Optical phase conjugation in difference-frequency generation

Mikhailov et al.

Vol. 20, No. 8 / August 2003 / J. Opt. Soc. Am. B

1715

Optical phase conjugation in difference-frequency generation Viktor N. Mikhailov Department of Photophysics of Holographic Processes, Research Center, S. I. Vavilov State Optical Institute, Birjevaya 12, St. Petersburg 199034, Russia

Maria Bondani, Fabio Paleari, and Alessandra Andreoni Dipartimento di Scienze Chimiche, Fisiche e Matematiche, Universita` degli Studi dell’Insubria and Istituto Nazionale di Fisica della Materia, Unita` di Como, Via Valleggio 11, Como 22100, Italy Received December 18, 2002; revised manuscript received April 10, 2003 We present a theoretical treatment of difference-frequency generation in conditions of phase mismatch for seed and generated fields that propagate at different angles to the pump. The analytical solution in the case of nondepleted plane-wave pump and experiments performed with seed fields strongly phase and amplitude modulated show that the pump-field wave fronts behave as efficient phase-conjugating mirrors. © 2003 Optical Society of America OCIS codes: 190.5040, 190.2620, 100.5070, 140.3280.

1. INTRODUCTION In the nonlinear optics literature, optical phase conjugation (OPC), a process whose earliest experimental realization was by stimulated Brillouin scattering,1 is more often associated with four-wave mixing than three-wave mixing interactions.2,3 As new photorefractive materials were developed, OPC by four-wave mixing at frequency degeneracy became a more and more popular technique for dynamic (real-time) holography. Similarly the increased availability of highly nonlinear ␹ ( 3 ) materials with fast responses stimulated many applications to ultrashort pulses. Among these we mention the correction of the temporal distortions caused by either dispersion at different orders or self-phase modulation of pulses during their propagation and amplification.4,5 However, since the 1970s it has been recognized that difference-frequency generation (DFG), a phenomenon that relies on ␹ ( 2 ) nonlinearity, can be used as a technique to produce OPC.3,6,7 The possibility of generating idler fields endowed with space-dependent phase reversal with respect to a signal field by DFG at frequency degeneracy was first exploited to recover wave fronts from subwavelength distortions,7 from strong disturbances or aberrations such as those that affect amplified laser beams,8 or from phase modulations imprinted by optical systems characterized by simple transfer functions.9 We also contributed to the latter field in recent years by showing that the idler field is strictly a holographic replica of the signal field, even for object, i.e., signal, fields with complicated wave-front shapes.10,11 Here we consider space-dependent phase conjugation for strongly diverging wave fronts affected by random phase disturbances. We develop a theory for noncollinear propagation of signal and idler fields and for a nondepleted plane-wave pump that includes the effects of 0740-3224/2003/081715-09$15.00

phase mismatch. We present experiments in which the pump field is provided by a frequency-doubled amplified Nd:YLF with a coherence length ⬎3 m. We show that OPC in a low-gain regime allows full recovery of a beam at the fundamental frequency that passes through a diffusing glass plate. Our research, together with the availability of nonlinear crystals with noticeable crosssectional sizes and wide acceptance angles, made of materials with high ␹ ( 2 ) nonlinearity, is a step forward in the use of DFG for image detection through scattering media. In fact complete cancellation of the distortion effects induced by a nonabsorbing scatterer is expected if the OPC wave is generated without loss or gain at an infinite phase-conjugate mirror.12

2. GENERAL THEORY The type I interaction depicted in Fig. 1 that occurs in a ␹ ( 2 ) nonlinear medium among three noncollinearly propagating amplitude-modulated plane waves, E1 (r, t), E2 (r, t), and E3 (r, t), with wave vectors k1,2,3 (unit vectors kˆ 1,2,3 ) can be described by Eqs. (2) in the paper by Bondani and Andreoni.10 In the parametric approximation, in which the amplitude of the field at the highest frequency (pump field E3 at ␻ 3 ) is virtually independent of r, we write the signal and idler fields, E1 (r, t) and E2 (r, t), with ordinary polarization parallel to xˆ , as

E1 共 r, t 兲 ⫽

冉

xˆ 2 ␩ 0 ប ␻ 1 2

n1

冊

1/2

⫻ 兵 a 1 共 r兲 exp关 ⫺i 共 k1 – r ⫺ ␻ 1 t 兲兴 ⫹ c.c.其 , (1.1) © 2003 Optical Society of America

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J. Opt. Soc. Am. B / Vol. 20, No. 8 / August 2003

E2 共 r, t 兲 ⫽

冉

xˆ 2 ␩ 0 ប ␻ 2 2

n2

冊

Mikhailov et al.

1/2

g eff ⫽ 共 d 22 cos ␣ ⫹ d 31 sin ␣ 兲

⫻ 兵 a 2 共 r兲 exp关 ⫺i 共 k2 – r ⫺ ␻ 2 t 兲兴 ⫹ c.c.其 , (1.2) where ␻ 1 ⫹ ␻ 2 ⫽ ␻ 3 for energy conservation, ␩ 0 is the vacuum impedance, and n j are the refractive indices at the corresponding frequencies ␻ j . Note that 兩 a j (r) 兩 2 has the dimension of a photon flux density. The equations that describe the interaction reduce to kˆ 1 • “a 1 共 r兲 ⫽ ig eff a 3 共 0 兲 a 2* 共 r兲 exp共 ⫺i⌬k – r兲 , (2.1) kˆ 2 • “a 2 共 r兲 ⫽ ig eff a 3 共 0 兲 a 1* 共 r兲 exp共 ⫺i⌬k – r兲 , (2.2) where r is the position inside the crystal with respect to an origin located on the entrance face where the constant pump-field amplitude, a 3 (0) ⫽ a 3 (r ⫽ 0), is taken. As shown in Fig. 1, with the z axis perpendicular to the crystal entrance face, r ⬅ (0, y, z) is in the ( y, z) plane, which is also the principal plane. Thus the gradient in Eqs. (2.1) and (2.2) is ⵜ ⫽ yˆ ⳵ / ⳵ y ⫹ zˆ ⳵ / ⳵ z with yˆ and zˆ as unit vectors along the axes. The fact that wave vectors k1,2,3 are at different angles ␽ 1,2,3 shown in Fig. 2 with respect to the z axis gives rise, in general, to an error in the phase-matching (PM) condition given by ⌬k ⫽ k3 ⫺ k1 ⫺ k2 . Finally g eff is the effective coupling coefficient that takes into account that the pump field enters the crystal as a transverse wave.10 For ␤ -BaB2 O4 in type I (BBO I) PM conditions, g eff has the following expression:

冉

2ប ␻ 1 ␻ 2 ␻ 3 ␩ 03 n 1n 2n 3

冊

1/2

,

(3)

where ␣ is the angle between the optical axis and the pump wave vector k3 (see Fig. 1) and d represents the relevant coupling coefficients.13 It is worth noting that Eqs. (2) include the conservation law, which is our Manley– Rowe relation: ⵜ 兩 a 1 共 r兲兩 2 • kˆ 1 ⫽ ⵜ 兩 a 2 共 r兲兩 2 • kˆ 2 ,

(4)

which links the amplitude square magnitude of the amplified (AMP) signal field E1 (r, t) with that of the DFG field E2 (r, t). The solutions to Eqs. (2) for boundary conditions a 1 (0) ⫽ 0 and a 2 (0) ⫽ 0 are

再冋冉冊册冋冉冊册冎 ⌬kˆ

a 1 共 r兲 ⫽ a 1 共 0 兲 cosh Q

⫹

⌬kˆ

i⌬k Q

sinh Q

冋冉

⫻ exp ⫺i

a 2 共 r兲 ⫽ a 1* 共 0 兲

⌬k 2

2

•r

2iA 3

冋冉

⌬k 2

•r

冊册

,

(5.1)

冋冉

sinh Q

共 ⌬kˆ • kˆ 2 兲 Q

⫻ exp ⫺i

•r

2

•r

冊册

⌬kˆ

•r

2

冊册

,

(5.2)

where Q⫽

Fig. 1. Geometry of the type 1 interaction among three noncollinearly propagating fields, E1 , E2 , and E3 , that propagate in different directions (wave vectors k1,2,3 ) in the ( y, z) plane; ␹ (2) nonlinear medium at z ⬎ 0.

冋

4 兩 A 3兩 2 共 ⌬kˆ • kˆ 1 兲共 ⌬kˆ • kˆ 2 兲

⫺ ⌬k 2

册

1/2

,

(6)

with A 3 ⫽ g eff a3(0) and ⌬kˆ ⫽ ⌬k/⌬k. To demonstrate the space-dependent phases of the fields we rewrite them [see Eqs. (1)] in the following form: E1 共 r, t 兲 ⫽ xˆ

E2 共 r, t 兲 ⫽ xˆ

冉冉

2 ␩ 0ប ␻ 1 n1 2 ␩ 0ប ␻ 2 n2

冊冊

1/2

兩 a 1 共 r兲兩 ⫻ cos关 ␸ 1 共 r兲 ⫹ ␻ 1 t 兴 ,

(7.1) 1/2

兩 a 2 共 r兲兩 ⫻ cos关 ␸ 2 共 r兲 ⫹ ␻ 2 t 兴

(7.2)

so that for the phases from Eqs. (5) we get

␸ 1 共 r兲 ⫽ ␸ 1 共 0 兲 ⫺ k1 –r ⫺ ⫹ tan⫺1 Fig. 2. Wave vectors k1,2,3 at angles ␽ 1,2,3 to the z axis, traces of the k1 and k2 surfaces (both spherical, i.e., type I interaction) in the ( y, z) plane, and ⌬k ⫽ k3 ⫺ k1 ⫺ k2 . The dashed line bisects the k1 -to-k2 angle.

再

⌬k Q

⌬k 2

冋冉

tanh Q

␸ 2 共 r兲 ⫽ ⫺␸ 1 共 0 兲 ⫹ ␸ 3 共 0 兲 ⫹ whose sum is

•r

␲ 2

⌬kˆ 2

•r

冊册冎

⫺ k2 –r ⫺

,

⌬k 2

(8.1)

•r

(8.2)

Mikhailov et al.


␸ 1 共 r兲 ⫹ ␸ 2 共 r兲

Moreover, after calculation of the time-averaged Poynting vectors, that is (see Appendix A),

⫽ ␸ 3 共 r兲 ⫹

␲ 2

⫹ tan⫺1

再

⌬k Q

冋冉

tanh Q

⌬kˆ

•r

2

冊册冎

,

(8.3) S1 共 r 兲 ⫽

where ␸ 3 (r) ⫽ ␸ 3 (0) ⫺ k3 – r is the phase of E3 (r, t) and, for the magnitudes of the amplitudes, 兩 a 1 共 r兲兩 ⫽

再

兩 a 1共 0 兲兩

Q

Q ⫹

冋冉

⫻ sinh Q 2

兩 a 2 共 r兲兩 ⫽ 兩 a 1 共 0 兲兩

4 兩 A 3兩 2

2

冊册冎冋冉 1/2

•r

2

2 兩 A 3兩

,

sinh Q

共 ⌬kˆ • kˆ 2 兲 Q

(9.1) ⌬kˆ 2

•r

冊册

.

