,	Argentina.		(*)	E-mail:	molinaprado@gmail.com			Received:	02/12/2016	 Accepted:	05/05/2017	DOI:	10.7149/OPA.50.2.49044	ABSTRACT:	The	aim	of	this	work	is	to	study	and	to	optimize	the	diffraction	efficiency	in	sillenite	BTO	crystals.	A	theoretical	 analysis,	where	 the	 effects	 of	 coupling	 beams	 are	 significant,	 is	 carried	 on.	 In	 addition,	non-mobile	transmission	gratings	are	considered.	The	diffraction	efficiency	is	obtained	by	using	the	coupled	wave	equations	in	on-Bragg	regime	using	Method	Runge-Kutta	of	order	4	in	Matlab.	Also,	the	non-uniformity	of	the	gratings	inside	the	crystal	sample,	the	optical	activity	of	the	material	and	the	polarization	of	the	recording	beams	are	taken	into	account.	Key	words:	Photorefractive	crystals,	diffraction	efficiency.	RESUMEN:	En	este	trabajo	se	estudia	la	eficiencia	de	difracción	en	cristales	silenitas	BTO.	Se	presenta	un	análisis	teórico	en	donde	los	efectos	de	acoplamiento	son	tenidos	en	cuenta.	Además,	se	consideran	redes	no-moviles	 de	 transmisión.	 La	 eficiencia	 de	 difracción	 es	 obtenida	 a	 través	 de	 la	 solución	 de	 las	ecuaciones	 acopladas	 en	 el	 regimen	 on-Bragg	 usando	 el	 método	 de	 Runge-Kutta	 de	 orden	 4	implementado	en	Matlab.	La	no	uniformidad	de	 las	 redes	dentro	del	 cristal,	 la	actividad	óptica	del	material	y	la	polarización	de	los	haces	son	tenidos	en	cuenta.	Palabras	clave:	Cristales	fotorrefractivos,	eficiencia	de	difracción.		REFERENCES	AND	LINKS	/	REFERENCIAS	Y	ENLACES	[1]	 Yeh,	“Introduction	to	Photorefractive	Nonlinear	Optics,”	Wiley,	New	York,	(1993)	[2]	 L.F.	 Magaña,	 I.	 Casar,	 J.G.	 Murillo,	 “Beam	 energy	 exchange	 in	 sillenite	 crystals	 (Bi12SiO20	 and	Bi12TiO20),	 considering	 the	 variation	 of	 light	modulation	 along	 sample	 thickness	 in	 a	 strong	non-linear	regime,”	Opt.	Mater.	30,	979–986	(2008).	 	https://doi.org/10.1016/j.optmat.2007.05.026	[3]	 Marrachi,	A.,	 Johnson,	R.V,	Tanguay,	 Jr.A.R	 “Polarization	properties	of	photorefractive	diffraction	 in	electrooptic	and	optically	active	sillenite	crystals	(Bragg	Regime),	J.	Opt.	Som.	A.	B,	(1986)	[4]	 Gonzales,	G.,	Zuñiga,	A.	“Optimization	of	the	diffraction	efficiency	in	nonuniform	gratings	in	sillenite	crystals	considering	the	variation	of	fringe	period,	optical	activity	and	polarization	angles	in	a	strong	non-linear	regime,”	Revista	Mexicana	de	Física,	(2009)	ÓPTICA PURA Y APLICADA. www.sedoptica.es  Opt. Pura Apl. 50 (2) 181-186 (2017)  © Sociedad Española de Óptica  182 [5]	Molina	Prado	M.L.,	et	al.	“Medida	de	la	eficiencia	de	difracción	fotorrefractiva	como	función	del	vector	de	onda	de	la	red	en	un	cristal	BTO,”	Bistua,	9	(2),	43-47	(2011).		[6]	Molina	M.L.	 et	 al,	 “Eficiencia	 de	 difracción	del	 registro	 de	 speckle	modulados	 generados	 a	 partir	 de	superficies	reflecto-difusoras,”	Bistua,	10	(2),	2012.		[7].	 J.G.	 Murillo,	 L.F.	 Magaña,	 M.	 Carrascosa	 and	 F.	 Agulló-López,	 “Temporal	 evolution	 of	 the	 physical	response	 during	 photorefractive	 grating	 formation	 and	 erasure	 for	 BSO”.	 J.	 Appl.	 Phys.	 78.	 5686	(1995)	 	https://doi.org/10.1063/1.359626	[8]	 C.L.	Woods,	C.L.	Matson	and	M.M.	Salour,	“Crystallographic	fringe	orientation	diffraction	efficiency	in	bismuth	silicon	oxide”.	J.	Appl.	Phys.	A	40,	177-182	(1986)	 	https://doi.org/10.1007/BF00617401	[9].	 J.P.	 Herriau,	 D.	 Rojas,	 and	 J.P.	 Huignard,	 “Highly	 efficient	 diffraction	 in	 photorefractive	 BSO-BGO	crystals	at	large	applied	fields”.	Ferroelectrics	75,	271-279	(1987)		 	https://doi.org/10.1080/00150198708008227		1.	Introduction	The	 photorefractive	 effects	 in	 electrooptical	 crystals	 have	 been	widely	 studied	 for	 applications	 to	 real-time	holography,	optical	data	storage,	and	more	recently	for	phase-conjugate	wavefront	generation	[1-6].	The	high	recording	sensitivities	combined	with	crystal	quality	and	availability	has	been	found	in	a	cubic,	paraelectric	and	electrooptic	Bi12TiO20.	The	 nonuniform	 pattern	 received	 by	 the	 crystal	 results	 in	 a	 refractive	 index	 change	 due	 to	 the	photorefractive	and	linear	electro-optic	effect.	The	optimum	diffraction	efficiency	has	been	observed	for	two	different	orientations	of	the	crystallographic	axes	with	respect	to	the	space	charge	field	Esc	[3].	The	sign	of	the	charge	carriers,	which	cause	the	space-charge	field,	can	be	determined	by	the	energy	transfer	direction	and	the	electro-optic	coefficient	measurements.	In	 this	 work	 we	 considered	 non-moving	 transmission	 gratings	 under	 an	 applied	 field,	 the	 variation	 of	fringe	 period	 and	 we	 included	 optical	 activity	 and	 birefringence	 in	 the	 Bragg-regime.	 One	 crystal	orientation	is	considered,	with	the	grating	vector	is	parallel	to	the	face	[001].										configuration	𝐾"	‖ < 001 >	.		Fig.1:	Longitudinal	holographic		2.	 Set	 Coupled	Wave	 Equations	The	BTO	is	a	material	with	electrooptic	coupling	and	optical	activity,	the	wave	equation	that	governs	light	propagation	is:		 𝛻C𝐸 + FGHIG𝐷 = 0;	 (1)	where	E	is	the	total	light	field,	kO = 2π λ	is	the	magnitude	of	the	optical	wave	vector,	λ	is	the	wavelength	and	ϵO	is	the	free	space	permittivity.	ÓPTICA PURA Y APLICADA. www.sedoptica.es  Opt. Pura Apl. 50 (2) 181-186 (2017)  © Sociedad Española de Óptica  183 For	a	birefringence	medium	with	optical	activity	and	electro-optic	coupling,	this	constitutive	relation	can	be	expressed	in	the	form:		 DZ = ϵO ϵZ[ + GZ[ + ∆ϵZ[ E[;	 (2)	where	GZ[	is	the	optical	activity	antisymetric	tensor,	∆ϵZ[	is	the	change	in	the	permittivity	tensor	induced	by	the	electrooptic	effect	and	ϵZ[	is	the	symmetric	optical	permittivity	tensor.		For	the	K_	‖ < 001 >	orientation,	the	x	axis	is	parallel	to	the	<001>	crystallographic	axis,	as	show	in	Fig.	1.	 The	 transformation	 matrix	 that	 converts	 from	 the	 crystallographic	 system	 to	 the	 light	 propagation	system	is:		 T =0 0 1−12−12012−120	 (3)	For	this	orientation,	the	permittivity	tensor,	exclusive	of	optical	activity	components,	is:		 ϵZ[ =ϵ 0 00 ϵ + ∆ϵ 00 0 ϵ − ∆ϵ	 (4)	If	∆ϵ = nObrcdEe,	where	nO	 is	 the	 index	of	 refraction,	 rcd	 is	electrooptic	 coefficient	and	Ee	 is	 the	electric	field.	