Frequency Doubling in Solid State Lasers

POWER LASERS

8.2 SOLID-STATE LASERS

8.2.2 Frequency Doubling in Solid State Lasers

High-power (upto 8 kW) infrared light is easily generated with a solid-state Nd:YAG.

More than a few milliwatts of infrared laser light is dangerous to the eyes because of the heat it can generate on a very small spot and failure to blink as the eye does not see infrared. Some military applications, such as range finding, target designators, countermeasures, and lidar, use a second harmonic generator to convert the IR to visi- ble green light. Nonmilitary applications include industrial, medical prostate surgery, and replacement of traffic police radar with lidar to reduce incidence of police cancer.

We derive equations for second harmonic generation (SHG) that doubles the laser frequencyf, or equivalently, asλ=c/f, halves the wavelengthλ. In this case, for a solid-state Nd:YAG laser, the IR wavelengthλ=1.064␮m is converted to green at λ=0.5␮m [67, 176]. A nonlinear crystal performs SHG.

8.2.2.1 Nonlinear Crystal for SHG Doubling is accomplished by passing the IR light into a nonlinear crystal [176]. In elastic media, the electrons are pulled back and forth by the alternating electric field of the IR light. The polarization of the crystal medium represents an induced electric field that opposes the IR field influence and can be detected by suitably arranging a system. Centrosymmetric crystals [176]

such as sodium chloride (common salt), NaCl, cannot be used for SHG because they have structural symmetry about a sodium ion (Na⁺), shown as dots in Figure 8.4a, which causes them to polarize linearly and equally for plus and minus electric fields (Figure 8.4b). Hence, there is no change in frequency between incident light and induced polarization.

In noncentrosymmetric crystals, such as potassium dihydrogen phosphate (KDP), or zinc sulfide (Figure 8.5a), the central sulfur ion does not see symmetry between NE and SW zinc ions because the zinc in this case is at different depths in these two places. Consequently, there is a nonlinear relation between incident electric field and polarization (Figure 8.5b). Electrons are more easily pulled to one side than the other.

An incident field is suppressed more in the positive polarization in the crystal than in the negative polarization. When decomposed into Fourier components, the distorted polarization sine wave shows presence of double frequencies. Nonlinear polarization

FIGURE 8.4 Symmetric crystal: (a) structure and (b) optical polarization.

SOLID-STATE LASERS 151

Optical polarization

Incident field

(a) (b)

Positive field suppressed more than negative

Input

O O

O O

O

½

½ ½

½ 1 1

3/4 3/4

Depth

Sulfur Zinc

+ -

FIGURE 8.5 Noncentrosymmetric crystal: (a) structure and (b) optical polarization.

(Pi) can be described by a Taylor series

P_i=E₀χ_ijE_j+2dijkE_jE_k+4χijklE_jE_kE_l+ · · · (8.1) whereχ_ij is the first-order or linear susceptibility,d_ijkis the second-order nonlinear susceptibility (allows SHG), andχ_ijklis the third-order nonlinear susceptibility (for different nonlinear effects). We are interested in the second-order nonlinear effects for SHG for which the polarization depends on the product of two electric fields that can be in different directions (j andk). In particular, from the second term on the right-hand side of equation (8.1), we desire the polarizationP_i^2ω, in theith direction at double frequency 2ω, resulting from the interaction of two incident IR electric fields, E^ω_j andE^ω_k, in thejandkorthogonal directions to be

Pi =2dijkEjEk (8.2)

We convert termsPandEto exponentials using

E=Re{E₀} = 1 2

E₀e^iωt+E₀e^−iωt

=1 2

E₀e^iωt+c.c.

(8.3)

where c.c. is complex conjugate and subscript 0 denotes a phasor. (Note that here we used the physics nomenclature time harmonic exp{iωt}and elsewhere we have used the Re exp{−jωt}.) Using equation (8.3), we have several ways to produce second harmonic (2ω) frequencies described in equation (8.2). Two ways are representative:

select the twoE₀fields so that their frequenciesω₁andω₂during nonlinear mixing will produce terms in 2ω1or select the two E₀fields with the same frequencyω, which is half the final double frequency 2ω.

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For the first case, with addition of fields atω1injdirection andω2inkdirection 1

2P0iω₁+ω₂

e^i(ω1+ω2)t+c.c.

= 2dijk

1 2

E^ω_0j¹e^iω1t+E^ω_0j²e^iω2t+c.c.

1 2

E^ω_0k¹e^iω1t+E^ω_0k²e^iω2t+c.c.