(9.2)

The solutions represent two waves, AMP and DFG, that amplify while they propagate in the medium for real Q. After squaring Eqs. (9) and summing we obtain 兩 a 1 共 r兲兩 2 ⫽ 兩 a 1 共 0 兲兩 2 ⫹

⌬kˆ • kˆ 2 ⌬kˆ • kˆ 1

兩 a 2 共 r兲兩 2

(10)

in agreement with Eq. (6). In fact, by calculating “ 兩 a 1,2(r) 兩 2 from Eqs. (9) we obtain “ 兩 a 1 共 r兲兩 2 ⫽

兩 a 1共 0 兲兩 2

4 兩 A 3兩 2

Q

共 ⌬kˆ • kˆ 1 兲共 ⌬kˆ • kˆ 2 兲

冋冉

⫻ sinh Q

“ 兩 a 2 共 r兲兩 2 ⫽

兩 a 1共 0 兲兩

Q

⌬kˆ 2

4 兩 A 3兩

2

•r

共 ⌬kˆ • kˆ 2 兲 2

冋冉

⫻ cosh Q

⌬kˆ 2

冊册冋冉 cosh Q

冋冉

2

•r

sinh Q

冊册

⌬kˆ

⌬kˆ 2

2

•r

•r

冊册

⌬kˆ .

冊册

⌬kˆ ,

(11.2)

⌬k 2

⫹

⌬k 2

⫻

兩 a 1共 0 兲兩 2 兩 a 1 共 r兲兩 2

,

(12.1)

whereas from Eq. (8.2) we obtained “ ␸ 2 共 r兲 ⫽ ⫺k2 ⫺

⌬k 2

ប␻2 k2

⫺

2

⌬k 2

冉

兩 a 2 共 r兲兩 2 k2 ⫹

S1 共 r兲 ⫽ ប ␻ 1 兩 a 1 共 0 兲兩 2 kˆ 1 ⫹

冉

⫻ k1 ⫹

⌬k 2

冊

兩 a 1共 0 兲兩 2

⫻

⌬k 2

冊

册

, 兩 a 1 共 r兲兩 2 (13.1)

,

(13.2)

ប␻1 k1

兩 a 2 共 r兲兩 2

⌬k • kˆ 2 ⌬k • kˆ 1

,

(14)

in which the former term is the contribution of the seed field at the crystal entrance and the latter is that due to seed amplification. It is worth noting at this point that, as in Ref. 10, in deriving Eqs. (2) from the Helmholtz equations with sources, we chose to neglect the second derivatives of a 1,2(r) with respect to y and z. We must then check a posteriori under which conditions the solutions in Eqs. (5) verify such a slowly varying envelope approximation. This is done in Appendix B where we show that such a condition is ⌬k/2 Ⰶ k 1 , k 2 ,

(15)

provided that both ⌬kˆ • kˆ 1 and ⌬kˆ • kˆ 2 are sufficiently different from zero. In the solutions in Eqs. (5), because of boundary condition a 2 (0) ⫽ 0, ⌬k appears as a free parameter, with kˆ 2 univocally linked to ⌬k as depicted in Fig. 2. This means that, given a pump field ( 兩 A 3 兩 and k3 assigned) for any seed field that enters the crystal with certain 兩 a 1 (0) 兩 and k1 , the system of Eqs. (2.1) and (2.2) provides solutions associated with any orientation of kˆ 2 (angle ␽ 2 ) corresponding to a real Q. Such an indetermination in kˆ 2 is obviously not encountered in PM conditions where k2 ⫽ k3 ⫺ k1 . When we solved Eqs. (2) in PM, we found10,14

冋

兩 A 3兩

兩 a 2PM共 r兲兩 ⫽ 兩 a 1 共 0 兲兩 sinh

(12.2)

and conclude that the fields in Eqs. (7), whose spacedependent complex amplitudes are constant in magnitude on planes perpendicular to ⌬k [see Eqs. (9) with constant ⌬kˆ • r values and Eqs. (11)], are such that field E2 (r, t) is always a plane wave with wave fronts that propagate along wave vector k2 ⫹ (⌬k/2) and that field E1 (r, t) also tends to become a plane wave with wave fronts that propagate along wave vector k1 ⫹ (⌬k/2) when the gain increases. Note that the sum of these wave vectors matches the pump-field wave vector k3 for any ⌬k.

⌬k

we find that for each field the power flows normally to the wave front although it grows maximally in a different direction 关 r ⫽ r⌬kˆ , see Eqs. (11)]. We finally note that, by inserting Eq. (10) into Eq. (13.1), this can be rewritten as

(11.1)

To find the propagation directions of the constant-phase surfaces we calculated “ ␸ 1 (r) and “ ␸ 2 (r). From Eq. (8.1) with the help of Eq. (9.1) we obtained “ ␸ 1 共 r兲 ⫽ ⫺k1 ⫺

k1

冋

兩 a 1 共 r兲兩 2 k1 ⫹

S2 共 r兲 ⫽

共 ⌬kˆ • kˆ 1 兲共 ⌬kˆ • kˆ 2 兲

⌬kˆ

ប␻1

1717

冉

⫻ sin

cos

␽1 ⫺ ␽2

␽1 ⫹ ␽2 2

2 y ⫹ cos

␽1 ⫹ ␽2 2

z

冊册

,

(16) where, as shown in Fig. 2, ( ␽ 1 ⫹ ␽ 2 )/2 is the angle formed by the bisector of the kˆ 1 -to-kˆ 2 angle with the z axis and ( ␽ 1 ⫺ ␽ 2 )/2 is one-half of the kˆ 1 -to-kˆ 2 angle. If we write Eq. (9.2) for ⌬k ⫽ 0,

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3. EXPERIMENTAL RESULTS AND DISCUSSION

Fig. 3. As in Fig. 2 with ⌬k parallel to the bisector (dashed line) of the k1 -to-k2 angle; see Eq. (18). The dotted circle (center C) is tangent to the k1 and k2 surfaces.

