Squeezing in traveling-wave second-harmonic generation
Ruo-Ding Li and Prem Kumar
Department of Electrical Engineering and Computer Science, Northwestern University,2145 Sheridan Road, Evanston, Illinois 60208-3118
Received July 8, 1993
We have analyzed squeezing of the fundamental field by means of traveling-wave, type-II phase-matched, second-harmonic generation, taking into account depletion of the fundamental field and phase mismatch between thefundamental and the harmonic fields. For the phase-matched case we show that the generated squeezing isS = 1 - y, where y is the harmonic conversion efficiency. In the case of a large phase-mismatch we haveidentified a new mechanism that generates squeezing.that is due to the cascaded X(2) nonlinearity.
Squeezed states of light have been successfullygenerated in experiments that involve the x (2)
process. Large amounts of quadrature squeezinghave been reported in subthreshold optical para-metric oscillation' and traveling-wave (TW) opticalparametric amplification.2 In these experimentsthe output of a laser is first frequency doubled topump the parametric device. There is considerableinterest in using the second-harmonic generation(SHG) process directly to produce squeezing becauseone then avoids having to frequency double a laser,thus greatly simplifying the experimental setup.To our knowledge, most of the experimental work3
and the associated theoretical studies4 on squeezinggeneration by means of SHG have concentrated onconfigurations in which the nonlinear medium isenclosed inside an optical cavity. We have exploredthe possibility of squeezing generation by means ofTW SHG because of its simplicity, large bandwidth,and potentially high obtainable squeezing, as theTW optical parametric amplification experimentssuggest.2 For the TW case the generation ofamplitude squeezing in both the fundamental andthe harmonic fields has been predicted.' However,these predictions are based on perturbation analysesthat iterate the quantum operator equations and arethus limited to short interaction lengths or smallharmonic conversion efficiencies.
In this Letter we analyze a commonly used TWSHG scheme, taking into account depletion of thefundamental field and phase mismatch betweenthe fundamental and the harmonic fields. Type-IIphase-matched SHG, such as that used to frequencydouble the output of a Nd:YAG laser employing aKTP crystal, is shown to generate squeezed vacuumin the mode that is polarized orthogonal to thefundamental beam. The nature of this schemeallows us to use a linearization method to analyze thequantum properties of the squeezed-vacuum modewhile treating the strong fundamental and harmonicfields classically. Since the classical solution forTW SHG with arbitrary phase mismatch and pumpdepletion is well known, 6 we avoid the limitationassociated with the perturbation methods.' Thelinearization methods have been used previously7
0146-9592/93/221961-03$6.00/0
It is the nonlinear phase shift of the fundamental field
to analyze the generation of squeezing by means ofself-phase modulation in optical fibers.
Besides the pairwise annihilation of the fundamen-tal photons, nonlinear phase shift of the fundamentalwave introduced during the SHG process also con-tributes to the generation of squeezing. In the caseof a large phase mismatch in which the fundamentalfield experiences little depletion with negligible har-monic being generated, this nonlinear phase shift isresponsible for the creation of squeezing. Recentlythis nonlinear phase shift, arising as a result of cas-cading of the second-order [x(2)] nonlinearity, has at-tracted considerable attention in attempts to developcompact all-optical switches.8 Squeezing created bythis process is similar to that generated by self-phasemodulation in a X(3) medium such as an optical fiber.9
We consider the TW, type-II phase-matched, SHGscheme shown in Fig. 1. This scheme is commonlyused to frequency double the output of a Nd:YAGlaser with a KTP crystal. An intense, linearly polar-ized, coherent-state, fundamental beam of frequencycv enters through one port of the polarization beamsplitter (PBS1). We are interested in squeezing ofthe vacuum-state mode with annihilation operator athat couples through the other input port of PBS1.In the Nd:YAG laser example cited above, d is theorthogonally polarized mode that copropagates withthe linearly polarized laser output. Because the fun-damental field is very intense, we treat it classically
A, Aou,
PBS _ I1 WP2 #
A + /Id [I U --- A HWPI PBS2
a A IPl____ A
A
A,
* S - polarization
P -polarization
Fig. 1. Schematic of a commonly employed, type-IIphase-matched, SHG scheme. The principal axes of theKTP crystal are parallel to the S- and P-polarizationdirections. The harmonic field, which is S polarizedbecause of the type-II phase matching, can be separatedby use of a prism or a dichroic beam splitter (not shown).