This	field	consists	of	two	components:		 Ee = EO + Efgcos	(K_. r + ϕ)	 (5)	EO	 is	 the	 applied	 electric	 field	 and	 Efg	 is	 the	 space	 charge	 field	 which	 depend	 of	 the	 diffusion	 and	saturation	fields:		 Efg = mEklGHmlnHlGHm lnmloHdC	 (6)	In	Eq.	(6)	EO	Is	the	applied	electric	field	of	eq.	(5),	Ed	is	the	diffusion	field	and	Eq	is	an	upper	limit	on	the	space	charge	field	corresponding	in	a	single	trap	model	to	saturation	of	available	acceptor	site	(assuming	monopolar	charge	transport	in	which	electrons	are	the	only	mobile	carriers).	The	electric	field	of	the	two	beams	can	be	written	as:		 E x, y, z = R(z)eZrs.t + S(z)eZrv.t	 (7)	The	polarization	states	of	the	incident	probe	beam	R	and	of	the	diffracted	signal	field	S	are	two	orthogonal	linear	polarization	states	which	components	is	given	by:		 R = yRl + (xcosθx − zsenθx)Ry	;	 (8)		 S = ySl + (xcosθz − zsenθz)𝑆y	 (9)	The	set	of	coupled	wave	equations	can	be	derived	now	that	structure	of	the	light	field	has	been	defined.	The	optical	activity	tensor	can	be	added	to	the	permittivity	tensor,	the	constitutive	relation	for	the	K_	‖ <001 >	orientation	is:		 D|Dl= ϵOϵ ia−ia ϵ + ∆ϵEyEl;	 (10)	where	a	is	a	parameter	defined	by	2nOρ kO.	Using	the	slowly	varying	envelope	approximation	derives	the	coupled	wave	equations.	The	resulting	set	of	coupled	wave	equation	is:		 ~x= − gfRl	dRldz=ρcosθRy + iCcosθRy + iCfgcosθSleZ	ÓPTICA PURA Y APLICADA. www.sedoptica.es  Opt. Pura Apl. 50 (2) 181-186 (2017)  © Sociedad Española de Óptica  184 dSydz= −ρcosθSl		 z= gfSy + igfSl + igfRleZ;	 (11)	where	ρ	is	the	optical	rotatory	power	and:	C =GrcdEO		Cfg =πnObλrcdEfg12	For	the	longitudinal	configuration	we	obtained	the	diffraction	efficiency	𝜂,	defined	by:	η =I(z)IZ(0)	where	I(z)	is	the	intensity	of	the	diffraction	light,	IZ(z)	is	the	intensity	of	the	incident	light.	For	theoretical	analysis	was	used	the	Runge-Kutta	method	and	its	algorithm	in	Matlab.	TABLE	1.	Parameter	for	BTO	[3,7-9]	taken	for	our	calculations		 BTO	𝜖	𝐷𝑖𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐	𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡	 47	𝑛		𝑟𝑒𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛	𝑖𝑛𝑑𝑒𝑥	 2.58	𝑟cd	𝑒𝑙𝑒𝑐𝑡𝑟𝑜	𝑜𝑝𝑡𝑖𝑐	𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡	(𝑚𝑉d)	𝑁¦	𝐴𝑐𝑐𝑒𝑝𝑡𝑜𝑟	𝑑𝑒𝑛𝑠𝑖𝑡𝑦	(𝑚b)	𝜌	𝑂𝑝𝑡𝑖𝑐𝑎𝑙	𝐴𝑐𝑡𝑖𝑣𝑖𝑡𝑦	º𝑐𝑚d	(𝜆 = 632𝑛𝑚)	5.1	x	10-12	4	x	10-22	65		 	3.	Results	and	Discussion	In	Fig.	2	we	show	the	result	 for	the	dependence	of	 the	diffraction	efficiency	on	the	sample	thickness	for	different	 applied	 field	 for	 a	BTO	 grating	 using	 red	 light	 (632	nm)	 for	 reading.	 For	 this	 case,	 KG∥<001>;	m=0.9	and	L	=	1µm;	10	µm;	the	largest	value	of	the	diffraction	efficiency	occurs	for	an	applied	field	of	10	kV/cm.	We	can	see	that	the	smallest	of	diffraction	efficiency	occurs	for	the	smallest	value	of	the	applied	field:	1	kV/cm.				Fig.	2.	BTO	crystal	diffraction	efficiency	in	terms	of	the	crystal	thicknesses	using	red	read-out	light	and	different	applied	electric	fields	Ea=	1kV/cm,	5kV/cm,	7kV/cm	y	10kV/cm,	ϵ = 47,	m=0.9,	at	various	values	(a)	Λ=1µm,	(b)	Λ=10µm.	 In	Fig.	3	we	show	the	result	for	the	dependence	of	the	diffraction	efficiency	on	the	sample	thickness	for	a	BTO	 grating	 using	 red	 light	 (632	nm)	 for	 reading	 for	 different	 grating.	 For	 this	 case,	KG∥<001>;	 Ea=	 10	Kv/cm;	m	=0.9	and	m=0.3;	the	largest	value	of	the	diffraction	efficiency	occurs	for	a	grating	period	of	10	microns	 and.	We	 can	 see	 that	 for	 all	 values	 of	 the	 grating	 period	 considered,	 the	 diffraction	 efficiency	reaches	a	maximum	value	and	the	smallest	of	these	occurs	for	the	smallest	value	of	the	grating	period:	1	micron.			ÓPTICA PURA Y APLICADA. www.sedoptica.es  Opt. Pura Apl. 50 (2) 181-186 (2017)  © Sociedad Española de Óptica  185 		Fig.	3.	BTO	crystal	diffraction	efficiency	in	terms	of	the	crystal	thicknesses	using	red	read-out	light	and	different	grating	period:	Λ=1µm,	2µm,	3µm,	5µm,	10µm,	at	various	values	of	(a)	m=0.9,	Ea=0kV/cm		(b)	m=0.9,	Ea=10	kV/cm	(c)	m=0.3,	Ea=10kV/cm	 In	Fig.	4	we	show	the	result	for	the	dependence	of	the	diffraction	efficiency	on	sample	thickness	for	a	BTO	grating	using	red-out	light	(632	nm)	for	reading	for	different	modulations.	For	this	case:	KG∥<001>;	Ea=10	Kv/cm;	L	=	1µm;	10	µm.	In	this	case,	the	largest	value	of	the	diffraction	efficiency	occurs	for	a	modulation	of	 0.9.	 For	 all	 the	 range	of	 values	of	 sample	 thickness	 considered,	 the	modulation	 corresponding	 to	0.9	maintains	the	largest	value	for	the	diffraction	efficiency.	  Fig	4.	BTO	crystal	diffraction	efficiency	in	terms	of	the	crystal	thicknesses	using	red	read-out	light	and	different	modulations,	m=0.1,	0.3,	0.6,	0.9	at	various	values	of	(a)	Λ=1µm,	(b)	Λ=10µm.	 4.	Conclusions	We	studied	the	diffraction	efficiency	in	non-moving	transmission	gratings,	in	Bragg	regime	in	longitudinal	configuration	 in	 BTO	 crystal.	 A	 set	 of	 coupled-wave	 equations	 has	 been	 derived	 for	 longitudinal	ÓPTICA PURA Y APLICADA. www.sedoptica.es  Opt. Pura Apl. 50 (2) 181-186 (2017)  © Sociedad Española de Óptica  186 holographic	orientation	in	crystal	BTO	of	sillenite	family.	These	equations	allow	an	analysis	to	be	made	of	the	polarization	properties	of	light	diffraction	including	the	effects	of	concomitant	optical	rotatory	power.	In	the	date	analytic	we	used	red	light	for	the	reading.	We	considered	the	birefringence,	coupling	electro-optic,	 optical	 activity,	 fringe	 period,	 different	 fields	 applied,	 different	modulations	 and	 the	 polarization	beams.		The	results	show	that	for	the	field	applied	of	1	Kv/cm	corresponded	to	the	small	diffraction	efficiency,	for	the	fringe	period	of	10	𝜇m	the	diffraction	efficiency	is	greater	and	for	a	modulation	of	0.1	the	diffraction	efficiency	is	smallest.		Acknowledgements	This	research	was	performed	under	the	grants:PR130-00-033	(GA160-BP-II-2013-2.1.2.2.1)	y	PR130-00-001	(GA160-CM-I-2014-2.1.2.2.1)-Unipamplona,	Astrid	Villamizar	acknowledges	 financial	support	of	 the	Vicerrectoría	de	 Investigaciones	of	 the	Universidad	de	Pamplona	(Pamplona-Colombia).	Portions	of	 this	work	were	presented	at	the	IX	RIAO/OPTILAS	in	Pucón,	Chile	2016"		

