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Forward and backward THz-wave difference

frequency generations from a rectangular

nonlinear waveguide

Yen-Chieh Huang,* Tsong-Dong Wang, Yen-Hou Lin, Ching-Han Lee, Ming-Yun

Chuang, Yen-Yin Lin, and Fan-Yi Lin

Institute of Photonics Technologies, Department of Electrical Engineering, National Tsinghua University, Hsinchu

30013, Taiwan

*ychuang@ee.nthu.edu.tw

Abstract: We report forward and backward THz-wave difference frequency

generations at 197 and 469 µm from a PPLN rectangular crystal rod with an

aperture of 0.5 (height in z) × 0.6 (width in y) mm2 and a length of 25 mm

in x. The crystal rod appears as a waveguide for the THz waves but as a

bulk material for the optical mixing waves near 1.54 µm. We measured

enhancement factors of 1.6 and 1.8 for the forward and backward THz-

wave output powers, respectively, from the rectangular waveguide in

comparison with those from a PPLN slab waveguide of the same length,

thickness, and domain period under the same pump and signal intensity of

100 MW/cm2.

©2011 Optical Society of America

OCIS codes: (190.4223) Nonlinear wave mixing; (190.4410) Parametric processes.

References and links

1. G. Kh. Kitaeva, S. P. Kovalev, A. N. Penin, A. N. Tuchak, and P. V. Yakunin, “A method of calibration of

terahertz wave brightness under nonlinear-optical detection,” Int. J. Infrared Millim. Waves 32(10), 1144–1156

(2011).

2. K. Suizu and K. Kawase, “Monochromatic-tunable terahertz-wave sources based on nonlinear frequency

conversion using lithium niobate crystal,” IEEE J. Sel. Top. Quantum Electron. 14(2), 295–306 (2008).

3. T. D. Wang, S. T. Lin, Y. Y. Lin, A. C. Chiang, and Y. C. Huang, “Forward and backward terahertz-wave

difference-frequency generations from periodically poled lithium niobate,” Opt. Express 16(9), 6471–6478

(2008).

4. K. L. Vodopyanov, “Optical THz-wave generation with periodically-inverted GaAs,” Laser Photonics Rev. 2(1–

2), 11–25 (2008).

5. Y. Takushima, S. Y. Shin, and Y. C. Chung, “Design of a LiNbO3 ribbon waveguide for efficient difference-

frequency generation of terahertz wave in the collinear configuration,” Opt. Express 15(22), 14783–14792

(2007).

6. A. C. Chiang, T. D. Wang, Y. Y. Lin, S. T. Lin, H. H. Lee, Y. C. Huang, and Y. H. Chen, “Enhanced terahertz-

wave parametric generation and oscillation in lithium niobate waveguides at terahertz frequencies,” Opt. Lett.

30(24), 3392–3394 (2005).

7. M. H. Chou, “Optical frequency mixers using three-wave mixing for optical fiber communications,” PhD thesis,

(Stanford University, 1999).

8. R. W. Boyd, Nonlinear Optics (Academic Press, 2008).

9. J. Hebling, A. G. Stepanov, G. Almasi, B. Bartal, and J. Kuhl, “Tunable THz pulse generation by optical

rectification of ultrashort laser pulses with tilted pulse fronts,” Appl. Phys. B 78, 593–599 (2004).

10. D. H. Jundt, “Temperature-dependent Sellmeier equation for index of refraction, ne, in congruent lithium

niobate,” Opt. Lett. 22(20), 1553–1555 (1997).

1. Introduction

Difference frequency generation (DFG) of two lasers beating at THz frequencies in a

nonlinear optical material is a popular optical technique to generate coherent THz wave

radiation. However, owing to the vast difference of the mixing wavelengths, the THz wave is

usually more absorptive in the material than the optical mixing waves. For example, lithium

niobate, while being transparent in the optical spectrum, has a typical absorption coefficient of

a few tens of cm−1 at THz frequencies [1]. The THz wave experiences a pure absorption loss

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as soon as it leaves the gain region of the optical pump and signal beams. To overcome this

problem, several schemes have been successfully implemented to couple out the THz wave as

soon as possible for non-collinearly phase matched THz-wave DFG in lithium niobate [2].

For parametric frequency mixing, however, collinear phase matching is a preferred

configuration to increase the parametric gain length and thus the wavelength conversion

efficiency. With quasi-phase-matching (QPM), we have previously demonstrated collinearly

phase matched THz-wave DFG in bulk periodically poled lithium niobate (PPLN) [3].

