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Figure 1: A beam of light accumulates phase and is attenuated as it propagates through a linear medium, but the photons of the beam do not interact with each other. Due to this, the input (E_in) and output (E_out) electric fields have the same frequency (ω) and are directly proportional.

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Figure 2: This nonlinear optical medium supports second harmonic generation and the input light is the same as in Figure 1. In addition to providing fundamental-frequency electric fields, as in Figure 1, the light output from this nonlinear medium also includes frequency doubled light, whose electric field E_out(2ω) is proportional to the square of the total input electric field.

Nonlinear optics refers to different optical phenomena that can occur when light alters the properties of the surrounding material. These phenomena are called nonlinear optics because the strength of the material's response is proportional to the square, cube, or other higher power of the light's total electric field. Since the light usually must be intense to produce a measurable nonlinear response, laser light is often used to generate desired nonlinear effects.

A material's nonlinear response occurs in addition to the linear response, which is proportional to only the first power of the light's electric field. The interaction between the light and the material is completely conventional (linear) when optical and material conditions are not appropriate for generating nonlinear phenomena. Then, light waves in a material propagate together without affecting one another, whether or not they are part of the same beam or pulse.

An example of a material's linear response to light is illustrated in Figure 1, in which an input beam is described by its electric field amplitude (E_in), frequency (ω), and polarization orientation (vertical). The beam accumulates phase and experiences conventional losses from effects like reflection and material absorption before it is output, but otherwise the beam is unaffected. The total electric field (E_out) output by the material has an amplitude that is directly proportional to the that of the input beam. In addition, the input and output light has the same frequency and polarization direction.

Nonlinear processes occur when the presence of the light in the material creates conditions in which different light waves can interact with one another. In some cases, light waves from the same beam or pulse interact, and in other cases light waves with significantly different frequencies and / or polarizations interact. Nonlinear phenomena include the generation of new frequencies of light, lensing of laser light propagating in uniform materials, and modulation effects. In the case of frequency generation, the light output by the nonlinear material will include a frequency of light different than any of the input beams' frequencies.

An example of nonlinear frequency generation is illustrated in Figure 2, in which the light is incident on a nonlinear optical material. While the incident light is the same as in Figure 1, the material supports second harmonic generation, and the electric field output from this crystal includes a new color, which has double the frequency of the input beams. The amplitude of this frequency-doubled light is proportional to the square of the total input electric field, and the losses accumulated by the two fundamental-frequency beams include the photons converted into frequency-doubled light.

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Figure 3: This nonlinear optical medium supports second harmonic generation and the input light is the same as in Figure 1. In addition to providing fundamental-frequency electric fields, as in Figure 1, the light output from this nonlinear medium also includes frequency doubled light, whose electric field E_out(2ω) is proportional to the square of the total input electric field. The efficiency of the SHG process is proportional to the square of the of the input light intensity multiplied by a material parameter χ⁽²⁾.

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Figure 4: This energy diagram shows the input photons on the left and the generated photons on the right. In the case of ideal second harmonic generation the frequency (ω) of the input photons is exactly twice that of the output photon.

Second harmonic generation (SHG) is a nonlinear optical process that effectively combines pairs of nominally similar photons into a single photon. This nonlinear process occurs due to interactions between the light and the surrounding material. When the input photons are identical, their frequencies are exactly equal, and this frequency (ω) is called the fundamental frequency. The frequency (2ω) of the generated photon is exactly double the fundamental. In other words, the output light wave's frequency is the second harmonic of the input wave's frequency. This relationship gives SHG its name.

While this seems to suggest the two input photons must have the same frequency, this is not the case. Photons in a beam or pulse of light have frequency values that span a non-zero spectral range around the fundamental frequency. It is possible to combine two photons with frequencies that are different, but within this spectral range. When this occurs, the nonlinear process is typically still called SHG, and the output photons are still described as being frequency doubled. This is also true for input photons from broad-bandwidth femtosecond pulses. However, to simplify the following discussion of SHG, it will be assumed that all input photons have the same fundamental frequency.