Optimization of the diffraction efficiency in photorefractive crystals

CONICET Digital

ÓPTICA PURA Y APLICADA. www.sedoptica.es  Opt. Pura Apl. 50 (2) 181-186 (2017)  © Sociedad Española de Óptica  181 Type:	Introduction	to	Research		Section:	Quantum	and	Non-Linear	Optics	 Optimization	of	the	diffraction	efficiency	in	photorefractive	crystals		Optimización	de	la	eficiencia	de	difracción	en	cristales	fotorrefractivos		Astrid	Lorena	Villamizar	Amado1,	Néstor	Alonso	Arias	Hernández2,	Martha	Lucía	Molina	Prado2	,	Myrian	Tebaldi3	1.	Estudiante	de	Física,	Semillero	COUP	–	Grupo	de	Óptica	Moderna,	Departamento	de	Física	y	Geología,	Universidad	de	Pamplona	2.	Grupo	de	Óptica	Moderna,	Departamento	de	Física	y	Geología,	Universidad	de	Pamplona	3.	Centro	de	Investigaciones	Ópticas	(CONICET	La	Plata,	CIC)	and	UIDET	OPTIMO,	Facultad	Ingeniería,	Universidad	Nacional	de	La	Plata,	Argentina.		(*)	E-mail:	molinaprado@gmail.com			Received:	02/12/2016	 Accepted:	05/05/2017	DOI:	10.7149/OPA.50.2.49044	ABSTRACT:	The	aim	of	this	work	is	to	study	and	to	optimize	the	diffraction	efficiency	in	sillenite	BTO	crystals.	A	theoretical	 analysis,	where	 the	 effects	 of	 coupling	 beams	 are	 significant,	 is	 carried	 on.	 In	 addition,	non-mobile	transmission	gratings	are	considered.	The	diffraction	efficiency	is	obtained	by	using	the	coupled	wave	equations	in	on-Bragg	regime	using	Method	Runge-Kutta	of	order	4	in	Matlab.	Also,	the	non-uniformity	of	the	gratings	inside	the	crystal	sample,	the	optical	activity	of	the	material	and	the	polarization	of	the	recording	beams	are	taken	into	account.	Key	words:	Photorefractive	crystals,	diffraction	efficiency.	RESUMEN:	En	este	trabajo	se	estudia	la	eficiencia	de	difracción	en	cristales	silenitas	BTO.	Se	presenta	un	análisis	teórico	en	donde	los	efectos	de	acoplamiento	son	tenidos	en	cuenta.	Además,	se	consideran	redes	no-moviles	 de	 transmisión.	 La	 eficiencia	 de	 difracción	 es	 obtenida	 a	 través	 de	 la	 solución	 de	 las	ecuaciones	 acopladas	 en	 el	 regimen	 on-Bragg	 usando	 el	 método	 de	 Runge-Kutta	 de	 orden	 4	implementado	en	Matlab.	La	no	uniformidad	de	 las	 redes	dentro	del	 cristal,	 la	actividad	óptica	del	material	y	la	polarización	de	los	haces	son	tenidos	en	cuenta.	Palabras	clave:	Cristales	fotorrefractivos,	eficiencia	de	difracción.		REFERENCES	AND	LINKS	/	REFERENCIAS	Y	ENLACES	[1]	 Yeh,	“Introduction	to	Photorefractive	Nonlinear	Optics,”	Wiley,	New	York,	(1993)	[2]	 L.F.	 Magaña,	 I.	 Casar,	 J.G.	 Murillo,	 “Beam	 energy	 exchange	 in	 sillenite	 crystals	 (Bi12SiO20	 and	Bi12TiO20),	 considering	 the	 variation	 of	 light	modulation	 along	 sample	 thickness	 in	 a	 strong	non-linear	regime,”	Opt.	Mater.	30,	979–986	(2008).	 	https://doi.org/10.1016/j.optmat.2007.05.026	[3]	 Marrachi,	A.,	 Johnson,	R.V,	Tanguay,	 Jr.A.R	 “Polarization	properties	of	photorefractive	diffraction	 in	electrooptic	and	optically	active	sillenite	crystals	(Bragg	Regime),	J.	Opt.	Som.	A.	B,	(1986)	[4]	 Gonzales,	G.,	Zuñiga,	A.	“Optimization	of	the	diffraction	efficiency	in	nonuniform	gratings	in	sillenite	crystals	considering	the	variation	of	fringe	period,	optical	activity	and	polarization	angles	in	a	strong	non-linear	regime,”	Revista	Mexicana	de	Física,	(2009)	ÓPTICA PURA Y APLICADA. www.sedoptica.es  Opt. Pura Apl. 50 (2) 181-186 (2017)  © Sociedad Española de Óptica  182 [5]	Molina	Prado	M.L.,	et	al.	“Medida	de	la	eficiencia	de	difracción	fotorrefractiva	como	función	del	vector	de	onda	de	la	red	en	un	cristal	BTO,”	Bistua,	9	(2),	43-47	(2011).		[6]	Molina	M.L.	 et	 al,	 “Eficiencia	 de	 difracción	del	 registro	 de	 speckle	modulados	 generados	 a	 partir	 de	superficies	reflecto-difusoras,”	Bistua,	10	(2),	2012.		[7].	 J.G.	 Murillo,	 L.F.	 Magaña,	 M.	 Carrascosa	 and	 F.	 Agulló-López,	 “Temporal	 evolution	 of	 the	 physical	response	 during	 photorefractive	 grating	 formation	 and	 erasure	 for	 BSO”.	 J.	 Appl.	 Phys.	 78.	 5686	(1995)	 	https://doi.org/10.1063/1.359626	[8]	 C.L.	Woods,	C.L.	Matson	and	M.M.	Salour,	“Crystallographic	fringe	orientation	diffraction	efficiency	in	bismuth	silicon	oxide”.	J.	Appl.	Phys.	A	40,	177-182	(1986)	 	https://doi.org/10.1007/BF00617401	[9].	 J.P.	 Herriau,	 D.	 Rojas,	 and	 J.P.	 Huignard,	 “Highly	 efficient	 diffraction	 in	 photorefractive	 BSO-BGO	crystals	at	large	applied	fields”.	Ferroelectrics	75,	271-279	(1987)		 	https://doi.org/10.1080/00150198708008227		1.	