(8.4) From equation (8.4), we extract those terms of the mixing that containω1+ω2(which add to become the double frequency 2ω1). Note that only the two cross terms give ω₁+ω₂. At the same time, we cancel the term e^iω1te^iω2t =e^i(ω1+ω2)tfrom each side to obtain the phasor equivalent of equation (8.2):

P_0i^ω1+ω2 =2dijkE^ω_0j¹E^ω_0k² (8.5) For the second case, we do not desire the sum of two different frequenciesω1andω2

to equal 2ω, (ω1+ω2=2ω1), but rather select both frequencies to beωorω1=ω2= ωin equation (8.4). Once again select only terms at the second harmonic frequency 2ω. Using the fact that the two brackets on the right-hand side of equation (8.4) equal each other gives us additional 2 factor (relative to the first case) and canceling the exponents gives

P_0i^2ω =d_ijkE_0j^ωE^ω_0k (8.6) As there are three rectangular coordinate directions (x, y, z) and six combinations of pairs of fields from three directions, the second-order nonlinear coefficientdijhis a 3×6 tensor (compressed from 3×9 by merging different index orders):

⎡

⎢⎣ P_0x^2ω P_0y^2ω P_0z^2ω

⎤

⎥⎦=

⎡

⎢⎣

d₁₁d₁₂d₁₃d₁₄d₁₅d₁₆ d21d22d23d24d25d26

d₃₁d₃₂d₃₃d₃₄d₃₅d₃₆

⎤

⎥⎦

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎣

E0xE0x

E_0yE_0y E0zE0z

2E0yE_0z 2E0zE_0x 2E0xE0y

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎦

(8.7)

In most crystals, only a few of the coefficients in the matrix are nonzero. For KDP, onlyd14, d25, andd36have significant values, giving

P_x^2ω =2d14E^ω_0yE_0z^ω P_y^2ω =2d25E_0x^ωE_0z^ω

P_z^2ω =2d36E^ω_0xE^ω_0y (8.8)

SOLID-STATE LASERS 153

Note that forE_z=0, the last equation of equation (8.8) represents an incident TEM mode in thezdirection, causing second harmonic in thezdirection. The angle of incidence of the incident wave at frequencyωrelative to the crystal axes must be selected carefully for phase matching between the input wave at frequencyωand the output second harmonic or frequency-doubled wave at frequency 2ωto achieve efficient SHG. Second-order nonlinear coefficients are provided in Ref. [176].

8.2.2.2 Electromagnetic Wave Formulation of SHG We follow the ap- proach of Ref. [176] and use labels E₁, E₂, andE₃for phasor fields x,y, andz, respectively, for a wave propagating in thezdirection. Thiszis not the same as the earlier usedzaxis of the crystal. Consider only the second case in Section 8.2.2.1, equation (8.6), there are two coupled waves, at frequenciesωand 2ω, propagating concurrently through the crystal in wave propagation directionzduring SHG. The fieldE₁from the inputs at frequencyω₁=ωdecays along the crystal according to dE1/dz <0 as power is converted to the second harmonic at 2ω. The desired wave E3at frequencyω3=2ω(double frequency) increases from zero along the crystal as its power builds up according to dE3/dz >0. Separating out polarization of each fieldE^ω_0i¹ =a1iE1andE_0i^ω3 =a3iE3, we write the instantaneous (a function ofzand t) nonlinear polarization for theωand 2ωwaves, using equations (8.5) and (8.6), and ω3−ω1=2ω1−ω1=ω1, as [176]

P_NL^ω3−ω1(z, t)

i =d_ijka_3ja_1kE₃E^∗₁e^{i[(ω3−ω1)t−(k3−k1)z]}+c.c.

P_NL^2ω1(z, t)

i = 1

2d_ijka_1ja_1kE₁E₁e^{i(2ω1t−2k1z)}+c.c. (8.9)

where summations over repeated indices are assignedd =

ijkd_ijka1ia2ja3k. Follow- ing a standard procedure, we substitute in turn the polarizations from equation (8.9) into the wave equation. The wave equation, assuming materials are not conductive (conductivityσ=0), is

∇²E1(z, t)=μ0

∂²

∂t²(1E1(z, t)+PNL)=μ01

∂²E₁(z, t)

∂t² +μ0

∂²

∂t²PNL (8.10) Substituting the first equation of equation (8.9) into equation (8.10) gives

∇²E^ω₁¹(z, t)=μ₀₁∂²E^ω₁¹(z, t)

∂t² +μ₀d ∂²

∂t²

E₃E^∗₁e^{i[(ω3−ω1)t−(k3−k1)z]}+c.c.

