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Vol. 24, No. 14 | 11 Jul 2016 | OPTICS EXPRESS 15666
Nonlinear Raman-Nath second harmonic
generation with structured fundamental wave
HAIGANG LIU,1,2 JUN LI,3 XIAOHUI ZHAO,1,2 YUANLIN ZHENG,1,2,4 AND
XIANFENG CHEN1,2,5
1
State Key Laboratory of Advanced Optical Communication Systems and Networks, Department of
Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
2
Key Laboratory for Laser plasma (Ministry of Education), Collaborative Innovation Center of IFSA
(CICIFSA), Shanghai Jiao Tong University, Shanghai 200240, China
3
Science and Technology College, Jiangxi Normal University, Jiangxi 330027, China
4
[email protected]
5
[email protected]
Abstract: We proposed and experimentally demonstrated that nonlinear Raman-Nath second
harmonic can be achieved in real time when a fundamental wave with the phase periodically
modulated, termed as structured fundamental wave, incident in a homogeneous nonlinear
medium. The diffraction of second harmonic originates from the structured fundamental
wave, rather than the grating of a nonlinear photonic crystal. Nonlinear second harmonic
generation, in forms of both one- and two-dimensional, was investigated in our experiment.
This method circumvents the limitation of nonlinear photonic crystals in some extend and has
potential applications in nonlinear frequency conversion, optical signal processing and beam
shaping, etc.
©2016 Optical Society of America
OCIS codes: (190.0190) Nonlinear optics; (090.1760) Computer holography; (050.1970) Diffractive optics;
(190.2620) Harmonic generation and mixing.
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#267074
Journal © 2016
http://dx.doi.org/10.1364/OE.24.015666
Received 27 May 2016; revised 24 Jun 2016; accepted 24 Jun 2016; published 30 Jun 2016
Vol. 24, No. 14 | 11 Jul 2016 | OPTICS EXPRESS 15667
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1. Introduction
Nonlinear photonic crystals (NPCs) [1,2], in which the second-order susceptibility χ (2) is
spatially modulated, open a door in the field of nonlinear wave mixing, like second-harmonic
generation (SHG), sum-frequency generation, difference-frequency generation and so on. The
phase matching of collinear and noncollinear wave mixing can be achieved by the
compensation of an reciprocal vector of NPCs without wavelength limitation. Both onedimensional (1D) and two-dimensional (2D) structures have widely been fabricated to realize
optical parametric processes in quasi-phase matching (QPM) manners [3–6]. More recently,
transverse schemes (perpendicular to traditional QPM direction) of noncollinear wave mixing
processes have attracted much attention. Various nonlinear processes, such as nonlinear
Cherenkov radiation [7–13], nonlinear Raman-Nath and nonlinear Bragg diffraction [9,14–
16] have been intensively studied in NPCs. And various patterns of 2D nonlinear photonic
structures have also been demonstrated, including square, hexagonal lattices, annular
periodical, or even random structures [17–21]. Especially, with holography concept
introduced to nonlinear optics, arbitrary two-dimensional nonlinear beam shaping, such as
vortex and airy beams, could be realized [22–24]. However, all these methods, which
manipulate the structure of the NPCs, have several drawbacks, including complex fabrication
and unchangeable second harmonic (SH) pattern. For desired spatial shaping of the SH beam,
an alternative approach has been used to control the incident FW [25,26].
The modulation of χ (2) structures periodically flips the phase of second-order nonlinear
polarization excited by the fundamental wave (FW). Considering a degenerate SHG process,
the nonlinear polarization P2ω excited by the FW in a nonlinear medium can be expressed as
P2ω (r , t ) = ε 0 χ (2) (r ) Eω (r , t ) 2 ,
(1)
where ε 0 is the vacuum permittivity, χ (2) is the second-order susceptibility and Eω is the
electric field of FW. In NPCs, the constructive interference of the radiation of polarization
P2ω is achieved by periodically modulating the sign (or value) of the coefficient χ (2) in
space. While, the physics behind Eq. (1) hints that periodically modulating the phase (or
amplitude) of FW can also achieve the phenomenon in NPCs.
In this Letter, we study the process of SHG using structured FW as the input in a
homogenous χ (2) medium. The wavefront of the FW is periodically modulated to provide a
Vol. 24, No. 14 | 11 Jul 2016 | OPTICS EXPRESS 15668
transverse structure in the process of SHG. In 1D geometry, due to the structure of the FW,
the SH exhibits as multiple discrete spatial spots. Extending this concept to 2D situation, SH
rings and squares can be attained. Under the condition of nondiffraction in a sufficient short
 