Introduction	The	 photorefractive	 effects	 in	 electrooptical	 crystals	 have	 been	widely	 studied	 for	 applications	 to	 real-time	holography,	optical	data	storage,	and	more	recently	for	phase-conjugate	wavefront	generation	[1-6].	The	high	recording	sensitivities	combined	with	crystal	quality	and	availability	has	been	found	in	a	cubic,	paraelectric	and	electrooptic	Bi12TiO20.	The	 nonuniform	 pattern	 received	 by	 the	 crystal	 results	 in	 a	 refractive	 index	 change	 due	 to	 the	photorefractive	and	linear	electro-optic	effect.	The	optimum	diffraction	efficiency	has	been	observed	for	two	different	orientations	of	the	crystallographic	axes	with	respect	to	the	space	charge	field	Esc	[3].	The	sign	of	the	charge	carriers,	which	cause	the	space-charge	field,	can	be	determined	by	the	energy	transfer	direction	and	the	electro-optic	coefficient	measurements.	In	 this	 work	 we	 considered	 non-moving	 transmission	 gratings	 under	 an	 applied	 field,	 the	 variation	 of	fringe	 period	 and	 we	 included	 optical	 activity	 and	 birefringence	 in	 the	 Bragg-regime.	 One	 crystal	orientation	is	considered,	with	the	grating	vector	is	parallel	to	the	face	[001].										 configuration	𝐾"	‖ < 001 >	.		Fig.1:	Longitudinal	holographic		2.	 Set	 Coupled	Wave	 Equations	The	BTO	is	a	material	with	electrooptic	coupling	and	optical	activity,	the	wave	equation	that	governs	light	propagation	is:		 𝛻C𝐸 + FGHIG 𝐷 = 0;	 (1)	where	E	is	the	total	light	field,	kO = 2π λ	is	the	magnitude	of	the	optical	wave	vector,	λ	is	the	wavelength	and	ϵO	is	the	free	space	permittivity.	ÓPTICA PURA Y APLICADA. www.sedoptica.es  Opt. Pura Apl. 50 (2) 181-186 (2017)  © Sociedad Española de Óptica  183 For	a	birefringence	medium	with	optical	activity	and	electro-optic	coupling,	this	constitutive	relation	can	be	expressed	in	the	form:		 DZ = ϵO ϵZ[ + GZ[ + ∆ϵZ[ E[;	 (2)	where	GZ[	is	the	optical	activity	antisymetric	tensor,	∆ϵZ[	is	the	change	in	the	permittivity	tensor	induced	by	the	electrooptic	effect	and	ϵZ[	is	the	symmetric	optical	permittivity	tensor.		For	the	K_	‖ < 001 >	orientation,	the	x	axis	is	parallel	to	the	<001>	crystallographic	axis,	as	show	in	Fig.	1.	 The	 transformation	 matrix	 that	 converts	 from	 the	 crystallographic	 system	 to	 the	 light	 propagation	system	is:		 T = 0 0 1−1 2 −1 2 01 2 −1 2 0 	 (3)	For	this	orientation,	the	permittivity	tensor,	exclusive	of	optical	activity	components,	is:		 ϵZ[ = ϵ 0 00 ϵ + ∆ϵ 00 0 ϵ − ∆ϵ 	 (4)	If	∆ϵ = nObrcdEe,	where	nO	 is	 the	 index	of	 refraction,	 rcd	 is	electrooptic	 coefficient	and	Ee	 is	 the	electric	field.	This	field	consists	of	two	components:		 Ee = EO + Efgcos	(K_. r + ϕ)	 (5)	EO	 is	 the	 applied	 electric	 field	 and	 Efg	 is	 the	 space	 charge	 field	 which	 depend	 of	 the	 diffusion	 and	saturation	fields:		 Efg = mEk lGHmlnHlGHm lnmlo H d C	 (6)	In	Eq.	(6)	EO	Is	the	applied	electric	field	of	eq.	(5),	Ed	is	the	diffusion	field	and	Eq	is	an	upper	limit	on	the	space	charge	field	corresponding	in	a	single	trap	model	to	saturation	of	available	acceptor	site	(assuming	monopolar	charge	transport	in	which	electrons	are	the	only	mobile	carriers).	The	electric	field	of	the	two	beams	can	be	written	as:		 E x, y, z = R(z)eZrs.t + S(z)eZrv.t	 (7)	The	polarization	states	of	the	incident	probe	beam	R	and	of	the	diffracted	signal	field	S	are	two	orthogonal	linear	polarization	states	which	components	is	given	by:		 R = yRl + (xcosθx − zsenθx)Ry	;	 (8)		 S = ySl + (xcosθz − zsenθz)𝑆y	 (9)	The	set	of	coupled	wave	equations	can	be	derived	now	that	structure	of	the	light	field	has	been	defined.	The	optical	activity	tensor	can	be	added	to	the	permittivity	tensor,	the	constitutive	relation	for	the	K_	‖ <001 >	orientation	is:		 D|Dl = ϵO ϵ ia−ia ϵ + ∆ϵ EyEl ;	 (10)	where	a	is	a	parameter	defined	by	2nOρ kO.	Using	the	slowly	varying	envelope	approximation	derives	the	coupled	wave	equations.	The	resulting	set	of	coupled	wave	equation	is:		 ~x = − gf Rl	dRldz = ρcosθ Ry + i Ccosθ Ry + i Cfgcosθ SleZ	ÓPTICA PURA Y APLICADA. www.sedoptica.es  Opt. Pura Apl. 50 (2) 181-186 (2017)  © Sociedad Española de Óptica  184 dSydz = − ρcosθ Sl		 z = gf Sy + i gf Sl + i gf RleZ;	 (11)	where	ρ	is	the	optical	rotatory	power	and:	C = G rcdEO		Cfg = πnObλ rcdEfg 12	For	the	longitudinal	configuration	we	obtained	the	diffraction	efficiency	𝜂,	defined	by:	η = I(z)IZ(0)	where	I(z)	is	the	intensity	of	the	diffraction	light,	IZ(z)	is	the	intensity	of	the	incident	light.	