Every photon has an energy that is proportional to its frequency. Energy diagrams, like the one shown in Figure 4, typically take advantage of this property to relate the frequencies of photons input to and output by nonlinear optical processes. Each arrow represents the energy of a single photon and is labeled using the photon's frequency. Arrows pointing up correspond to input photons, and arrows pointing down correspond to generated photons. The dotted horizontal lines are typically called virtual energy levels and are not related to energy levels in the material.

The energy diagram illustrates the remarkable fact that nonlinear processes like second harmonic generation are lossless, so that the material does not gain or lose energy as a result of the process. The material provides the specific environment required for the process to occur, and there is interaction between the material and the photons, but no photons are absorbed or emitted by the material during the process. This is the reason the sum of the input photons' energies exactly equals the sum of the output photon's energy.

Not all of the input photons will undergo second harmonic generation as the light propagates through the material. As illustrated in Figure 3, the efficiency of the second harmonic generation process is proportional to the square of the of the input light intensity multiplied by a material specific parameter (χ⁽²⁾, which is pronounced "kai two"). Values of χ⁽²⁾ are usually extremely small (on the order of 10^-13 m/V), and large input intensities (I_o) are typically required for second harmonic generation to be efficient. As the input light intensity increases, the intensity of the frequency doubled (I_2ω) light increases relative to the fundamental-frequency output light (I_ω).

SHG is a Specific Case of Sum-Frequency Generation
Sum-frequency generation (SFG) is a nonlinear optical process during which input photons are combined into output photons. There are no restrictions on the maximum frequency difference between the input photons. Usually, the term SFG is reserved for cases in which the input photons do not come from the same light source, and / or when the spectra of the input beams or pulses do not overlap.

Second harmonic generation is a specific case of SFG. The nonlinear optical process is typically described as SHG when both photons originate from the same beam or pulse, and the term is especially preferred when frequency doubling the input light. Since SHG combines input photons that have nominally the same energy, second harmonic generation is a degenerate case of SFG.

Only optical materials which have a non-zero χ⁽²⁾ parameter support SHG, and not all crystals fulfill this requirement. An example of a crystal that can be used for SHG is β-BBO. A material is crystalline when its atoms, ions, or molecules are arranged in a repeating pattern. If there is no central point around which the crystal pattern is symmetric (i.e. if the crystal is not centrosymmetric), the crystal has a non-zero χ⁽²⁾ parameter and can be used for SHG.

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Figure 6: When light's oscillating electric field interacts with a centrosymmetric material, an electron's induced potential energy, which is illustrated as a function of the electron's displacement (x) from its equilibrium position, is symmetric and described by a polynomial function consisting of even powers of x.

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Figure 5: This cubic crystal structure includes two different ions, A and B, in a repeating pattern. The crystal structure is centrosymmetric since the crystal at point (x, y, z) and its inverse (-x, -y, -z) are identical.

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Figure 8: When light's oscillating electric field interacts with a non-centrosymmetric material, an electron's induced potential energy, which is illustrated as a function of the electron's displacement (x) from its equilibrium position, is asymmetric for larger displacements. This curve is described by a polynomial function consisting of even and odd powers of x.

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Figure 7: This wurtzite crystal structure includes two different ions, A and B, in a repeating pattern. The crystal structure is not centrosymmetric since there is at least one point (u, v, w) which is not identical to its inverse (-u, -v, -w).

Centrosymmetric Crystal
The repeating crystalline pattern of one type of centrosymmetric crystal is illustrated in Figure 5. As is true for all centrosymmetric crystals, the structure at point (x, y, z) is identical to the structure at the inverse location (-x, -y, -z). Due to the crystal's structural symmetry, the material's response is the same whether the incident light's electric field vector points in one direction or in the opposite direction.

The material’s response to light occurs due to interactions between the light’s oscillating electric field and the electrons in the material. An electron bound to an atom in the material experiences forces contributed by the atom and the light’s electric field, as well as the surrounding atoms in the material. The blue curve in Figure 6 was calculated using the Lorentz model of an atom, which takes these forces into account using a classical mechanics approach. This model can also be used to estimate the χ⁽²⁾ parameter.