medium and G m k 1 ≤ 0.1 , the generated SH appears in nonlinear Raman-Nath diffraction
direction. The SHG process is assisted by the structure of the FW itself, rather than any
nonlinear grating or structure in the nonlinear medium.
2. The coupled wave equation

 

The incident FW is defined in the form of E1 = A1 exp[−i (k 1 ⋅ r − ωt )] , where A1 and k 1 are
the amplitude and wavevector of the FW, respectively. Suppose that the FW propagates along
the y-axis of the crystal. When the wavefront is periodically modulated, the FW can be more
intuitively written as an expansion of Fourier series:

 
 
E1 = A1 ⋅  Cm exp(−iG m ⋅ r ) exp[−i (k ym ⋅ r − ωt )]
(2)

where Cm and G m = 2π m Λ are the Fourier coefficients and reciprocal vectors of the FW, Λ
2
is the period of the FW modulation. The relation between Gm and k ym are Gm2 + k ym
= k12 . In
order to obtain SH in nonlinear Raman-Nath diffraction directions, some restrictions need to
 
be done. Under the condition of G m k 1 ≤ 0.1 (for larger values of m, Cm are small enough to
be negligible), the diffraction of FW is negligible, which holds in a sufficient short medium
(the critical length L ≈ k1Λ Gm ≥ 10Λ ). The crystal is a sufficient short medium when its
thickness Lc is shorter than twice the critical length L . For simplicity, under this assumption,
the modulation function can be written as:

 
 
E1 = A1 exp[−i (k 1 ⋅ r − ωt )] ⋅  Cm exp(−iG m ⋅ r )
(3)

 
We define the expression of SH wave E 2 = A2 exp[−i (k 2t ⋅ r + k 2 y y − 2ωt )] . Under the
assuming that the FW is undepleted, the evolution of the SH wave is directly given by [22]

 
dA2
= κ A12 ⋅  bq exp[i (k2 y − 2k1 ) y ]exp[i (k 2t − G q ) ⋅ r ],
dy
where κ is the nonlinear coupling coefficient. bq =  m , n
m+ n=q
(4)
Cm Cn is the Fourier coefficient of
the nonlinear polarization and the summation runs from −∞ to +∞ . In Eq. (4), the first
exponential term stands for the longitudinal phase mismatch. The second exponential term is
the nonlinear Raman-Nath diffraction phase mismatch, in which the structure function
originates from the FW. Thus, we can get SH in nonlinear Raman-Nath diffraction directions
under the assumption mentioned above. If the phase of incident FW period is sharply
modulated from 0 to φ and the duty cycle D is 0.5. The Fourier coefficients can be derived as

C0 = 1 + eiφ

. m = 0, ±1, ±2,
(5)