For	theoretical	analysis	was	used	the	Runge-Kutta	method	and	its	algorithm	in	Matlab.	TABLE	1.	Parameter	for	BTO	[3,7-9]	taken	for	our	calculations		 BTO	𝜖	𝐷𝑖𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐	𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡	 47	𝑛		𝑟𝑒𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛	𝑖𝑛𝑑𝑒𝑥	 2.58	𝑟cd	𝑒𝑙𝑒𝑐𝑡𝑟𝑜	𝑜𝑝𝑡𝑖𝑐	𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡	(𝑚𝑉d)	𝑁¦	𝐴𝑐𝑐𝑒𝑝𝑡𝑜𝑟	𝑑𝑒𝑛𝑠𝑖𝑡𝑦	(𝑚b)	𝜌	𝑂𝑝𝑡𝑖𝑐𝑎𝑙	𝐴𝑐𝑡𝑖𝑣𝑖𝑡𝑦	º𝑐𝑚d	(𝜆 = 632𝑛𝑚)	 5.1	x	10-12	4	x	10-22	65		 	3.	Results	and	Discussion	In	Fig.	2	we	show	the	result	 for	the	dependence	of	 the	diffraction	efficiency	on	the	sample	thickness	 for	different	 applied	 field	 for	 a	BTO	 grating	 using	 red	 light	 (632	nm)	 for	 reading.	 For	 this	 case,	 KG∥<001>;	m=0.9	and	L	=	1µm;	10	µm;	the	largest	value	of	the	diffraction	efficiency	occurs	for	an	applied	field	of	10	kV/cm.	We	can	see	that	the	smallest	of	diffraction	efficiency	occurs	for	the	smallest	value	of	the	applied	field:	1	kV/cm.				Fig.	2.	BTO	crystal	diffraction	efficiency	in	terms	of	the	crystal	thicknesses	using	red	read-out	light	and	different	applied	electric	fields	Ea=	1kV/cm,	5kV/cm,	7kV/cm	y	10kV/cm,	ϵ = 47,	m=0.9,	at	various	values	(a)	Λ=1µm,	(b)	Λ=10µm.	 In	Fig.	3	we	show	the	result	for	the	dependence	of	the	diffraction	efficiency	on	the	sample	thickness	for	a	BTO	 grating	 using	 red	 light	 (632	nm)	 for	 reading	 for	 different	 grating.	 For	 this	 case,	KG∥<001>;	 Ea=	 10	Kv/cm;	m	=0.9	and	m=0.3;	the	largest	value	of	the	diffraction	efficiency	occurs	for	a	grating	period	of	10	microns	 and.	We	 can	 see	 that	 for	 all	 values	 of	 the	 grating	 period	 considered,	 the	 diffraction	 efficiency	reaches	a	maximum	value	and	the	smallest	of	these	occurs	for	the	smallest	value	of	the	grating	period:	1	micron.			ÓPTICA PURA Y APLICADA. www.sedoptica.es  Opt. Pura Apl. 50 (2) 181-186 (2017)  © Sociedad Española de Óptica  185 		Fig.	3.	BTO	crystal	diffraction	efficiency	in	terms	of	the	crystal	thicknesses	using	red	read-out	light	and	different	grating	period:	Λ=1µm,	2µm,	3µm,	5µm,	10µm,	at	various	values	of	(a)	m=0.9,	Ea=0kV/cm		(b)	m=0.9,	Ea=10	kV/cm	(c)	m=0.3,	Ea=10kV/cm	 In	Fig.	4	we	show	the	result	for	the	dependence	of	the	diffraction	efficiency	on	sample	thickness	for	a	BTO	grating	using	red-out	light	(632	nm)	for	reading	for	different	modulations.	For	this	case:	KG∥<001>;	Ea=10	Kv/cm;	L	=	1µm;	10	µm.	In	this	case,	the	largest	value	of	the	diffraction	efficiency	occurs	for	a	modulation	of	 0.9.	 For	 all	 the	 range	of	 values	of	 sample	 thickness	 considered,	 the	modulation	 corresponding	 to	0.9	maintains	the	largest	value	for	the	diffraction	efficiency.	  Fig	4.	BTO	crystal	diffraction	efficiency	in	terms	of	the	crystal	thicknesses	using	red	read-out	light	and	different	modulations,	m=0.1,	0.3,	0.6,	0.9	at	various	values	of	(a)	Λ=1µm,	(b)	Λ=10µm.	 4.	Conclusions	We	studied	the	diffraction	efficiency	in	non-moving	transmission	gratings,	in	Bragg	regime	in	longitudinal	configuration	 in	 BTO	 crystal.	 A	 set	 of	 coupled-wave	 equations	 has	 been	 derived	 for	 longitudinal	ÓPTICA PURA Y APLICADA. www.sedoptica.es  Opt. Pura Apl. 50 (2) 181-186 (2017)  © Sociedad Española de Óptica  186 holographic	orientation	in	crystal	BTO	of	sillenite	family.	These	equations	allow	an	analysis	to	be	made	of	the	polarization	properties	of	light	diffraction	including	the	effects	of	concomitant	optical	rotatory	power.	In	the	date	analytic	we	used	red	light	for	the	reading.	We	considered	the	birefringence,	coupling	electro-optic,	 optical	 activity,	 fringe	 period,	 different	 fields	 applied,	 different	modulations	 and	 the	 polarization	beams.		The	results	show	that	for	the	field	applied	of	1	Kv/cm	corresponded	to	the	small	diffraction	efficiency,	for	the	fringe	period	of	10	𝜇m	the	diffraction	efficiency	is	greater	and	for	a	modulation	of	0.1	the	diffraction	efficiency	is	smallest.		Acknowledgements	This	research	was	performed	under	the	grants:PR130-00-033	(GA160-BP-II-2013-2.1.2.2.1)	y	PR130-00-001	(GA160-CM-I-2014-2.1.2.2.1)-Unipamplona,	Astrid	Villamizar	acknowledges	 financial	support	of	 the	Vicerrectoría	de	 Investigaciones	of	 the	Universidad	de	Pamplona	(Pamplona-Colombia).	Portions	of	 this	work	were	presented	at	the	IX	RIAO/OPTILAS	in	Pucón,	Chile	2016"		