In Figure 6, the X-axis corresponds to the position of the electron, which is in equilibrium at x = 0. The input light’s oscillating electric field moves the electron back-and-forth between positive and negative displacements. The greater the intensity of the electric field, the larger the displacement of the electron. The central and surrounding atoms exert a restoring force on the displaced electron, returning it to equilibrium. This restoring force provides the electron with potential energy, which is plotted on the Y-axis and increases as the electron moves further from equilibrium.

At moderate displacements, the central atom exerts the majority of the restoring force, and the electron’s corresponding potential energy is symmetric and approximately parabolic (the dashed gray curve is a parabolic fit). At larger displacements, achieved using higher-intensity light, surrounding atoms contribute more to the restoring force, as well as the electron’s potential energy. At these larger displacements, the electron’s potential energy deviates from parabolic but remains symmetric.

The symmetric material response is a consequence of the centrosymmetric structure of the crystal. In order to support SHG, this curve would have to be asymmetric. Mathematically, a non-zero χ⁽²⁾ parameter requires the polynomial describing the curve to include odd powers of x, while the polynomial describing the curve in Figure 6 includes only even powers of x.

Non-Centrosymmetric Crystal
An example of a non-centrosymmetric material is shown in Figure 7. In the case of these materials, there is aways at least one location (u, v, w) for which the crystal structure differs from the crystal structure at the inverse location (-u, -v, -w).

The potential energy of an electron, as a function of its displacement from its equilibrium position, has been calculated for a non-centrosymmetric material using the Lorentz model of the atom (Figure 8). When the incident light has low intensity, the displacement of the electron from its equilibrium position is small, and the majority of the restoring force is provided by the central atom. Within this limited region, the electron's corresponding potential energy function is symmetric and approximately parabolic, which indicates a linear optical material response. When illuminated by higher-intensity light, the electron's displacement is greater and the surrounding atoms contribute more to the restoring force. Since the distribution of the surrounding atoms is not symmetric, neither is the restoring force nor the potential energy well confining the electron for these higher displacements.

The polynomial needed to model this asymmetric curve includes even, as well as odd, powers of x. It is the non-zero cubic (x³) terms in this polynomial that provide a non-zero χ⁽²⁾ parameter. A third-order polynomial term is required for the generation of the second-order nonlinearities quantified by the χ⁽²⁾ parameter, since the x³ term is the lowest-order correction to the linear optics model. Because the asymmetric response only becomes significant for larger electron displacements, high-intensity light is required to induce SHG.

Uniaxial Crystal
A secondary requirement for efficient production of second harmonic light, which will be discussed in a section below, is the so-called phase matching condition. At a high level, phase matching is matching of the phase velocities between the input fundamental frequency with the second harmonic frequency. This is frequently accomplished by utilizing the birefringence of anisotropic optical materials.

Uniaxial crystals are a specific type of anisotropic material, where two of the three principal refractive indices are the same. The name “uniaxial” is derived from there being a single propagation axis in these crystals, known as the optical axis or c-axis, on which the index of refraction is the same for all polarization orientations. Many common optical materials belong to the uniaxial class: calcite, quartz, sapphire, and β-BBO.

Reference:
Robert W. Boyd, Nonlinear Optics (Academic Press, New York, 1992) pp. 17 - 52.

Certain materials have a refractive index that depends on the orientation of the light's electric field polarization. An example is birefringent materials, which include uniaxial crystals like β-BBO. Uniaxial crystals characteristically have two principal refractive indices, which are called the ordinary (n_o) and extraordinary (n_e) refractive indices. The principal refractive indices are the maximum and minimum the material can provide. Since light’s electric field (E) is always polarized perpendicular to the propagation direction (k), both the propagation direction and polarization state must be known to determine the refractive indices experienced by the light's polarization components.

A Cartesian coordinate system, with the Z-axis aligned along the crystal’s optic axis (c), can be used to relate the light's polarization components (E_x, E_y, and E_z) to a birefringent crystal's geometry.

In uniaxial crystals, the refractive index can be determined by referencing the angles of the propagation direction and polarization orientation relative to the crystal's optic axis (c).