with its amplitude A as a c-number quantity. Toachieve type-II phase matching, we rotate the polar-ization of the fundamental field by 450, using of ahalf-wave plate (HWP1). At the output of HWP1 wethus have two fundamental fields with amplitudes Aland A2, which have the same frequency a)1 = cv2 = CObut orthogonal polarizations. They can be written as
Al(2)(0) = 2-" 2(A + AA +c d), (1)
where AA represents the coherent-state fluctuationsof the strong fundamental input beam. For simplic-ity, in what follows we ignore AA.10
The interaction of three waves in a X(2) nonlinearmedium is described by6
dAI() = KA2(1,)A3 exp(-iAkz),
dA 3_
d3 -KA 1 A 2 exp(iAkz), (2)
where the electric field Ej = i(2Acvj/eoVnj)Vj2Aj, j1, 2, 3, V is the mode volume, nj is the refractiveindex of the medium at angular frequency cv, K =
deff(cvl 2 (c32fi/nn 2 n3eoc 2V)- 2 , deff is the effectivesecond-order nonlinearity, cv3 = c 1 + cv2, and thephase mismatch Ak = (k, + k2 - k3) * i,.
In general the solution of Eq. (2) can be written as6
AI(L)= 77,A,(O)exp(iDNLj),
A2(L) = 92A2(0)exp(i(,NL2), (3)
where 77j IAj(L)/Aj(0)1, j = 1, 2 are the depletioncoefficients for the two fundamental fields, and cNLjare the nonlinear phase shifts accumulated duringthe SHG process. Because for the SHG process wehave A3 (z = 0) = 0, it can be shown6 that the de-pletion coefficients qj and the nonlinear phase shiftsc1NLj depend only on the intensities of the inputfundamental fields and not on their phases. If wedefine the input field intensities as ujJIAj(0) I2, then77j = 7?j[jAl(0)12 , IA2(0)12] =?7j(Ul, U2 ), and cINLj =(INLj[IA,(0)11, JA 2(0)12] == (NLj(Ul, U2), for j = 1, 2.After substituting the input fields given by Eq. (1)into Eqs. (3), we can linearize the resulting solutionwith respect to the vacuum-state operation a whiletreating the strong fundamental fields classically.To do so, we need to linearize the depletion coeffi-cients 77j and the nonlinear phase shifts INLj aroundthe strong fundamental fields. For example,
where we have used relations such as a771/aAj(0)(an7,/auj)Aj(O)*. Similar expansions for 772, (DNL1,
and 'FNL2 are given by replacing 7, in Eq. (4) with772, (DNLI, and (DNL2, respectively.
Because Eqs. (2) are symmetric with respect toan interchange of Al with A2, and we chose theinput fundamental fields to be equal in amplitude,the exchange symmetry holds at all z as the SHGprocess evolves through the nonlinear medium. This
can be straightforwardly proved from the solution ofEq. (2) of Ref. 6. This exchange symmetry impliesthat the coefficients appearing in Eq. (4) satisfythe relations 771(1A12 /2, 1A1 2 /2) = 77 2 (1A12 /2, IA1 2 /2),ad7 1 /aul = a772 /au2, a771 /au2 = a77 2/aul, with similarexpressions holding for the nonlinear phases PN1and (NL2-
Substituting the linear expansion of 77j and DNT j
[see Eq. (4)] into Eqs. (3) and further linearizing withrespect to the vacuum-state operator d, we obtain atthe output of the nonlinear medium
Al(2)(L) = 2- 2(77A ± pAd ± ve)exp(iDNL), (5)
where because of the exchange symmetry we haveset 771 = 772 = 77 and ONLY = (DNL2 = DNL. The coeffi-cients ,A and v are given by
.L= 77 +12 71 a ±d\ NLl (a77 kmU1,
v 2L'k7a( 2 a=,,/L1+ \3u2 auJj(6)
Next we prove that 1/12 - Iv12 = 1. On first glanceat Eqs. (6), it seems that one needs to know thedependencies of the coefficients 77j on the input pumpintensities up. However, we find that &Ip
2- IV = 1
can be proved without calculation of the detailedsolution of the SHG problem. From its definition inEqs. (3), we can rewrite 771 as 771 IA,(L)/A,(0)l =
[IA,(L)12/u,1v2 , from which it is easy to show that
a 771 a 77,- I[ + ir A (L)12 -(7au2 au,1A12 [a au2 auAl _2
From the Manley-Rowe relation for the three-waveinteraction given by Eqs. (2), we have IA,(L)12 -IA2 (L)I 2