兩 a 2 共 r兲兩 ⫽ 兩 a 1 共 0 兲兩

冉

⌬kˆ • kˆ 1 ⌬kˆ • kˆ 2

冋

⫻ sinh

冊

1/2

兩 A 3兩

冑共 ⌬kˆ • kˆ 1 兲共 ⌬kˆ • kˆ 2 兲

册

共 ⌬kˆ • r兲 ,

(17) 兩 a 2PM(r) 兩

and pretend that this expression converges to in Eq. (16) on approaching the PM condition, we find that ⌬kˆ must lie along the bisector of the kˆ 1 -to-kˆ 2 angle: ⌬kˆ • kˆ 1 ⫽ ⌬kˆ • kˆ 2 ,

(18)

as depicted in Fig. 3. In Fig. 3 the shaded triangle of base ⌬k is an isosceles as emphasized by the dotted circle whose point C, being centered at the opposite vertex, is tangent to both the k1 and the k2 surfaces. Thus, when k1 and k2 almost obey momentum conservation, k1 ⫹ k2 ⬵ k3 , ⌬k is certainly far from being perpendicular to either k1 or k2 and the fulfillment of the condition in Eq. (15) is enough to grant the validity of our slowly varying envelope approximation. Moreover, by using Eqs. (15) and (18) in Eqs. (14) and (13.2) we obtain 兩 S2 共 r 兲兩

ប␻2

⬵

兩 S1 共 r 兲兩

ប␻1

⫺ 兩 a 1共 0 兲兩 2,

(19)

which shows that for each DFG photon created a photon is added to the AMP field. Since this property is general for this ␹ ( 2 ) interaction and we found it by making assumptions that are independent of the proximity to PM, we assume that the constraint on ⌬kˆ and kˆ 2 in Eq. (18) also applies when the phase mismatch is noticeable. We thus conclude that any DFG experiment with seed-field amplification in which the pump depletion is negligible can be interpreted by use of Eqs. (5), namely, Eqs. (9) for the magnitudes of the field space-dependent amplitudes and Eqs. (8) for the field space-dependent phases, together with Eq. (18) provided Eq. (15) is fulfilled and Q is real, that is, from Eq. (6): 共 ⌬k • kˆ 1 兲共 ⌬k • kˆ 2 兲 ⭐ 4 兩 A 3 兩 2 .

The experiments presented in this paper were performed at frequency degeneracy ( ␻ 1 ⫽ ␻ 2 ⫽ ␻ ). The intense plane wave to be used as the pump field ( ␻ 3 ⫽ 2 ␻ ) was provided by the frequency-doubled output of a highcoherence pulsed Nd:YLF laser (Green Star, Neva Technologies, Ltd., St. Petersburg, Russia). The residual fundamental pulse provided the seed field. This Q-switched source emits 25-ns pulses at the fundamental (1053 nm) and includes a double-pass Nd:glass amplifier. A phaseconjugating mirror based on Brillouin scattering reflects the pulse back to the amplifier between the two passes. After frequency doubling in a KTP crystal the coherence length is ⬎3 m and the energy is well above 1 J. Both fundamental and frequency-doubled beams have flat-top profiles with 9-mm diameter. The polarization of the output at 527 nm (pump field at ␻ 3 ) was in the horizontal plane and that of the 1053 nm (seed field at ␻ 1 ) was in the vertical plane, so that the ( y, z) plane in Fig. 1 was horizontal. By using the setup in Fig. 4, we performed two distinct series of experiments in BBO. The first, which we designate as exp-I in what follows, was based on a type I interaction and used the configuration depicted in Fig. 4(a). The second, exp-II, was based on a type II interaction and implied the superimposition of a strong phase disturbance on the signal through the scheme sketched in Fig. 4(b). In exp-I we used a BBO crystal (Fujian Castech Crystals, Inc., Fuzhou, China) with a 5 mm ⫻ 5 mm cross section and thickness of d ⫽ 2 mm, cut for phase-matched collinear secondharmonic generation at normal incidence by use of Nd:YAG [BBO I in Fig. 4(a)]. In exp-II the BBO crystal (10 mm ⫻ 10 mm cross section, 3-mm thickness, same manufacturer) was cut for collinear Nd:YAG secondharmonic generation at normal incidence in a type II PM condition [BBO II in Fig. 4(b)]. The space filter on the

(20)

Finally note that the extension of the theory to other nonlinear materials as well as to type II interactions requires only minor changes [see, for example, Eq. (3)].

Fig. 4. (a) Layout of the experimental setup (general) and details of (b) exp-I and (c) exp-II. See text for the nonlinear crystals BBO I (exp-I) and BBO II (exp-II). Lenses L1–L7, pinholes PH1 and PH2, mirrors M1–M4, neutral and bandpass filters F1– F3, photodiodes PD1 and PD2; BS1, high-energy harmonic separator (wedged substrate); DM, high energy dichroic mirror; DIFF, diffusing glass plate; P, film polarizer.

Mikhailov et al.


1719

Canada]. The pump intensity values were then calculated from which we obtained the 兩 A 3 兩 2 [see Section 1, 2 ⫽ (8.83019 ⫻ 10⫺14) 2 s]. In Fig. 5 we plot the integ eff grals that represent the ratio of DFG field photon flux (at BBO I output) to seed photon flux (at BBO I input) as a function of 兩 A 3 兩 2 . The points in Fig. 5 are well fitted by a straight line with slope (4.52 ⫾ 0.1) ⫻ 10⫺6 m2 up to the maximum pump intensity value of 203 MW/cm2. To interpret this result we first observe that, in the case at hand, where we can assume that kˆ 3 ⫽ zˆ , because of Eq. (18) ⌬kˆ ⫽ kˆ 3 (⫽zˆ ). Therefore when we square Eq. (9.2), we obtain

Fig. 5. Photon flux in the DFG field at BBO I output ⌽ 2 (r), normalized to seeded photon flux ⌽ 1 (0), as a function of pump photon flux density 兩 A 3 兩 2 (lower scale). Upper scale, pump intensity; straight line: least-squares fit of experimental points; inset, typical fluence map of a DFG spot obtained with the highest pump intensity.