Since the crystal is uniaxial, the refractive index for fields polarized orthogonal to the crystal's optic axis is n_o. For fields parallel to the crystal's optic axis, the refractive index is n_e. The two components (E_x and E_y) of light polarized orthogonal to the crystal's optic axis have refractive indices equal to n_o, since the crystal is uniaxial. Light polarized parallel (E_z) to the crystal’s optic axis has a refractive index equal to n_e.

It is also possible for birefringent materials to provide a refractive index value between n_o and n_e, which is discussed in a following section.

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Figure 9: The diagram on the left indicates the propagation direction (k) of the incident light is along the optic axis (c) of the crystal. The incident light can be decomposed into orthogonal polarization components E_x and E_y. The diagram on the right is a refractive index ellipsoid, with a conic section (red circle) at the XY-plane. This constant-radius circle shows the refractive index is always n_o in this case and the crystal does not affect the light's polarization.

Light Propagating Parallel to the Optic Axis
(Example: Optical Windows)

Light propagating along the optic axis (Z-axis) of a birefringent material always has a refractive index equal to n_o. This is a defining trait of the optic axis.

The left side of Figure 9 illustrates the electric field components of light propagating along the crystal's optic axis. These polarization state components can include E_x and E_y, but not E_z. In other words, the electric field vector is orthogonal to the Z-axis, so the angle φ equals 90°. The relative magnitude of E_x and E_y depends on angle α.

The right side of Figure 9 uses an ellipsoidal volume, the so-called index ellipsoid, to map all possible refractive indices provided by the material for all combinations of light propagation directions and polarization orientations. In the case of this material, the maximum possible refractive index is n_e and the minimum is n_o. The opposite can be true for other materials. The red circle drawn on the ellipsoid is a conic section that marks the refractive indices that light propagating along the optic axis will experience. The refractive index of both E_x and E_y components equals n_o. Due to this, E has a refractive index of n_o, regardless of the orientation of the E vector in the XY-plane. Therefore, the E_x and E_y components travel with the same phase velocity, and the polarization state stays the same as the light propagates.

Optical windows made from uniaxial crystals, such as sapphire (for example Item # WG31050), are typically cut or polished so that normally incident light propagates parallel to the optic axis. This is done so that the window has no effect on the polarization state of the transmitted light, as described above. When the end faces are orthogonal to the optic axis like this, the crystal is often described as Z-cut or C-cut.

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Figure 10: The diagram on the left indicates the incident light's propagation direction (k) and polarization state orientation (E), which can be decomposed into orthogonal components E_x and E_z. The propagation direction is aligned to the Y-axis and is orthogonal (θ = 90°) to the optic axis (c) of the crystal. The diagram on the right is a refractive index ellipsoid, with conic sections (red ellipses) drawn in the XZ- and XY-planes. The refractive index (n_e) of the extraordinary component (E_e = E_z) is indicated by the intersection of the Z-axis and ellipse, while the refractive index (n_o) of the ordinary component (E_o = E_x) is indicated by the intersection of the X-axis and the ellipse. The angle φ determines the two components' relative magnitudes.

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Figure 11: The diagram on the left indicates the incident light's propagation direction (k), which is aligned to the Y-axis, which is orthogonal (θ = 90°) to the optic axis (c) of the crystal. Since φ is 0°, the only polarization component is E_z, which is also the extraordinary component. The diagram on the right indicates this component has a refractive index of n_e.

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Figure 12: The diagram on the left indicates the incident light's propagation direction (k), which is aligned to the Y-axis, which is orthogonal (θ = 90°) to the optic axis (c) of the crystal. Since φ is 90°, the only polarization component is E_^x, which is also the ordinary component. The figure on the right indicates this component has a refractive index of n_o.

Light Propagating Perpendicular to the Optic Axis
(Example: Wave Plate)

When linearly polarized light incident on a birefringent material propagates in a direction orthogonal to the optic axis (θ = 90°), the light may have components polarized both parallel and perpendicular to the optic axis (Z-axis). This is illustrated in Figure 10, in which the light’s propagation direction is aligned to the Y-axis for convenience. The refractive index of the electric field component parallel to the optic axis (E_z) is n_e, and the component perpendicular to the optic axis (E_x) is n_o. This is the largest refractive index difference that can exist between two polarization components in the material.