= IA,(0)12 - IA 2 (0)12= U, - U2. Directly
taking the partial derivative with respect to U2 yields
alA,(L)12 aIAj(L)I2 _ 1,au2 au,
(8)
where we have used the symmetry relationalA 2(L)12/au2 = alA,(L)12/aul as pointed out afterEq. (4) above. Finally, substituting Eqs. (7) and (8)into Eqs. (6), we get
A 2 1 7 a ( (DNL1 (1 + Y7)]
1 A2 a a) A2 (' 1V = LL77A 2 (~7I)NLI + 1~2
(9)
from which it easily follows that ,412 - Iv12 = 1.If the strong fundamental field is extracted by
using HVWP2 and PBS2 as shown in Fig. 1, then atthe a0 ,t port of PBS2 we obtain [absorbing exp(iINL)in the definition of &out]
aout = 2- 2 [A,(L) - A2(L)l = A' + v't (10)
which is a Bogoliubov transformation that generatessqueezing"; hence the output operator a0out is in asqueezed vacuum state if d is in the vacuum state.
Fig. 2. Obtainable squeezing as a function of theintensity of the fundamental field (ocIAI2) for variousphase-mismatch parameters.
Use of a homodyne detector will yield the maximumsqueezing,
S = (11p - 17,I)2
= {[(771 + 77)2 + 4Ier 2] 2
4_ [(71-1 - 71)2 + 4D eff2 ]'/2 }2 , (11)
where we have defined the effective nonlinear phaseshift (IDeff =_7A12 (aIDNL/aU2 - aDNLl/,au)/2.
Since no further approximation was invoked be-yond those used to derive Eqs. (2), Eq. (11) is a gen-eral result; it is valid for arbitrary phase mismatchand high SHG conversion efficiencies. One inter-esting case is that of perfect phase matching, forwhich Ak = 0. In this case it can be shown that(Deff = 0; Eq. (11) thus leads to the simple expressionfor squeezing,
S = 772 = 1 - y, (12)
where y 21A3(L) 2/[A12 = 1 - 772 is the efficiency ofSHG. This is a remarkable result-the squeezingin the orthogonally polarized mode is equal to thefundamental intensity depletion coefficient. Also, inthis case, the solution of Eqs. (2) is well known6; itdirectly yields
S = 72 = [sech(KIAIL/2"2)]2. (13)
Now we consider the large phase-mismatch casein which the depletion of the pump field is verysmall and we can set Y7 7 1. We then calculate theeffective nonlinear phase shift and find that it isgiven by (Deff- K2 L2 1A12 /2AkL, which is the sameas that given in Ref. 8. For the general situation,we have numerically integrated Eqs. (2) to calculate77 and (eff. The resulting squeezing, Eq. (11), isdisplayed in Fig. 2 as a function of the fundamentalintensity (ocIAl2). Maximum squeezing is obtainingfor the case of perfect phase matching (Ak = 0). Fornonzero phase mismatch, the squeezing undergoessome up-and-down variation because of oscillation inthe SHG conversion efficiency with the input funda-mental intensity. In this case, both the upconver-sion process and the nonlinear phase experienced bythe fundamental field contribute to the squeezing.
The squeezing generated from the TW SHG processdoes not require phase matching as a necessary
condition. This implies that one may use it as asource for spatially broadband squeezing that canbe used for imaging faint objects with sensitivitybetter than the shot-noise limit."2 As the phasemismatch becomes large, one needs a correspondinglygreater pump intensity to get greater squeezing (seeFig. 2). The obtainable squeezing is thus limited bythe damage threshold of the nonlinear crystal ratherthan by the phase-matching condition. TW SHGis a widely used, standard method for upconversion,and high conversion efficiencies have been reported.' 3
It should not be difficult to observe squeezing withthe scheme proposed in this Letter. The schemeis also simpler to set up than optical parametricamplification or SHG in an optical cavity.
This research was supported in part by the U.S.Office of Naval Research and the National ScienceFoundation.
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