527-nm beam consisting of lenses L4 and L5 plus pinhole PH2 shapes the laser output into a Gaussian beam of approximately 3 mm FWHM with a Rayleigh range much greater than any relevant distance for our experiments. Lenses L2 and L3 and pinhole PH1 produced a similar beam at the fundamental frequency. To detect fluence maps we used a CCD camera [pixel size 8.6 ␮m (H) ⫻ 8.3 ␮m (V), Model TM-6CN Pulnix Europe, Ltd., Basingstoke, UK] with suitable color filters in front to reject the green pump-field light. In the first experiment of series exp-I, in which we measured the ratio of the energy of the DFG pulse to that of the seed pulse at the crystal entrance for different pump-field intensities, the pump beam at 2␻ and the seed beam at ␻ were set to be collinear and centered on each other after the dichroic mirror (DM). The seed field that impinges onto the crystal [BBO I in Fig. 4(a)] propagated from the focal spot of a lens (L7 in the figure, focal length f ⫽ 110 mm) positioned at distance 2f from BBO I to illuminate it over the same area covered by the pump beam. Lens L7 was carefully aligned for axial kˆ 1 coinciding with kˆ 3 and, because of the nonideal crystal cut (YAG versus YLF), preliminary to the DFG experiment the crystal was angle tuned to generate a second-harmonic spot of the seed beam on axis when the pump beam was stopped. This procedure ensured the fulfillment of the (collinear) PM I condition for the axial kˆ 1 . The CCD was located on axis and moved longitudinally until, at 110 mm from BBO I, the DFG spot could be detected with minimum size over the background of the AMP seed beam (see the intensity map inserted in Fig. 5). A series of CCD frames were also acquired in the absence of pump and the corresponding pixel values normalized to the signal of the photodiode that monitored the seed pulse (PD1 in Fig. 4). The same normalization was performed for the frames that display the DFG spot over the AMP background and the pixel-to-pixel differences were integrated over the DFG spot. For each measurement the pump pulse was also monitored by photodiode PD2 as shown in Fig. 4. The PD2 response was calibrated off-line by using an energy meter [Model DE-500 (2.2 V/J), Gentec, Quebec,

兩 a 2 共 r兲兩 ⫽ 兩 a 1 共 0 兲兩兩 A 3 兩 2

2

2

共 zˆ • r兲 2

cos2 ␽ 2

冋

冦

sinh

Q 2

Q 2

共 zˆ • r兲

共 zˆ • r兲

册

冧

2

(21)

to be substituted into Eq. (13.2) for calculating S2 (r). Note that this Poynting vector and 兩 a 2 (r) 兩 2 in Eq. (21) refer to a seed constituted by a single plane wave of given kˆ 1 and 兩 a 1 (0) 兩 , whereas we measured integral photon fluxes in the spot formed by the DFG field on the CCD sensor plane when many kˆ 1 unit vectors symmetrically spread in angle about the single kˆ 3 of the pump impinged onto the crystal. Moreover we normalized these integral photon fluxes to those of the seed field entering the crystal. Such normalized integral fluxes were found to increase linearly with 兩 A 3 兩 2 as shown in Fig. 5. Since we set the CCD sensor perpendicular to kˆ 3 , we use Eq. (A1.4) with ˆ ⫽ kˆ ⫽ zˆ and Eq. (13.2), together with Eq. (21), with r A 3 such that ⌬kˆ • r ⫽ zˆ • r( ⫽ d) is constant to calculate the photon flux density ⌽ 2 (r) generated in response to a single spatial Fourier component (i.e., single kˆ 1 at angle ␽ 1 to kˆ 3 ) of seed field 1. At degeneracy (k 1 ⫽ k 2 ⫽ k ␻ ), we find

⌽ 2 共 r兲 ⫽ 兩 a 1 共 0 兲兩 2 兩 A 3 兩 2

冉

⫻ kˆ 2 ⫹

d2 cos ␽ 1 2

⌬k 2k ␻

⬵ 兩 a 1共 0 兲兩 2兩 A 3兩 2

冊

冦

冋

sinh

Q 2

Q 2

共 zˆ • r兲

册

冧

2

册

冧

2

共 zˆ • r兲

zˆ • zˆ

d2 cos2 ␽ 1

冦

冋

sinh

Q 2

Q 2

共 zˆ • r兲

共 zˆ • r兲

kˆ 1 • zˆ ,

(22) since ⌬k/2 Ⰶ k ␻ and kˆ 1 • zˆ ⫽ kˆ 2 • zˆ (namely, ␽ 2 ⫽ ⫺␽ 1 ) according to Eqs. (15) and (18), respectively. By recognizing that 兩 a 1 (0) 兩 2 kˆ 1 • zˆ ⫽ ⌽ 1 (0) is the photon flux density associated with our single plane-wave component at angle ␽ 1 , we can recast Eq. (22) into the following form:

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Mikhailov et al.