The vertical, red circle on the right side of Figure 10 is the conic section of the index ellipsoid in a plane perpendicular to the k-vector. The horizontal, red elliptical conic section indicates a difference between the refractive indices of the orthogonal principal polarization axes of and .

Due to different refractive indices, the two polarization components travel at different velocities as they propagate through the material. This causes a phase shift between the two polarization states E_z and E_x.

Wave plates are often fabricated from uniaxial birefringent materials, with the optic polished and mounted so that normally incident light propagates orthogonal to the optic axis. Markings on the wave plate housing, such as the ones on Item # WPQ05M-266, or flat features on the perimeter of the optic, identify the orientations of the ordinary and extraordinary axes, or for a typical waveplate made of crystal quartz the so-called fast and slow axes, respectively.

Light polarized parallel to the fast axis travels with a faster velocity in the material due to the lower refractive index, compared with light polarized parallel to the slow axis. Quartz, a commonly used waveplate material, is a positively uniaxial crystal with a smaller ordinary refractive index compared to its extraordinary index. For quartz, the fast axis is the ordinary axis.

Tuning a wave plate requires rotating the optic in the XZ-plane, around the Y-axis, so that the light's propagation direction is maintained perpendicular to the optic axis. This rotation changes the angle φ, while keeping θ = 90°.

When the angle φ is between 0° and 90°, the linear polarization state has both orthogonal E_x and E_z components, and the wave plate will delay one component relative to the other. However, note that since θ is unchanged, there is no change in the indices of refraction of the two principal orthogonal polarization orientations along and .

When φ = 0°, as illustrated in Figure 11, the light's polarization state is oriented along the optic axis. This provides an extraordinary component (E_z = E_e) without an ordinary component. This component has a refractive index equal to n_e. Since there is only one polarization component, the wave plate does not affect the polarization state.

The polarization state is similarly unaffected when the linear polarization state is aligned with the X-axis (φ = 90°), as seen in Figure 12. In this case, there is an ordinary component (E_x = E_o) without an extraordinary component and the refractive index is n_o.

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Figure 13: In the diagram on the left, the propagation direction (k) of the incident light and the optic axis (c) of the uniaxial birefringent crystal are in the same plane (XZ plane). The angle (θ) between k and c is arbitrary. The input fundamental field, represented by the red vector, is parallel to the Y-axis. The SHG light is polarized perpendicularly, represented by the green vector in the XZ plane. Rather than decompose the SHG polarization into components E_x and E_z, the light is typically described as an extraordinary wave (E'_e) with a refractive index, n'_e. In the diagram on the right, n'_e, which depends on θ, is indicated by the intersection of the conic section illustrated by the red ellipse with the XZ plane.

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Figure 15: How θ and γ are defined in a laboratory setup.

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Figure 14: In the diagram above, the propagation direction (k) of the incident light and the optic axis (c) of the uniaxial birefringent crystal are in the same plane (XZ-plane). The angle (θ) between k and c is arbitrary. In this case, the incident fundamental field (E), represented by the red vector, is rotated off of the laboratory reference frame by an angle γ, and thus only a portion of the fundamental (E_oγ) is converted by SHG into the resulting polarization E'_e, represented by the green vector.

Light Propagating at an Arbitrary Angle
(Example: Nonlinear Crystals)

In uniaxial crystals, the field polarizations are arranged as in Figure 13 for Type-I second harmonic generation (SHG). The k-vector is at an angle θ from the optic axis, , and in the XZ plane. The input fundamental electric field (E_o) is polarized parallel to one of the principal ordinary axes, in this case the Y-axis, and has a refractive index of n_o. The index of refraction for light polarized in the ordinary direction, parallel to , is independent of changes to θ.

The SHG light is polarized in the XZ plane and is orthogonal to both the E_o and the k-vector. Rather than decompose the polarization of the SHG light into E_x and E_z components, the electric field is typically described as an extraordinary wave (E'_e). The refractive index for this extraordinary wave, n'_e, changes as a function of θ. n'_e(θ) can be calculated using the equation in Figure 13. The ellipse defining the extraordinary index of refraction as a function of θ is a conic section of the index ellipsoid. The two extreme cases are 1) θ = 0°, where the field is polarized along the other principal ordinary axis, , and the index of refraction is n_o, or 2), θ = 90°, where the field is polarized along the principal extraordinary axis, , and the index of refraction is n_e. The prime (') notation is used to denote index of refraction for cases in between the two extreme cases of θ.