Unfortunately, the much longer wavelength of the THz wave could still make the generated

THz wave quickly diffracted and absorbed outside the gain region of the optical pump and

signal beams. GaAs is also a demonstrated QPM material for THz-wave DFG with much less

absorption in the THz spectrum [4]. However, LN has a three-time larger nonlinear

coefficient. If the diffraction induced absorption in LN can be alleviated, LN is still a

promising material for high-efficiency THz DFG.

To reduce the diffraction-induced absorption, one could in principle design a nonlinear

optical waveguide that guides the THz wave as well as the optical waves [5]. However,

guiding an optical wave requires a waveguide aperture comparable to the optical wavelengths,

which severely restricts the input and output powers of the mixing waves. Previously we have

shown an experimental evidence of enhanced, non-collinearly phase matched THz-wave DFG

from a 0.5-mm thick crystal slab of lithium niobate [6]. This crystal slab guides the THz wave

in one transverse direction but behaves like a bulk crystal to the optical mixing waves. For

what follows, we call such a waveguide a one-dimensional (1-D) nonlinear optical semi-

waveguide (NOSW). In this paper, we compare collinearly phase matched THz-wave DFG in

rectangular (2-D) and 1-D NOSWs of the same length made of PPLN. As will be shown

below, the additional confinement of the THz wave in the other transverse dimension of the

crystal indeed enhances the THz-wave output power under the same pump condition.

2. Theory

To be consistent with our following experiment using type-0 phase matched PPLN crystals,

we choose z as the direction of polarization for all mixing waves and + x as the propagation

direction for the pump and signal waves. Forward or backward THz-wave DFG refers to the

propagation of the THz wave in the +x or −x direction, respectively. In a NOSW, the

unguided optical component has a varying beam size along the propagation direction x.

However, in our case, the optical mixing waves have a depth of focus significantly longer

than the crystal length and the mode-radius variation is less than 2% over the whole crystal

length. Therefore, it is a good approximation to write the electric fields of the collinearly

propagating optical pump and signal waves with a constant Gaussian field profile in the

=

subscript i = p, s denote the pump and signal waves, respectively, β is the propagation

constant in the x direction, η = η0/n is the intrinsic wave impedance in a material of refractive

index n, and

( , )

e y z

is the transverse field profile with the normalization

∫∫

, so that

( )( )

A xP x

is the power of the wave at x. For the THz-

wave DFG, the optical pump and signal wavelengths are nearly the same and

( , )( , )

ps

ey ze y z

≈

for a fundamental Gaussian beam can be written as

transverse direction, given by

()

( , , , )

E t y z x

Re[ 2 ( , )

e y z A x e

η

( )]

ii

jtx

iiii

ωβ

−

, where the

2

( , )

e x y

1

dxdy

∞

−∞

=

2

=

222

( )/

,

1

w π

2

( , )

y z

,

yzw

p s

ee

−+

=

(1)

where w is the average mode radius of the pump and signal waves in the nonlinear optical

material. Likewise, the electric field of the guided THz wave can be expressed as

ω

η

=

propagation constant βTHz denotes a THz wave propagating along ± x.

()

( , , , )

t y z x

Re[ 2( , )

y z A

( )

x e

],

THzTHz

jtx

THzTHzTHz

e

THz

E

β

∓

where the sign ∓ preceding the

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In our experiment, pump depletion was negligible. Without pump depletion and signal

absorption, the coupled-mode equations for continuous-wave, collinear DFG are given by [7]

*

THz

,

jx

s

sp

A

∂

jA Ae

x

β

κ

− ∆

∂

= −

(2)

*

s

,

2

jx

THz

x

∂

THz

THzpTHz

A

jA A eA

β

α

κ

− ∆

∂

=

∓∓

(3)

where αTHz is the absorption coefficient of the THz wave, κi is the nonlinear coupling

coefficient, and

ps THz QPM

k

ββββ

∆=−−

∓

is the wave-number mismatch among the

collinear pump, signal, THz waves, and the quasi-phase-matching (QPM) grating kQPM.