(8.11)

154 POWER LASERS

AsE1in equation (8.11) is dependent on the product ofz-dependent terms, the left- hand side can be written in terms of phasorE1as

∇²E^(ω₁¹⁾(z, t)= 1 2

∂²

∂z²

E₁(z)e^{i(ω1t−k1z)}+c.c.

= −1 2

k²₁E₁(z)+2ik1

dE1

dz (z)

e^{i(ω1t−k1z)}+c.c. (8.12) where we assumed the slowly varying amplitude approximation to eliminate d²E₁(z)/(dz²) with (see Section 2.1.1)

d²E1(z)

dz² ≤k₁∂E1

∂z (z) (8.13)

Substituting equation (8.12) for the left-hand side of equation (8.11) and using

∂²/∂t²≡ −ω²₁,

−1 2

k²₁E₁(z)+2ik1

dE1

dz (z)

e^{i(ω1t−k1z)}+c.c.

= −μ01ω²₁ E₁(z)

2 e^{i(ω1t−k1z)}

+c.c.

−μ₀dω₁²

E₃E^∗₁e^{i{(ω3−ω1)t−(k3−k1)z}}

+c.c. (8.14)

As k²₁=μ₀₁ω²₁, the terms like this on both sides cancel. Also, asω₃=2ω and ω₁=ω,ω₃−ω₁=ω, the term e^i(ω1t)at left cancels with e^{i(ω3−ω1)t}at right:

−ikdE1

dz e^−i(k1z)+c.c.= −μ0dω²₁

E3E^∗₁e^{−i(k3−k1)z}+c.c.

(8.15) or the rate of decay of the incident field at frequencyωwith distancezalong the crystal

dE1

dz = −iω1

μ₀

₁dE3E^∗₁e^{−i(k3−2k1)z} (8.16) where we used (μ0ω₁²)/(ik)≡ −iω₁√

μ₀/₁withk=ω√μ₀₁.

Similarly, substituting the second equation of equation (8.9) into the wave equa- tion (8.10) gives the rate of increase of the second harmonic at 2ωwith distancez along the crystal: (ω3=2ω)

dE3

dz = −iω₃ 2

μ₀

₃dE1E^∗₁e^i(k3−2k1)z (8.17)

SOLID-STATE LASERS 155

Equations (8.16) and (8.17) describe how the incident waves at frequencyωdecay and the wave at frequency 2ωgrows. It can be shown that in the absence of loss, the power of combined waves is constant; that is, the power lost with distance in the input waves at frequencyωequals the growth in power with distance in the 2ωwave [176]:

d dz(√

₁|E₁|²+√

₃|E₃|²)=0 (8.18)

For high-intensity lasers and phase matching by angle adjustment, conversion effi- ciency of 100% is possible.

8.2.2.3 Maximizing Second Harmonic Power Out Following Ref. [176], phase matching for theωand 2ωwaves over sufficient distance enhances efficiency.

Its effect can be seen by initially assuming that negligible power is lost forω, which occurs after only a short distance into the crystal or for lower intensity beams. In this case,E₁is constant and only equation (8.17) showing the growth of second harmonic (2ω) from zero needs be considered:

d

dzE^(2ω)₃ = −iω μ₀

₃d|E1|²e^{i kz} (8.19) where we defined k=k₃−2k1andω=ω₃/2. The field of the second harmonic E3is obtained by integration along the crystal lengthL:

E^(2ω)₃ (L)≡ _L

z=0

d

dzE^(2ω)₃ dz= −iω n²

μ0

0

d|E₁|²

e^{i kL}−1 i k

(8.20)

where we used3=n²0. Intensity of the second harmonic generated is

I^(2ω)(L)=E^2ω₃ (L)E^∗₃^(2ω)(L)= ω²d² n²

μ_{0 0}

L²|E1|²sin²( kL/2)

( kL/2)² (8.21) where we used exp{i kL} −1=exp{i kL/2}(exp{i kL/2} −exp{−i kL/2})= exp{i kL/2}(−2isin( kL/2)) and multiplied top and bottom byL². From intensity of positive-goingωwave and intrinsic impedanceη=√

μ0/0,

I^(ω)= 1 2n

₀

μ₀|E1|² or |E1|²=2I^(ω) μ₀

₀ 1

n (8.22)

156 POWER LASERS

Dividing equation (8.21) by equation (8.22) gives the efficiency of second harmonic generation:

ηSHG= I^(2ω)

I^(ω) = w²d² n²

μ_{0 0}L²

2I^(ω)

μ_{0 0} 1 n

sin² kL/2 ( kL/2)²

= 2ω²d² n³

μ0

0

3/2

L²sin² kL/2

( kL/2)² I^ω (8.23)

Thus, the efficiency of second harmonic generation is proportional to the intensity of the input beam. So for maximizing the intensity of the second harmonic, we should select material with a high value ford(KDP is good), use a longer lengthL, use a large intensity input, and have theωwave propagate at the same propagation constant (wave number)kas the 2ωwave. The last condition gives k=k^(2ω)−k^(ω)=0 for which the phases of the two waves are matched. Unfortunately, waves copropagating through a crystal at frequenciesωand 2ωdo not have the same propagation constant, so kcannot equal zero. The next section addresses this issue.