Figure 14 shows a departure from an ideal polarization orientation for Type-I SHG, where the incident fundamental field has a linear polarization but is rotated by a degree γ from the principal ordinary axis . The fundamental field now driving the Type-I SHG process is the E_oγ component of the incident field. At γ = 90°, there would be zero SHG output, since the drive field E_oγ = 0.

In practice, γ is nonzero when the polarization states of light in a laboratory reference frame are not matched to the intrinsic principal axes in the crystal frame. To orient the two reference frames together, e.g. the vertically polarized light from the laser on the optical table is parallel to in the crystal, we mount the nonlinear crystal in a rotation stage to finely adjust γ to zero, first by eye using the markings on the housing and then further, if necessary, using the SHG output power as feedback. Figure 15 shows the orientation of γ in a laboratory frame rotation of the crystal. In this example case, the nonlinear crystal is mounted in an RSP1 rotation mount on top of an RP01 rotation stage. Changes in γ and θ adjust the orientation of the optic axis, . In this case, adjustments in γ have been used to align with the horizontal plane and one of the principle ordinary axes vertically. For a vertically polarized input beam, this orientation allows the rotation stage to be used to adjust θ.

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Figure 17: Type-I phase matching can be used with a negative uniaxial crystal like β-BBO. This requires the fundamental frequency (ω) to be polarized perpendicular to the plane that includes the light's propagation vector (k) and the crystal's optic axis (c). The second harmonic light is then polarized parallel to the plane. Angle tuning adjusts θ and is performed by rotating the crystal around the Y-axis.

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Figure 16: Without phase matching the intensity of the generated second harmonic light, I(2ω), remains below a low threshold. This intensity can increase exponentially when phase matching is performed.

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Figure 19: The refractive index (n_e') of the extraordinary wave can be varied between n_e and n_o by tuning the angle (θ) between the crystal's optic axis and the propagation direction of the input ordinary wave. At an angle of 23.37°, fundamental light at 1030 nm is phase matched with SHG light at 515 nm.

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Figure 18: The ordinary and extraordinary refractive indices of β-BBO vary with frequency and wavelength due to a material property called dispersion. Dispersion exists in all media except vacuum.

Wave Type	Angle	Refractive Index
Ordinary	k ⊥	no
Extraordinary	θ = 0°	no
	0° < θ < 90°	ne'(θ )
	θ = 90°	ne

Phase matching significantly improves the conversion efficiency and can be required to obtain sufficient frequency-doubled light for applications, but second harmonic generation (SHG) will occur even if phase matching is not performed. The effect of phase matching on the output intensity of SHG light (I_2ω) is illustrated in Figure 16. When phase matching is not performed, the intensity of the SHG light alternately increases and decreases, but generally remains at a low level. In contrast, the intensity of phase-matched SHG light can increase exponentially with propagation distance through the material. Phase matching is performed by appropriately orienting the input fundamental frequency (ω) light with respect to the SHG crystal (Figure 17).

With ideal phase matching, the fundamental (ω) and second harmonic (2ω) frequencies travel at the same phase velocity, so that all of the second harmonic light created along the beam path adds constructively. The phase velocity is the rate at which the field carrier, i.e. the sine wave, travels in a material. The rate at which a pulse envelope travels in a material is called the group velocity and defines the speed of the energy packet comprising the optical pulse traveling through the material.

The two frequencies' phase velocities are equal when they have the same refractive index. The refractive indices for these two frequencies are generally not equal, because all media except vacuum are dispersive. This means that a material's refractive index depends on the light's frequency. As an example, β-BBO's principal ordinary and extraordinary refractive indices are graphed, in Figure 17, over a frequency range that extends from the visible into the IR. Over this range, the refractive index increases for higher frequencies.