Without any initial THz-wave power at x = 0, the forward THz-wave output power at x = L

normalized to the input signal power Ps(0) is given by

2

2

/2

2

( )

L

sinh() ,

(0)

THzL

α

−

s

THz

P

P

f

s THz

f

e g L

g

λ

λ

Γ

=

(4)

where λ is the wavelength in vacuum, Γ is the parametric gain coefficient, and

αβ

≡ − ∆+Γ with

f

ββ

∆=

of the parametric gain coefficient Γ is given by

22

( / 4/ 2)

f THzf

gj

.

ps THzQPM

k

ββ

−−−

The specific expression

22

eff

λ λ

0

22

0

22

8

(0)(0),

sTHzpppp

psTHzsTHz

d

PI

n n n

πη

κ κϑϑ

Γ == Γ=

AA

(5)

where Pp(0) and Ip(0) are the initial pump power and intensity at x = 0, respectively, Γ0 is the

free-space plane-wave parametric gain coefficient [8], deff is the effective nonlinear

coefficient,

p

A is the pump-mode area, and

A

plane-wave parametric gain coefficient Γ0 with the mode-overlapping integral ϑ defined as

pϑ

is a modification factor to the free-space

( , ) ( , )

y z e y z e

( , )

y z dydz

.

psTHz

e

ϑ

∞

−∞

=∫∫

(6)

It is straightforward to show that in the limit of a much larger THz-wave mode size than

the optical one,

ϑ approaches the inverse of the THz-wave mode area or

Γ is reduced from its free-space plane-wave value by a factor of

2

2

1/

THz

ϑ →

A

and

2

/.

pTHz

AA

Since the THz

wavelength is much longer than that of the optical mixing waves, this parametric gain

reduction can be very significant due to fast diffraction of the THz wave in a bulk nonlinear

optical material.

For the case of a backward THz wave propagating in the −x direction, we derive the THz-

wave output power at x = 0 with zero initial THz-wave power at x = L, given by

2

2

sin()(0)

,

(0)

sin()cos()

4

sb

THz

P

P

THz

sTHz

bbb

g L

g L gg L

λ

λ

α

=Γ

+

(7)

where

22

( / 4/ 2)

bTHzb

gj

αβ

≡ Γ −− ∆

with

.

bpsTHzQPM

k

ββββ

∆=−+−

It can be seen from

Eq. (7) that the THz absorption increases the oscillation threshold and broadens the

parametric gain bandwidth.

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3. Experiment

We fabricated three 2-D PPLN NOSWs from congruent lithium niobate, as shown in Fig.

1(a), but unfortunately broke the two shorter ones during polishing. For comparison, we also

fabricated a 1-D PPLN NOSW with no wave confinement in the y direction. Both NOSWs are

25 mm long and 0.5 mm thick in the crystallographic x and z directions, respectively. The 2-D

NOSW has a width of 0.6 mm along the y direction. All the guiding surfaces of the two

crystals were optically polished. The ±x faces of the crystals were coated with anti-reflection

layers at the pump and signal wavelengths. The QPM domain period of the two PPLN

NOSWs is 65 µm, which permits phase matching for the generation of forward and backward

THz waves at λTHz = 197 and 469 µm, respectively, at room temperature with a pump

wavelength at λp = 1538.9 nm.

In our experiment, the waist radius of the optical pump and signal waves was 127-µm at

the center of the PPLN crystals. Since the aperture of the PPLN crystals is 4-5 times the waist

radius of the signal and pump waves, the PPLN crystals appear as a bulk material to the

optical mixing waves. However, the very same crystals act as a waveguide for the THz waves,

because the diffraction angles of the forward and backward THz waves are 0.1 and 0.23 rad,

respectively, for an initial beam radius comparable to that of the optical waves. The crystal

apertures are relatively large and capable of accommodating many waveguide modes for the

THz waves. However, the fundamental THz-wave mode overlaps well with the Gaussian

optical mode and is the dominant mode to build up in such a highly absorptive NOSW [6].

Figure 1(b) shows the schematic of the THz-wave DFG experiment. The pump and signal

waves are first combined from a fixed-frequency distributed-feedback diode laser (DFBDL) at

1538.9 nm and an external-cavity diode laser (ECDL) with its wavelength tuned to the phase

matching one for the downstream THz-wave DFG. An amplifier system, consisting of an

Erbium-doped fiber amplifier (EDFA) followed by a pulsed two-color optical parameter

amplifier (OPA), boosts each of the pump and signal energy to 9.7 µJ/pulse in a 360-ps pulse

width. The forward propagating signal and pump pulses were then focused to the center of the

PPLN NOSW. A silicon bolometer was installed before and after the PPLN NOSW to detect

the backward and forward THz waves, respectively [3].

Fig. 1. (a) Photograph of the three 2-D NOSWs fabricated from PPLN for our experiment.

Experimental data in this paper were taken from the longest one. (b) Schematic of the forward

and backward THz-wave DFG in PPLN NOSW. The pump and signal are initially combined

from a distributed feed-back diode laser (DFBDL) and a tunable external cavity diode laser

(ECDL), and then boosted up in power by an Erbium-doped fiber amplifier (EDFA) and a

pulsed optical parametric amplifier (OPA). A 4K silicon bolometer detects the backward and

forward THz waves before and after the PPLN NOSW, respectively.