8.2.2.4 Phase Matching for SHG Phase matching to achieve k=0 is accomplished by selecting the angle at which the incident wave at frequencyωstrikes the crystal. A uniaxial anisotropic crystal such as KDP exhibits birefringence as shown in its index ellipsoid (Figure 8.6). The equation for the index ellipsoid is

x² n_o +y²

n_o+ z²

n_e =1 (8.24)

x

y n_e

z Ellipse with

axes 2n and 2n

o e

x

y n_o

n₀

z Ellipse with

axes 2n_o and 2n_e(θ)

n_e(θ)

Incoming wave Incoming

wave

(a) (b)

θ

FIGURE 8.6 Process for phase matching for SHG: (a) index ellipsoid and (b) finding angle θfor phase matching.

SOLID-STATE LASERS 157

An incoming wave sees refractive indicesnofor itsxandypolarized components and nefor itszpolarized one. In a uniaxial crystal, subscriptsnoandnerefer to ordinary and extraordinary directions, respectively, the latter being extraordinary because its refractive index differs from the other two orthogonal directions that have the ordinary refractive indexno. If the incoming wave propagating in thez–yplane is tilted at an angleθwith respect to thezaxis (Figure 8.6b), the wave strikes an elongated ellipse (marked by dashes) at normal incidence. The refractive indexnois unchanged for the perpendicular component of the incoming wave, while the refractive index for the parallel component is stretched fromn_eton_e(θ). The stretch is computed as follows.

For the ellipse including thez,y, and incident wave directions, y²

n²_o+ z²

n²_e =1 (8.25)

Dropping a perpendicular fromne(θ) to theyaxis gives

y=n_e(θ) cos θ (8.26)

and to thezaxis gives

z=ne(θ) sinθ (8.27)

Substituting equations (8.26) and (8.27) into equation (8.25) gives an equation for determiningn_e(θ) in order to achieve phase matching with a specific crystal:

1

n²_e(θ) =cos² θ

n²_o +sin² θ

n²_e (8.28)

We write equation (8.28) for the second harmonic wave at frequency 2ωas 1

n^(2ω)e

2 = cos²θ

n^(2ω)o

2+ sin²θ

n^(2ω)e

2 (8.29)

For phase matching ofωand 2ωwaves, we require

n^2ω_e (θ)=n^(ω)_o (8.30)

so that velocities and propagation constants for the two waves are equal, velocity= c/n^(2ω)_e (θ)=c/n^(ω)_o . Substituting equation (8.30) into the left-hand side of equa- tion (8.29) gives an equation for which the optimum angleθfor phase matching can

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Brewster plate 0.5 m

Nd : YAG

18.5 cm 2.5 cm

0.532 µm a 10 m

c b

Ba₂Nb₅O₁₅ 46.5 cm

FIGURE 8.7 Configuration for second harmonic generation, crystal inside laser cavity.

be computed as

1 n^(ω)o

2 = cos² θ

n^(2ω)o

2 + sin² θ

n^(2ω)e

2 (8.31)

Solving forθby relating cos²θ=1−sin²θgives

sin²θ=

n^(ω)_{o −}2

− n^(2ω)_o

₋2

n^(2ω)_e

₋2

− n^(2ω)_o

₋2 (8.32)

Figure 8.7 shows a simple configuration for SHG [67] that allows a large TEM00

mode volume in a crystalline Nd:YAG rod and high intensity in a nonlinear noncen- trosymmetric barium sodium niobate crystal. The laser cavity for the Nd:YAG laser is formed by two concave mirrors that highly reflect the YAG infrared wavelength of 1.064␮m. The right-hand side mirror is transparent to the half-wavelength green 532␮m (double frequency) generated in the barium sodium niobate crystal. The latter is placed at the location of maximum intensity in the cavity. Green light that travels left from this crystal is absorbed by the Nd:YAG rod. KDP, potassium dideuterium phosphate (KH2PO4), is commonly used for power applications because it has a high damage threshold and high optical quality [67].