A. L. Villamizar Amado

N. A. Arias Hernández

M. L. Molina Prado

M. Tebaldi

Crossref

ÓPTICA PURA Y APLICADA. www.sedoptica.es  Opt. Pura Apl. 50 (2) 181-186 (2017)  © Sociedad Española de Óptica  181 Type:	Introduction	to	Research		Section:	Quantum	and	Non-Linear	Optics	 Optimization	of	the	diffraction	efficiency	in	photorefractive	crystals		Optimización	de	la	eficiencia	de	difracción	en	cristales	fotorrefractivos		Astrid	Lorena	Villamizar	Amado1,	Néstor	Alonso	Arias	Hernández2,	Martha	Lucía	Molina	Prado2	,	Myrian	Tebaldi3	1.	Estudiante	de	Física,	Semillero	COUP	–	Grupo	de	Óptica	Moderna,	Departamento	de	Física	y	Geología,	Universidad	de	Pamplona	2.	Grupo	de	Óptica	Moderna,	Departamento	de	Física	y	Geología,	Universidad	de	Pamplona	3.	Centro	de	Investigaciones	Ópticas	(CONICET	La	Plata,	CIC)	and	UIDET	OPTIMO,	Facultad	Ingeniería,	Universidad	Nacional	de	La	Plata,	Argentina.		(*)	E-mail:	molinaprado@gmail.com			Received:	02/12/2016	 Accepted:	05/05/2017	DOI:	10.7149/OPA.50.2.49044	ABSTRACT:	The	aim	of	this	work	is	to	study	and	to	optimize	the	diffraction	efficiency	in	sillenite	BTO	crystals.	A	theoretical	 analysis,	where	 the	 effects	 of	 coupling	 beams	 are	 significant,	 is	 carried	 on.	 In	 addition,	non-mobile	transmission	gratings	are	considered.	The	diffraction	efficiency	is	obtained	by	using	the	coupled	wave	equations	in	on-Bragg	regime	using	Method	Runge-Kutta	of	order	4	in	Matlab.	Also,	the	non-uniformity	of	the	gratings	inside	the	crystal	sample,	the	optical	activity	of	the	material	and	the	polarization	of	the	recording	beams	are	taken	into	account.	Key	words:	Photorefractive	crystals,	diffraction	efficiency.	RESUMEN:	En	este	trabajo	se	estudia	la	eficiencia	de	difracción	en	cristales	silenitas	BTO.	Se	presenta	un	análisis	teórico	en	donde	los	efectos	de	acoplamiento	son	tenidos	en	cuenta.	Además,	se	consideran	redes	no-moviles	 de	 transmisión.	 La	 eficiencia	 de	 difracción	 es	 obtenida	 a	 través	 de	 la	 solución	 de	 las	ecuaciones	 acopladas	 en	 el	 regimen	 on-Bragg	 usando	 el	 método	 de	 Runge-Kutta	 de	 orden	 4	implementado	en	Matlab.	La	no	uniformidad	de	 las	 redes	dentro	del	 cristal,	 la	actividad	óptica	del	material	y	la	polarización	de	los	haces	son	tenidos	en	cuenta.	Palabras	clave:	Cristales	fotorrefractivos,	eficiencia	de	difracción.		REFERENCES	AND	LINKS	/	REFERENCIAS	Y	ENLACES	[1]	 Yeh,	“Introduction	to	Photorefractive	Nonlinear	Optics,”	Wiley,	New	York,	(1993)	[2]	 L.F.	 Magaña,	 I.	 Casar,	 J.G.	 Murillo,	 “Beam	 energy	 exchange	 in	 sillenite	 crystals	 (Bi12SiO20	 and	Bi12TiO20),	 considering	 the	 variation	 of	 light	modulation	 along	 sample	 thickness	 in	 a	 strong	non-linear	regime,”	Opt.	Mater.	30,	979–986	(2008).	 	https://doi.org/10.1016/j.optmat.2007.05.026	[3]	 Marrachi,	A.,	 Johnson,	R.V,	Tanguay,	 Jr.A.R	 “Polarization	properties	of	photorefractive	diffraction	 in	electrooptic	and	optically	active	sillenite	crystals	(Bragg	Regime),	J.	Opt.	Som.	A.	B,	(1986)	ÓPTICA PURA Y APLICADA. www.sedoptica.es  Opt. Pura Apl. 50 (2) 181-186 (2017)  © Sociedad Española de Óptica  182 [4]	 Gonzales,	G.,	Zuñiga,	A.	“Optimization	of	the	diffraction	efficiency	in	nonuniform	gratings	in	sillenite	crystals	considering	the	variation	of	fringe	period,	optical	activity	and	polarization	angles	in	a	strong	non-linear	regime,”	Revista	Mexicana	de	Física,	(2009)	[5]	Molina	Prado	M.L.,	et	al.	“Medida	de	la	eficiencia	de	difracción	fotorrefractiva	como	función	del	vector	de	onda	de	la	red	en	un	cristal	BTO,”	Bistua,	9	(2),	43-47	(2011).		[6]	Molina	M.L.	 et	 al,	 “Eficiencia	 de	 difracción	del	 registro	 de	 speckle	modulados	 generados	 a	 partir	 de	superficies	reflecto-difusoras,”	Bistua,	10	(2),	2012.		[7].	 J.G.	 Murillo,	 L.F.	 Magaña,	 M.	 Carrascosa	 and	 F.	 Agulló-López,	 “Temporal	 evolution	 of	 the	 physical	response	 during	 photorefractive	 grating	 formation	 and	 erasure	 for	 BSO”.	 J.	 Appl.	 Phys.	 78.	 5686	(1995)	 	https://doi.org/10.1063/1.359626	[8]	 C.L.	Woods,	C.L.	Matson	and	M.M.	Salour,	“Crystallographic	fringe	orientation	diffraction	efficiency	in	bismuth	silicon	oxide”.	J.	Appl.	Phys.	A	40,	177-182	(1986)	 	https://doi.org/10.1007/BF00617401	[9].	 J.P.	 Herriau,	 D.	 Rojas,	 and	 J.P.	 Huignard,	 “Highly	 efficient	 diffraction	 in	 photorefractive	 BSO-BGO	crystals	at	large	applied	fields”.	Ferroelectrics	75,	271-279	(1987)		 	https://doi.org/10.1080/00150198708008227		1.	Introduction	The	 photorefractive	 effects	 in	 electrooptical	 crystals	 have	 been	widely	 studied	 for	 applications	 to	 real-time	holography,	optical	data	storage,	and	more	recently	for	phase-conjugate	wavefront	generation	[1-6].	