The largest possible THz mode area is the aperture area of the 2-D PPLN NOSW, which

=×

rectangular function with a unit amplitude inside and zero amplitude outside the range of −l/2

can be described by

( , )

y z

rect( / ) rect( / )/,

THz

e

yz y z

l ly lz l

where rect( / )

r l

is a

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< r < l/2. The maximum gain reduction factor thus estimated from Eq. (6) is

0.29

pϑ ≈

A

,

which is about the square root of the area ratio

mismatch. Therefore, the actual parametric gain coefficient Γ for the 2-D NOSW could be

reduced to about 1/3 of its free-space plane-wave value. Given deff = 168 × 2/π = 107 pm/V

[9] and np = ns = 2.14 [10], the reduced parametric gain is estimated to be

0.65, 0.44

Γ = Γ=

A

cm−1 for the forward and backward THz waves with refractive

/

p THz

AA

, as expected for a large mode-area

0

pϑ

indices of 5.22 and 5.05 [3], respectively. For the 1-D NOSW, the aforementioned theory is

not valid due to the fast variation of the THz wave beam size along the propagation direction.

However, one would expect a smaller effective parametric gain coefficient Γ and thus a

smaller growth rate for the THz wave resulting from worsened mode mismatch and

diffraction-induced absorption in the 1-D NOSW.

Figure 2(a) shows the measured DFG tuning curves for the forward THz waves at 197 µm

generated from the 1-D (crosses) and 2-D (dots) PPLN NOSWs with a pump intensity of 102

MW/cm2. In the plot, the dashed and continuous lines are fitting curves using Eq. (4) with Γ =

0.53 and 0.65 cm−1 for the 1-D and 2-D NOSWs, respectively, given αTHz = 40 cm−1 at 1.5

THz for congruent lithium niobate [1]. As expected, the effective parametric gain coefficient

for the 1-D NOSW is smaller due to diffraction of the THz wave in the y direction. The

parametric gain of Γ = 0.65 cm−1 for the 2-D NOSW agrees well with the theory for a large

mode-area mismatch. The measured tuning curves clearly show an enhancement factor of 1.6

at the phase matching wavelength for the THz-wave output power from the 2-D NOSW.

Figure 2(b) shows the THz-wave output power versus pump intensity from the 1-D (squares)

and 2-D (circles) NOSWs at the phase matching wavelength. During the measurement, the

ratio of the pump to signal intensity remained one. The power enhancement factor of the THz

wave is nonlinearly increased with the pump intensity, which indicates some exponential gain

for the THz wave as predicted by Eq. (4). The estimated forward THz-wave pulse energies

generated inside the 1-D and 2-D NOSWs are 41 and 63 pJ, respectively.

180 180190

wavelength (microns)

(a)(a)

200 200210 210220 220

00

0.2 0.2

0.4 0.4

0.60.6

0.80.8

11

forward THz-wave power (arbitrary unit)

1-D fitting curve

1-D data

2-D fitting curve

2-D data2-D data

2020 40

pump/signal intensity (MW/cm2)

(b)(b)

6060 8080 100 100

00

0.2 0.2

0.40.4

0.60.6

0.80.8

11

forward THz-wave power (arbitrary unit)

190

wavelength (microns)

forward THz-wave power (arbitrary unit)

1-D fitting curve

1-D data

2-D fitting curve

40

pump/signal intensity (MW/cm2)

forward THz-wave power (arbitrary unit)

Fig. 2. (a) Measured forward THz-wave tuning curves from the 1-D (dots) and 2-D (crosses)

PPLN NOSWs. The dashed and continuous lines are fitting curves of Eq. (4) with Γ = 0.53

and 0.65 cm−1 for the 1-D and 2-D NOSW, respectively, given an attenuation coefficient of 40

cm−1. (b) The Measured THz-wave output power versus pump intensity from the 1-D (squares)

and 2-D (circles) NOSWs, indicating some nonlinear THz-wave power enhancement in the 2-

D NOSW.

Figure 3(a) shows the measured DFG tuning curves for the backward THz wave at 469

µm generated from the 1-D (cross) and 2-D (dot) PPLN NOSWs with a pump intensity of 104

MW/cm2. The dashed and continuous lines are fitting curves using Eq. (7) with Γ = 0.34 and

0.44 cm−1 for the 1-D and 2-D NOSWs, respectively, and with an assumed attenuation

coefficient of 6 cm−1. The reduced parametric gain coefficient for the 1-D NOSW also

indicates a poorer beam overlap between the backward THz and forward optical waves due to

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