Several different phase-matching approaches can be used to equalize the refractive indices of the fundamental and second harmonic light. One method, which uses linearly polarized light and the difference between the ordinary and extraordinary refractive indices, is Type-I critical phase matching. This is a common approach for negative uniaxial crystals in which n_o > n_e.

Type-I Phase Matching
The fundamental-frequency input beam and the generated second-harmonic-frequency output beam orientations for Type-I phase matching are illustrated in Figure 17. This is the so-called oo-e arrangement, as the two photons from the fundamental are polarization-oriented along the ordinary axis and the second harmonic photon's polarization is in the extraordinary plane.

The linearly polarized fundamental-frequency wave is aligned to the crystal so that it is input as the ordinary wave. This wave is polarized along the ordinary refractive index axis (Y-axis), which is perpendicular to the plane that includes both the light's propagation vector (k) and the crystal's optic axis (c). The second harmonic frequency wave is generated as an extraordinary wave, which is polarized orthogonal to the fundamental-frequency ordinary wave.

Rotating the crystal around the Y-axis tunes the angle (θ) between the propagation vector and the crystal's optic axis. This varies the extraordinary wave's refractive index (n'_e(θ)) between the values of n_e and n_o, as illustrated in Figure 19. The red curve corresponds to n'_e(90°) = n_e, while the blue curve plots the values for n_o. The position of the green curve depends on θ. In this case, the angle θ has been tuned to provide a refractive index for 515 nm light that matches the refractive index for 1030 nm light. This tuning angle is therefore optimized for Type-I SHG phase matching and a fundamental-frequency input beam of 1030 nm.

Some manufacturers' SHG crystals, including Thorlabs', provide near-phase-matched conditions for normally incident input light for a specified fundamental frequency. These manufacturers polish the faces of the SHG crystals such that the angle between the normally incident light and the optic axis is the phase-matched angle for the specified frequency. The phase matching angle can be adjusted by rotating the crystal around the y axis. An adjustment that is typically within a few degrees can either optimize the performance of the crystal in the setup or match another fundamental frequency in the crystal's operating range.

Other phase matching conditions are also possible. Critical Type-II phase matching uses an oe-o or eo-e for positive and negative uniaxial crystals. Non-critical phase matching uses crystal temperature to achieve phase matching rather than angular tuning. In quasi-phase matching, the crystal material is “poled” to have up and down domains to counteract the phase-mismatch of the material, where the poling period is specific to the wavelengths of interaction.

When choosing an SHG crystal, the pulse duration of the laser source must be taken into consideration. A mismatch between the duration and crystal thickness will lead to SHG pulses longer in time and narrower in spectral content than possible under ideal conditions.

The issue that arises when using ultrashort pulses for second harmonic generation (SHG) is that shorter pulses have larger bandwidths, but the ideal phase-matched condition can be achieved for only a single fundamental wavelength at a time. Due to this, the quality of the phase match can be significantly worse towards the spectral edges of broad-bandwidth pulses. A second concern is that the group velocity mismatch between the second harmonic and fundamental-frequency pulses will result in one pulse lagging the other as the two propagate. This is called temporal walk-off and will occur even under optimal phase-matched conditions.

Both broad-bandwidth pulses and temporal walk-off cause a reduction in the conversion efficiency, as well as worsen the optical quality of the second harmonic pulses. The appropriate crystal thickness should be chosen to minimize both effects.

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Figure 20: On the left, the fundamental input pulse (red) has a spectral width narrower than the reference pulse spectral width (dotted black line). Therefore, the second harmonic pulse spectrum, shown on the right, has a spectral width approximately the same as the ideal case. Ideally, every frequency in the fundamental pulse spectrum would be simultaneously phase matched.

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Figure 21: On the left, the fundamental input pulse (red) has a spectral width wider than the reference pulse spectral width (dotted black line). Therefore, the second harmonic pulse spectrum (blue), shown on the right, has a spectral width smaller than in the ideal case. Ideally, every frequency in the fundamental pulse spectrum would be simultaneously phase matched.