The	high	recording	sensitivities	combined	with	crystal	quality	and	availability	has	been	found	in	a	cubic,	paraelectric	and	electrooptic	Bi12TiO20.	The	 nonuniform	 pattern	 received	 by	 the	 crystal	 results	 in	 a	 refractive	 index	 change	 due	 to	 the	photorefractive	and	linear	electro-optic	effect.	The	optimum	diffraction	efficiency	has	been	observed	for	two	different	orientations	of	the	crystallographic	axes	with	respect	to	the	space	charge	field	Esc	[3].	The	sign	of	the	charge	carriers,	which	cause	the	space-charge	field,	can	be	determined	by	the	energy	transfer	direction	and	the	electro-optic	coefficient	measurements.	In	 this	 work	 we	 considered	 non-moving	 transmission	 gratings	 under	 an	 applied	 field,	 the	 variation	 of	fringe	 period	 and	 we	 included	 optical	 activity	 and	 birefringence	 in	 the	 Bragg-regime.	 One	 crystal	orientation	is	considered,	with	the	grating	vector	is	parallel	to	the	face	[001].		Fig.1:	Longitudinal	holographic	configuration	𝐾"	‖ < 001 >	.			2.	Set	Coupled	Wave	Equations	The	BTO	is	a	material	with	electrooptic	coupling	and	optical	activity,	the	wave	equation	that	governs	light	propagation	is:	ÓPTICA PURA Y APLICADA. www.sedoptica.es  Opt. Pura Apl. 50 (2) 181-186 (2017)  © Sociedad Española de Óptica  183 	 𝛻C𝐸 + FGHIG𝐷 = 0;	 (1)	where	E	is	the	total	light	field,	kO = 2π λ	is	the	magnitude	of	the	optical	wave	vector,	λ	is	the	wavelength	and	ϵO	is	the	free	space	permittivity.	For	a	birefringence	medium	with	optical	activity	and	electro-optic	coupling,	this	constitutive	relation	can	be	expressed	in	the	form:		 DZ = ϵO ϵZ[ + GZ[ + ∆ϵZ[ E[;	 (2)	where	GZ[	is	the	optical	activity	antisymetric	tensor,	∆ϵZ[	is	the	change	in	the	permittivity	tensor	induced	by	the	electrooptic	effect	and	ϵZ[	is	the	symmetric	optical	permittivity	tensor.		For	the	K_	‖ < 001 >	orientation,	the	x	axis	is	parallel	to	the	<001>	crystallographic	axis,	as	show	in	Fig.	1.	 The	 transformation	 matrix	 that	 converts	 from	 the	 crystallographic	 system	 to	 the	 light	 propagation	system	is:		 T =0 0 1−12−12012−120	 (3)	For	this	orientation,	the	permittivity	tensor,	exclusive	of	optical	activity	components,	is:		 ϵZ[ =ϵ 0 00 ϵ + ∆ϵ 00 0 ϵ − ∆ϵ	 (4)	If	∆ϵ = nObrcdEe,	where	nO	 is	 the	 index	of	 refraction,	 rcd	 is	electrooptic	 coefficient	and	Ee	 is	 the	electric	field.	This	field	consists	of	two	components:		 Ee = EO + Efgcos	(K_. r + ϕ)	 (5)	EO	 is	 the	 applied	 electric	 field	 and	 Efg	 is	 the	 space	 charge	 field	 which	 depend	 of	 the	 diffusion	 and	saturation	fields:		 Efg = mEklGHmlnHlGHm lnmloHdC	 (6)	In	Eq.	(6)	EO	Is	the	applied	electric	field	of	eq.	(5),	Ed	is	the	diffusion	field	and	Eq	is	an	upper	limit	on	the	space	charge	field	corresponding	in	a	single	trap	model	to	saturation	of	available	acceptor	site	(assuming	monopolar	charge	transport	in	which	electrons	are	the	only	mobile	carriers).	The	electric	field	of	the	two	beams	can	be	written	as:		 E x, y, z = R(z)eZrs.t + S(z)eZrv.t	 (7)	The	polarization	states	of	the	incident	probe	beam	R	and	of	the	diffracted	signal	field	S	are	two	orthogonal	linear	polarization	states	which	components	is	given	by:		 R = yRl + (xcosθx − zsenθx)Ry	;	 (8)		 S = ySl + (xcosθz − zsenθz)𝑆y	 (9)	The	set	of	coupled	wave	equations	can	be	derived	now	that	structure	of	the	light	field	has	been	defined.	The	optical	activity	tensor	can	be	added	to	the	permittivity	tensor,	the	constitutive	relation	for	the	K_	‖ <001 >	orientation	is:		 D|Dl= ϵOϵ ia−ia ϵ + ∆ϵEyEl;	 (10)	where	a	is	a	parameter	defined	by	2nOρ kO.	Using	the	slowly	varying	envelope	approximation	derives	the	coupled	wave	equations.	The	resulting	set	of	coupled	wave	equation	is:	ÓPTICA PURA Y APLICADA. www.sedoptica.es  Opt. Pura Apl. 50 (2) 181-186 (2017)  © Sociedad Española de Óptica  184 	 ~x= − gfRl	dRldz=ρcosθRy + iCcosθRy + iCfgcosθSleZ	dSydz= −ρcosθSl		 z= gfSy + igfSl + igfRleZ;	 (11		where	ρ	is	the	optical	rotatory	power	and:	C =πnObλrcdEO	Cfg =πnObλrcdEfg12		For	the	longitudinal	configuration	we	obtained	the	diffraction	efficiency	𝜂,	defined	by:	η =I(z)IZ(0)	where	I(z)	is	the	intensity	of	the	diffraction	light,	IZ(z)	is	the	intensity	of	the	incident	light.	For	theoretical	analysis	was	used	the	Runge-Kutta	method	and	its	algorithm	in	Matlab.	TABLE	1.	