i[cos(mπ ) − 1]
(1 − eiφ )
Cm =
2mπ

Note that the Fourier coefficients Cm are non-zero only for odd values of m (except C0 ).
For a certain order of q, the SH intensity is proportional to bq and can be expressed as
I 2 ∝ bq
2
2
χ (2) I12 , where I1 and I 2 are the FW intensity and the SH intensity, respectively.
Vol. 24, No. 14 | 11 Jul 2016 | OPTICS EXPRESS 15669
3. Experiment results and discussion
In our proof-of-principle experiment, the FW at a wavelength of 1064 nm (Nd:YAG
nanosecond laser) was phase modulated by a spatial light modulator (SLM). The SLM had a
resolution of 512 × 512 pixels, each with a rectangular area of 19.5 × 19.5μ m 2 . The light was
then imaged by a 4-f system (magnification of 0.25) to imprint the modulated wavefront
pattern to the onset of a χ (2) crystal. The beam waist was reduced to 2 mm. After the crystal,
a shortpass filter was used to filter out the FW. Finally, the generated SH beam was projected
on a screen in the far-field and recorded by a camera. For simplicity and without loss of
generality, a 5mol% MgO:LiNbO3 bulk crystal ( 10 × 0.3 × 10mm3 in x × y × z dimensions)
was used in our experiment. The FW was kept as o-polarized and propagated along the y-axis
of the crystal at room temperature. In this situation, m should smaller than 9. In the following
experiment Λ ≥ 10 μ m was used ( L ≥ 0.4mm ), hence the thickness of the medium in our
experiment ( L = 0.3mm < 2 Lmin ) could be treated as a sufficient short medium.
In order to illustrate the role of such structure of the FW in the process of SHG, we did a
comparison experiment using a uniform and a wavefront modulated FW input, as shown in
Fig. 1. For simplicity, the hologram loaded on the SLM is used to represent the wavefront
profile. Firstly, due to the near birefringent phase matching (BPM) condition, there is only
one collinear phase-mismatched SH spot observed in the case of FW without any phase
structure (shown in Fig. 1(a)-1(c)). The type of SH interaction is oo-e and the phase-matching
geometry is shown in Fig. 1(b). Secondly, the phase of the FW is periodically sharply
modulated from 0 to π 2 in 1D structure. The hologram loaded on SLM is illustrated in Fig.
1(d), which exhibits 1:1 duty cycle (D). The phase-matching geometry is shown in Fig. 1(e).
The output is a set of symmetrically distributed SH spots, in which 0, ±1 , ±2 and ±3 orders
were experimentally observed, shown in Fig. 1(f). From the Fourier coefficient of Eq. (5) and
the relation of bq =  m , n
m+n=q
Cm Cn , in this case, the Fourier coefficients bq are non-zero for
each value of q and this is in consistence with the experiment results. Figure 1(g) show the
experimental image of the FW from the output end of the nonlinear crystal, as can be seen the
diffraction of FW was not noticeable.
Fig. 1. (a)(d) Holograms loaded on SLM representing the phase structure of the FW. (b)(e)
Phase-matching geometries of the nonlinear Raman-Nath diffraction SHG process,
corresponding to (a) and (d) under the assumption of nondiffraction of FW, respectively. (c)(f)
Observed SH diffraction patterns of (a) and (d), respectively. (g) The experimental image of
the FW from the output end of the nonlinear crystal.
For precise determination of the angle of the SH emission, we used various modulation
periods ranging from 50 to 250 μm. The angle of the qth diffraction order nonlinear RamanNath diffraction SH spot is defined by the transverse phase-matching condition:
Vol. 24, No. 14 | 11 Jul 2016 | OPTICS EXPRESS 15670