⌽ 2 共 r兲 ⌽ 1共 0 兲 sinh2 ⫽ 兩 A 3兩 2d 2

再冋冋

cos ␽ 1 2

4 兩 A 3兩 2 cos2 ␽ 1 4 兩 A 3兩 2 cos2 ␽ 1

册冎册冉冊 1/2

⫺ 4k ␻2 共 1 ⫺ cos ␽ 1 兲 2 ⫺

4k ␻2 共 1

⫺ cos ␽ 1 兲

2

d

2

d

2

,

2 (23)

where we made use of Eq. (6) with k 1 ⫽ k 2 ⫽ k ␻ . The maximum deviation of the fraction on the right-hand side of Eq. (23) from the value 1 occurs for ␽ 1 ⫽ 0. For the 兩 A 3 兩 value corresponding to the highest pump intensity used in the experiment (203 MW/cm2), the fraction value is ⬃1.06. It is then reasonably correct to fit the experimental points in Fig. 5 by a straight line and interpret its slope as d 2 . Our fitting gives a d value of 2.13 mm, to be compared with the nominal thickness of 2 mm. The fact that the fitting line in Fig. 5 is slightly shifted with respect to the origin of the plot might be due to (small) offsets either in the measurements of the pump-pulse energies from which we calculated the 兩 A 3 兩 2 values (abscissa in Fig. 5) or in those performed with CCD and photodiodes on the DFG and seed beams that were necessary to construct the ratios ⌽ 2 (r)/⌽ 1 (0) that were plotted on the vertical scale. Lefort and Barthelemy conducted a similar experiment9 and, although they worked with a 7-mmthick KTP crystal in type II PM conditions, they obtained a conversion efficiency that agrees with Eq. (23). As to the phase relations among the three interacting fields we note that the space-dependent phase of pump, DFG, and AMP fields are linked to each other by Eq. (8.3), which is well approximated by

␸ 1 共 r兲 ⫹ ␸ 2 共 r兲 ⫽ ␸ 3 共 r兲 ⫹ ␲ /2,

(24)

when tan⫺1兵(⌬k/Q)tanh关Q(⌬kˆ • r)/2兴其 Ⰶ ␲ /2. In this condition, if point r inside the nonlinear crystal is chosen on plane ⌸ of Fig. 6, perpendicular to kˆ 3 , we would find spatial phase reversal between fields 1 and 2 on ⌸. This property leads to a number of effects that are observable outside the crystal. We note, for example, that the DFG field (field 2), after propagation from r苸⌸ to r⬘苸⌸ in the

Fig. 6. Phase relations according to Eq. (24). The pump-field phase on plane ⌸ is constant, (e.g., ␸ 3 (P) ⫽ ⫺␲ /2). See text for the values of ␸ 1 and ␸ 2 at P and ␸ 1 and ␺ 2 at origin O.

Fig. 7. Fluence maps of the seed field after removal of the diffusing glass plate, DIFF in Fig. 4(c), and in the absence of the pump as detected by the CCD camera at distances of (a) 51 and (b) 291 cm from BBO II.

absence of interaction, would achieve a phase value ␺ 2 (r⬘ ) ⫽ ␸ 2 (r) ⫺ k2 • (r⬘ ⫺ r). By substituting Eq. (8.2), it is easy to show that, if r⬘ coincides with the origin (see Figs. 4 and 1), then ␺ 2 (0) ⫽ ⫺␸ 1 (0) ⫹ ␸ 3 (0) ⫹ ␲ /2 ⫺ (⌬k/2) • r. Since within the limit Q 关 (⌬kˆ /2) • r兴 → 0, that is, in a regime of linear amplification [see Eqs. (5)], the condition for the validity of Eq. (24) amounts to a request for tan⫺1关(⌬k/2) • r兴 Ⰶ ␲ /2. Under this condition, if the DFG field 2 is backpropagated by means of a plane mirror, it will reach the origin with a phase value ␺ 2 (0) ⫽ ⫺␸ 1 (0) ⫹ const. Note that the above condition for frequency degeneracy and for kˆ 3 ⫽ zˆ becomes tan⫺1关(⌬k/2)(zˆ • r) 兴 ⫽ tan⫺1关k␻(1 ⫺ cos ␽1)d兴 Ⰶ ␲/2, which is easily fulfilled. For the experiment presented below (d ⫽ 3 mm, ␽ 1 ⭐ 0.25 deg), we calculated a maximum value of ⬃0.087␲. Although by operating with a different laser source we already demonstrated that the simple fact in Eq. (5.2), a 2 (r) ⬀ a 1* (0), allows us to obtain DFG fields that are capable of reconstructing holographic replicas of the wave fronts of the seed field with high spatial/angular resolution.10,11,14 The experiment (exp-II) we present here is aimed at demonstrating that the phase reversal expected from Eq. (24) is so accurate as to allow corrections of remarkable phase disturbances deliberately introduced in the seed field that enters the crystal. We substituted pinhole PH1 (see Fig. 4) with a micromask that, in the absence of interaction, produced a beam at ␻ with the intensity distributions displayed in Figs. 7(a) and 7(b). The two maps were detected by the CCD at distances of 51 and 291 cm from the nonlinear crystal, respectively (see below for details). With the beams at ␻ and at 2␻ aligned as in the previous experiment, in exp-II we passed such a seed beam through a diffusing glass plate, DIFF in Fig. 4(b), which was located 55 mm in front of the nonlinear crystal. The DIFF plate introduced a beam divergence of 0.5 deg (full angle at 1/e 2 ) and produced the intensity distribution shown in Fig. 8. The ex-

Mikhailov et al.

traordinarily polarized DFG field that emerged from BBO II was reflected back to the DIFF plate by a plane mirror (M4) located behind the BBO II crystal as near as possible (2 mm) to its exit face and perpendicular to the pump beam. Obviously the phase distortions operated by the diffusing plate on the reflected DFG field are identical to those operated on the beam at ␻ used as the seed for the nonlinear interaction.12 The reflected DFG field, behind the DIFF plate, was deflected by a beam splitter [BS2 in Fig. 4(b)] toward the CCD camera. To prevent the DFG light transmitted by BS2 from reentering the laser amplifier, we chose PM II and inserted a plastic polarizer (P in the figure) to stop it. The cross-sectional intensity distribution of the reflected DFG field after backpropagation through the diffusing plate detected at distances of 51 and 291 cm from BBO II are displayed in Figs. 9(a) and 9(b), respectively. The corresponding measurements performed on the seed beam reflected by mirror M4 after removal of the diffusing plate are those in Figs. 7(a) and 7(b). We note that the DFG beam in Figs. 9(a) and 9(b) is well collimated and has a profile similar in shape and size to that of the nondiffused seed beam in Figs. 7(a) and 7(b). Note that after propagation over meters, the diffused seed beam would exhibit a speckle pattern similar to that in Fig. 8 but enlarged to many-centimeter diameters. The accurate restoration produced by OPC of the nondiffused beam profile shows that, at the crystal output,

Fig. 8. Speckle pattern of the seed field measured at a distance of 6 cm from the diffusing glass plate.