Parameter	for	BTO	[3,7-9]	taken	for	our	calculations		 BTO	𝜖	𝐷𝑖𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐	𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡	 47	𝑛		𝑟𝑒𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛	𝑖𝑛𝑑𝑒𝑥	 2.58	𝑟cd	𝑒𝑙𝑒𝑐𝑡𝑟𝑜	𝑜𝑝𝑡𝑖𝑐	𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡	(𝑚𝑉d)	𝑁¢	𝐴𝑐𝑐𝑒𝑝𝑡𝑜𝑟	𝑑𝑒𝑛𝑠𝑖𝑡𝑦	(𝑚b)	𝜌	𝑂𝑝𝑡𝑖𝑐𝑎𝑙	𝐴𝑐𝑡𝑖𝑣𝑖𝑡𝑦	º𝑐𝑚d	(𝜆 = 632𝑛𝑚)	5.1	x	10-12	4	x	10-22	65		 	3.	Results	and	Discussion	In	Fig.	2	we	show	the	result	 for	the	dependence	of	 the	diffraction	efficiency	on	the	sample	thickness	for	different	 applied	 field	 for	 a	BTO	 grating	 using	 red	 light	 (632	nm)	 for	 reading.	 For	 this	 case,	 KG∥<001>;	m=0.9	and	L	=	1µm;	10	µm;	the	largest	value	of	the	diffraction	efficiency	occurs	for	an	applied	field	of	10	kV/cm.	We	can	see	that	the	smallest	of	diffraction	efficiency	occurs	for	the	smallest	value	of	the	applied	field:	1	kV/cm.				Fig.	2.	BTO	crystal	diffraction	efficiency	in	terms	of	the	crystal	thicknesses	using	red	read-out	light	and	different	applied	electric	fields	Ea=	1kV/cm,	5kV/cm,	7kV/cm	y	10kV/cm,	ϵ = 47,	m=0.9,	at	various	values	(a)	Λ=1µm,	(b)	Λ=10µm.	 In	Fig.	3	we	show	the	result	for	the	dependence	of	the	diffraction	efficiency	on	the	sample	thickness	for	a	BTO	 grating	 using	 red	 light	 (632	nm)	 for	 reading	 for	 different	 grating.	 For	 this	 case,	KG∥<001>;	 Ea=	 10	ÓPTICA PURA Y APLICADA. www.sedoptica.es  Opt. Pura Apl. 50 (2) 181-186 (2017)  © Sociedad Española de Óptica  185 Kv/cm;	m	=0.9	and	m=0.3;	the	largest	value	of	the	diffraction	efficiency	occurs	for	a	grating	period	of	10	microns	 and.	We	 can	 see	 that	 for	 all	 values	 of	 the	 grating	 period	 considered,	 the	 diffraction	 efficiency	reaches	a	maximum	value	and	the	smallest	of	these	occurs	for	the	smallest	value	of	the	grating	period:	1	micron.					Fig.	3.	BTO	crystal	diffraction	efficiency	in	terms	of	the	crystal	thicknesses	using	red	read-out	light	and	different	grating	period:	Λ=1µm,	2µm,	3µm,	5µm,	10µm,	at	various	values	of	(a)	m=0.9,	Ea=0kV/cm		(b)	m=0.9,	Ea=10	kV/cm	(c)	m=0.3,	Ea=10kV/cm	 In	Fig.	4	we	show	the	result	for	the	dependence	of	the	diffraction	efficiency	on	sample	thickness	for	a	BTO	grating	using	red-out	light	(632	nm)	for	reading	for	different	modulations.	For	this	case:	KG∥<001>;	Ea=10	Kv/cm;	L	=	1µm;	10	µm.	In	this	case,	the	largest	value	of	the	diffraction	efficiency	occurs	for	a	modulation	of	 0.9.	 For	 all	 the	 range	of	 values	of	 sample	 thickness	 considered,	 the	modulation	 corresponding	 to	0.9	maintains	the	largest	value	for	the	diffraction	efficiency.	  Fig	4.	BTO	crystal	diffraction	efficiency	in	terms	of	the	crystal	thicknesses	using	red	read-out	light	and	different	modulations,	m=0.1,	0.3,	0.6,	0.9	at	various	values	of	(a)	Λ=1µm,	(b)	Λ=10µm.	ÓPTICA PURA Y APLICADA. www.sedoptica.es  Opt. Pura Apl. 50 (2) 181-186 (2017)  © Sociedad Española de Óptica  186 4.	Conclusions	We	studied	the	diffraction	efficiency	in	non-moving	transmission	gratings,	in	Bragg	regime	in	longitudinal	configuration	 in	 BTO	 crystal.	 A	 set	 of	 coupled-wave	 equations	 has	 been	 derived	 for	 longitudinal	holographic	orientation	in	crystal	BTO	of	sillenite	family.	These	equations	allow	an	analysis	to	be	made	of	the	polarization	properties	of	light	diffraction	including	the	effects	of	concomitant	optical	rotatory	power.	In	the	date	analytic	we	used	red	light	for	the	reading.	We	considered	the	birefringence,	coupling	electro-optic,	 optical	 activity,	 fringe	 period,	 different	 fields	 applied,	 different	modulations	 and	 the	 polarization	beams.		The	results	show	that	for	the	field	applied	of	1	Kv/cm	corresponded	to	the	small	diffraction	efficiency,	for	the	fringe	period	of	10	𝜇m	the	diffraction	efficiency	is	greater	and	for	a	modulation	of	0.1	the	diffraction	efficiency	is	smallest.		Acknowledgements	This	research	was	performed	under	the	grants:PR130-00-033	(GA160-BP-II-2013-2.1.2.2.1)	y	PR130-00-001	(GA160-CM-I-2014-2.1.2.2.1)-Unipamplona,	Astrid	Villamizar	acknowledges	 financial	support	of	 the	Vicerrectoría	de	 Investigaciones	of	 the	Universidad	de	Pamplona	(Pamplona-Colombia).	Portions	of	 this	work	were	presented	at	the	IX	RIAO/OPTILAS	in	Pucón,	Chile	2016"		

http://ri.conicet.gov.ar/bitstream/11336/49335/2/CONICET_Digital_Nro.09b8ebbc-ecb0-436d-b270-073c9f55d5fe_A.pdf

Optimization of the diffraction efficiency in photorefractive crystals

Abstract

Similar works

Full text

Available Versions

El Servicio de Difusión de la Creación Intelectual

LAReferencia - Red Federada de Repositorios Institucionales de Publicaciones Científicas Latinoamericanas

CONICET Digital

Crossref

LAReferencia - Red Federada de Repositorios Institucionales de Publicaciones Científicas Latinoamericanas