sin α q = G q

k 2 , q = 0, ±1, ±2, . According to the Snell’s law n2 e sin α q = sin β q , the
external radiation angles are β q = arcsin(qλ2 Λ ) , q = 0, ±1, ±2, , where n2 e and λ2 are the
refractive index and wavelength of the SH. The external angles of the ±1 , ±2 and ±3 order
nonlinear SH spots are shown in Fig. 2(a). The SH diffraction angles are increasing while the
modulation period of the FW decreases. The theoretical prediction and experimental results
are also in well agreement with each other.
Apart from the angular information, the influence of parameters of FW such as φ and the
duty cycle D on the SH pattern are also investigated. In case of D = 0.5 , from the Fourier
coefficient of Eq. (5), the value of φ determines the existence of C0 and the relative values
of Cm (the value of Cm is 0 if m is even). According to the relation of bq =  m , n
m+ n=q
Cm Cn ,
the value of φ only influence the relative value of bq . However, it is interesting when the
parameter φ = π , the Fourier coefficients bq becomes non-zero only for even q ' s . The
corresponding experimental result is described in Fig. 2(b), in which only 0 and ±2 orders
are present. When the duty cycle D ≠ 0.5 ( C0 = 2 D − 1, Cm = 2sin(mπ D) mπ ), each order of
bq becomes non-zero again. The missing odd orders of SH spots reappear as observed in the
experiment.
It is worth noticing that the excited nonlinear polarization wave experiences a π phase in
two adjacent opposite domains, which is caused by the sharp χ (2) modulation from + 1 to −1.
And it only relates to once Fourier transform of the structure of the crystal. While, in our case,
the nonlinear polarization wave involves the product of two Fourier expansion series of the
FW [see Eq. (1)].
Fig. 2. (a) The external angles of + 1, + 2, and + 3 order SH as a function of the FW
modulation period. Theoretical prediction (solid curves) and measured ones (signs) are in well
agreement with each other. (b) Observed SH pattern with D = 0.5 and φ = π in 1D
structure.
In addition to 1D geometry, we also extended such structured FW to 2D situations. The
phase of FW period was sharply modulated from 0 to π 2 in 2D patterns and the period of
FW corresponding to 40 μm with the duty cycle D = 0.5 . For the annular structure of FW
(Fig. 3(a)), the phase-matching geometry with radial ‘reciprocal vectors’ is shown in Fig.
3(b). The SH forms multiple rings structure in Fig. 3(c). The 1-, 2-, 3- and 4-order SH rings
were observed in experiment. The SH of different order rings are circularly symmetric with
uniform intensity distribution and lower orders of the SH rings are brighter due to larger
Fourier coefficients bq . Moreover, a square patterned FW phase was applied (Fig. 3(d)) with
the phase-matching geometry shown in Fig. 3(e). The ‘reciprocal vectors’ of the FW
decreases with increase of the FW period around the azimuthal direction (perpendicular to the
beam propagation direction) of the FW to form a square structure. Different orders of SH
form multiple squares structure (Fig. 3(f)). However, unlike annular structure of the FW, the
SH intensity distribution is not uniform with the corner direction of SH square pattern
exhibiting higher intensity. This is caused by the symmetric property of this structure, in
Vol. 24, No. 14 | 11 Jul 2016 | OPTICS EXPRESS 15671
which most of the FW only feel the structure along horizontal and vertical directions. When
the phase of FW period was sharply modulated from 0 to π , the odd orders of the SH rings,
dots and squares were not presented. This is similar to those in 1D cases. The Fourier
coefficients bq = 0 make odd orders of the SH rings, dots and squares disappear.
Fig. 3. (a)(d) Holograms loaded on SLM representing the phase structure of the FW. (b)(e) The
phase-matching geometries of the nonlinear Raman-Nath diffraction SHG process with
different structures of the FW, corresponding to (a) and (d) under the assumption of
nondiffraction of FW, respectively. (c)(f) SH pattern observed in experiment, corresponding to
(a) and (d), respectively.
The phase modulation of the FW itself for nonlinear wave mixing features many
advantages. Actually, compared to χ (2) modulation, it is more flexible to modulate the FW
itself [26]. Firstly, the proposed method does not require any special fabrication. Secondly,
the FW can be controlled by an SLM in real time, whereas SHG in NPCs is fundamentally
restricted by their predesigned structures. Moreover, the modulated FW can propagate along
any direction of bulk nonlinear medium. This is important for efficient SHG by utilizing
larger components of the χ (2) tensor. Unlike the work in [26], which dealt with on-axis
shaping, the present paper deals with off-axis nonlinear Raman-Nath diffraction. The phase of
FW can be continually changed in our method while the values of modulated χ (2) are only +
1 and −1 in NPCs. This method will largely enrich the SHG process, but not limited to. The
concept can be adopted in other schemes as well, for example the investigation of nonlinear
Cerenkov radiation in bulk media. Other potential applications may lie in nonlinear frequency
conversion, optical signal processing, beam shaping and so on.
4. Conclusion
In summary, under the condition of nondiffraction of FW, nonlinear Raman-Nath SH can be
achieved when a structured FW incident in a homogeneous nonlinear medium. The change of
pattern can be done in real time. In the 1D case, the SH exhibits as multiple discrete spatial
spots. And in 2D cases, SH rings and squares were also observed in experiment. The scheme
simplifies the procedure where NPCs are needed, and provides much more flexibility in
generating complex SH patterns. The concept can be adopted in other schemes and has
potential applications in nonlinear frequency conversion, optical signal processing and beam
shaping, etc.
Acknowledgment
This work is supported in part by the National Natural Science Foundation of China (NSFC)
under Grant Nos. 61125503, 61235009 and 11421064, the Foundation for Development of
Science and Technology of Shanghai under Grant No. 13JC1408300.