1721

the DFG field has phase values that are point by point equal to the opposite of those of the AMP seed beam [see ␸ 2 (P) ⫽ ⫺␸ 1 (P) in Fig. 6]. In fact it is only in this case that backpropagation through the diffusing plate allows the DFG field to recover the phase disturbances of the seed-field wave front that the nonlinear interaction has oppositely copied onto the DFG wave front.

4. CONCLUSIONS By working with pump intensities below the damage threshold of BBO for long pulses we obtained OPC idler fields with 10% photon conversion efficiency per millimeter of BBO thickness (see Fig. 5). The high quality of the phase conjugation should allow one to use the technique to correct severe phase distortions imprinted on fields that carry image information by passage through scattering media or by transmission through fiber optics and optical components that cause aberrations. The feasibility of such an OPC with pulses of tens of nanosecond duration is relevant for application to the correction of phase aberrations brought about by laser amplification.8 Our theory shows that the OPC quality can be preserved on increasing ⌬k if a pump beam of suitable intensity were available. In fact, the error in phase conjugation [see the last term in Eq. (8.3)], which in the linearamplification regime of our experiments reduces to tan⫺1关⌬k(d/2) 兴 , becomes at most tan⫺1(⌬k/Q) in the highgain regime. Also this expression can represent a negligible phase error for sufficiently high pump intensities. Thus suitably short laser pulses could be used to obtain space–phase conjugation in DFG processes in which the seed-field spectrum may have a wider angular spread than that in our experiments. Our theory should still apply and could also be fully tested: properties that concern the DFG wave-vector propagation [see Eq. (18) and Fig. 3], the magnitude and propagation of the Poynting vectors [see Eqs. (13) and (14)], and the limit of validity in Eq. (20) can be investigated only at high pump intensity. Finally we note that, although we are interested in investigating space–phase conjugation by DFG with shorter pulses for the above reasons, others have recently proposed temporal–phase conjugation by DFG in BBO as an efficient method for compensating the effects of dispersion and nonlinearity on ultrashort pulses that propagate through optical fibers.15 This is an application that could greatly benefit from the use of a ␹ ( 2 ) instead of a ␹ ( 3 ) nonlinear interaction,5 with the further advantage that ␹ ( 2 ) media with negligible group-velocity dispersion are available and that they can often be operated in conditions of exact group-velocity matching.16,17 From this standpoint, BBO is an ideal material.17

APPENDIX A The time-averaged Poynting vectors, S1 and S2 , are given by Fig. 9. Fluence maps of the DFG field after reflection by mirror M4 in Fig. 4(c) and backpropagation through DIFF. The CCD camera is at distances of (a) 51 and (b) 291 cm from BBO II.

冋

S1,2共 r兲 ⫽ Re

1 2

册

* 共 r兲 , E1,2共 r兲 ⫻ H1,2

(A1)

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Mikhailov et al.

where E1,2(r) is the space-dependent part of fields * (r) is the complex conjugate of H1,2(r), E1,2(r, t) and H1,2 with “ ⫻ E1,2共 r兲 ⫽ ⫺i ␻ 1,2␮ 0 H1,2共 r兲 .

(A2)

Since, according to Eqs. (2) and (7) E1,2共 r兲 ⫽ xˆ

冉

2 ␩ 0 ប ␻ 1,2 n 1,2

冊

1/2

兩 a 1,2共 r兲兩 exp关 i ␸ 1,2共 r兲兴 ,

(A3)

if H1,2(r) is calculated from Eq. (A2) and substituted into Eq. (A1), we obtain the S1 and S2 vectors reported in Eqs. (13.1) and (13.2), respectively. Note that, according to Eq. (A1), the intensities, in watts per square meter, on planes perpendicular to the directions of S1 and S2 are given by 兩 S1,2兩 whereas the photon flux densities ⌽ 1,2(r) through an elementary area dA located at r with arbitrary orientation are given by ˆ /ប ␻ , ⌽ 1,2共 r兲 ⫽ S1,2共 r兲 • A 1,2

(A4)

ˆ is the unit vector perpendicular to area dA. where A

APPENDIX B

2ik 1

2ik 2

The authors are grateful to Adriano Brega for his technical assistance and to Fondazione Cariplo–Landau Network for the support granted to V. N. Mikhailov. The Ph.D. position of F. Paleari is funded by Istituto di Fisica del Plasma–Piero Caldirola, Consiglio Nazionale delle Ricerche, Milano, Italy. The e-mail address for A. Andreoni is andreoni @uninsubria.it.

1.

“ 2 a 1 共 r兲 ⫹ kˆ 1 • “a 1 共 r兲

2.

⫽ ig effa 3 共 0 兲 a 2* 共 r兲 exp共 ⫺i⌬k–r兲 , 1

ACKNOWLEDGMENTS

REFERENCES AND NOTE

The complete equations would be 1

former we observe that, according to Eq. (6), (Q 2 ⫹ ⌬k 2 ) ⫽ 4 兩 A 3 兩 2 / 关 (⌬kˆ • kˆ 1 )(⌬kˆ • kˆ 2 ) 兴 . Considering the damage threshold intensities of nonlinear crystals the former inequality cannot be violated in any practical situation. We can safely conclude that the a 1,2(r) solutions in Eqs. (5) that correspond to ⌬k/2 values negligible with respect to k 1 and k 2 fulfill the slowly varying envelope approximation in which Eq. (B1) can be substituted by Eqs. (2),

(B1)

3.

“ 2 a 2 共 r兲 ⫹ kˆ 2 • “a 2 共 r兲 ⫽ ig effa 3 共 0 兲 a 1* 共 r兲 exp共 ⫺i⌬k–r兲 .

(B2)

A more than safe criterion for accepting our a 1,2(r) solutions is the request

冏

1 2ik 1,2

冏

“ 2 a 1,2共 r兲 Ⰶ 兩 kˆ 1,2 • “a 1,2共 r兲兩 ,

(B3)

4. 5. 6. 7.

which, after substitution of Eqs. (5) and squaring, corresponds to

冋冉

共 Q ⫹ ⌬k 兲 sinh Q 2

2 2

2

⌬kˆ 2

•r

冊册

⫹ 16Q

2

冉

⌬k 2

• ⌬kˆ

冋冉

Ⰶ 16共 ⌬kˆ • k1,2兲 2 共 Q 2 ⫹ ⌬k 2 兲 sinh2 Q

冊

⌬kˆ 2

8.

2

•r

冊册

9.

(B4)

10.

This ordering relation is certainly true if we simultaneously require that

11.

⫹ 16Q 2 共 ⌬kˆ • k1,2兲 2 .

共 Q 2 ⫹ ⌬k 2 兲 Ⰶ 16共 ⌬kˆ • k1,2兲 2 ,

冉

⌬k 2

• ⌬kˆ

冊

2

Ⰶ 共 ⌬kˆ • k1,2兲 2 .

(B5)

If we take a 1,2(r) solutions that correspond to ⌬k/2 Ⰶ k 1,2 we fulfill the latter relation provided that ⌬kˆ is at angles to k1 and k2 , sizably different from ␲/2. As to the

12. 13. 14.

B. Y. Zeldovich, V. I. Popovichev, V. V. Ragulskii, and F. S. Faisullov, ‘‘Connection between the wavefronts of the reflected and exciting light in stimulated Mandel’shtam– Brillouin scattering,’’ Sov. Phys. JETP 15, 109–110 (1972). A. Yariv, ‘‘Phase conjugate optics and real-time holography,’’ IEEE J. Quantum Electron. QE-14, 650–660 (1978). D. M. Pepper and A. Yariv, ‘‘Optical phase conjugation using three-wave and four-wave mixing via elastic photon scattering in transparent media,’’ in Optical Phase Conjugation, R. A. Fisher, ed. (Academic, San Diego, Calif., 1983), pp. 23– 78. J. H. Marburger, ‘‘Optical pulse integration and chirp reversal in degenerate four-wave mixing,’’ Appl. Phys. Lett. 32, 372–374 (1978). A. Yariv, D. Fekete, and D. M. Pepper, ‘‘Compensation for channel dispersion by nonlinear optical phase conjugation,’’ Opt. Lett. 4, 52–54 (1979). A. Yariv, ‘‘Three-dimensional pictorial transmission in optical fibers,’’ Appl. Phys. Lett. 28, 88–89 (1976). P. V. Avizonis, F. A. Hopf, W. D. Bomberger, S. F. Jacobs, A. Tomita, and K. H. Womack, ‘‘Optical phase conjugation in a lithium formate crystal,’’ Appl. Phys. Lett. 31, 435–437 (1977). A. M. Gorlanov, N. I. Grishmanova, N. A. Sventsitskaya, and V. D. Solov’yov, ‘‘Angular characteristics of radiation from neodymium laser with wavefront conjugation under three-wave parametric interaction,’’ Kvant. Elektron. (Moscow) 5, 415–417 (1982), in Russian. L. Lefort and A. Barthelemy, ‘‘Revisiting optical phase conjugation by difference-frequency generation,’’ Opt. Lett. 21, 848–850 (1996). M. Bondani and A. Andreoni, ‘‘Holographic nature of threewave mixing,’’ Phys. Rev. A 66, 033805 (2002). M. Bondani, A. Allevi, A. Brega, E. Puddu, and A. Andreoni, Dipartimento di Scienze Chimiche, Fisiche e Matematiche, Universita` degli Studi dell’Insubria and Istituto Nazionale di Fisica della Materia, Unita` di Como, Via Valleggio 11, Como 22100, Italy, are preparing a manuscript to be called ‘‘Difference-frequency-generated holograms of 2D-objects.’’ G. S. Agarwal, A. T. Friberg, and E. Wolf, ‘‘Elimination of distortions by phase conjugation without losses or gains,’’ Opt. Commun. 43, 446–450 (1982). V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, Berlin, 1997). M. Bondani, A. Allevi, and A. Andreoni, ‘‘Holography by

Mikhailov et al.

15. 16.

nondegenerate ␹ (2) interactions,’’ J. Opt. Soc. Am. B 20, 1–13 (2003). M. R. Fewings and A. L. Gaeta, ‘‘Compensation of pulse distortions by phase conjugation via difference-frequency generation,’’ J. Opt. Soc. Am. B 17, 1522–1525 (2000). R. Danielius, A. Piskarskas, P. Di Trapani, A. Andreoni, C.


17.

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Solcia, and P. Foggi, ‘‘Matching of group velocities by spatial walkoff in collinear three-wave interaction with tilted pulses,’’ Opt. Lett. 21, 973–975 (1996). A. Andreoni, M. Bondani, and M. A. C. Potenza, ‘‘Ultrabroadband and chirp-free frequency doubling by ␤-barium borate,’’ Opt. Commun. 154, 376–382 (1998).