Phase Matching Bandwidth Limitations
As discussed in another section, Is phase matching (angle tuning) necessary for SHG?, the angle between the light's propagation direction and the crystal's optic axis can be tuned so that the refractive index of a particular frequency (ω) in the fundamental input pulse exactly equals the refractive index of twice the input frequency (2ω). It is typical to optimize phase matching conditions for the frequency at approximately the center of the pulse's bandwidth. Since the refractive index does not vary linearly with frequency in dispersive crystals like β-BBO, it is not possible to optimally phase match every frequency in the input pulse's spectral range with the corresponding second harmonic frequency in the generated pulse.

Assuming a crystal of a given thickness, it is possible to define a reference pulse with a spectral range that is the maximum that can be adequately phase matched. This spectral bandwidth limitation can also be thought of in terms of an SHG spectral filter whose passband depends on both the specific crystal and the pulse's center frequency. Frequencies outside of this filter’s passband will be rejected from undergoing the SHG process.

When an input pulse's spectral range is equal to or narrower than the reference pulse, the generated second harmonic pulse has a duration that is shorter by a factor of 1/√2 and a spectral width that is broader by a factor of √2. Physically, when the fundamental spectrum is well within the reference pulse spectrum, the SHG pulse intensity profile is the square of the fundamental intensity profile, effectively truncating the duration. This case is illustrated in Figure 20.

When the input pulse has a spectral range that exceeds the width of the reference pulse (Figure 21), the second harmonic pulse spectrum (blue curve) is narrower than it would be if all frequencies in the input pulse were phase matched (gray curve). The spectrally narrowed SHG pulse spectrum is said to be apodized, since an effect of the fundamental pulse exceeding the maximum input bandwidth limit is the removal of the highest and lowest frequencies from the SHG pulse spectrum.

The length of the SHG crystal significantly affects the spectral width of the reference pulse, or alternatively the bandwidth of the SHG filter. As illustrated in the following table, the shorter the crystal, the wider the reference pulse spectrum. The table below assumes a fundamental center wavelength in the near-IR.

Temporal Walk-Off
There are two velocities that define the propagation of light in optical media: the phase and group velocities. The phase velocity is the speed of the carrier, the oscillation of the electric field, and the velocity that is matched between the fundamental and second harmonic frequencies for phase matching. The phase velocity relative to the speed of light in a vacuum is defined by the index of refraction.

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Figure 22: The diagram at the top shows two pulses with different center frequencies incident on a material. The group velocity delay between the pulses when they enter the material is zero. The four lower figures illustrate that the group velocity delay (temporal walk-off) increases as the pulses propagate through the material. The temporal walk-off is illustrated for distances Z₀ through Z₃, which are indicated on the diagram at top.

The group velocity is the speed of the pulse, or more formally the envelope of the electric field. The group index similarly governs the speed of the envelope relative to the speed of light in a vacuum and is dependent on the dispersion of the material. The group velocity and the phase velocity are not equal, meaning the sine wave of the carrier and the envelope of the pulse move through optical media at different speeds.

The group velocities of the fundamental and second harmonic pulses are also different, even when phase-matched, and the magnitude of the group velocity mismatch depends on the crystal properties and the optical frequencies in the pulses. In some cases, the mismatch can be small. For example, when a β-BBO crystal is used to frequency double 1550 nm light and provide 775 nm light, the group velocity mismatch is only a few femtoseconds per millimeter.

However, in many cases, the difference in the fundamental and second harmonic pulse velocities can cause one pulse to significantly lag the other. The temporal walk-off of the slower pulse with respect to the faster can be a substantial fraction of the pulse duration. This is illustrated in Figure 22.

The temporal walk-off is often considered severe when the temporal lag exceeds the duration of the fundamental pulse. Severe walk-off can result in temporal lag that irreparably increases the duration of the output second harmonic pulse. Since the dynamics of this process are highly subject to experimental parameters, it is difficult to arbitrarily predict the precise effects on the second harmonic pulse. These effects can include back conversion, in which second harmonic light is converted back to fundamental light, and spectral narrowing of the SHG pulse.

Choosing Crystal Thickness
The length of the crystal is an important parameter to consider when optimizing the SHG process for the parameters of particular input pulses. Absent phase-matching and temporal walk-off effects, the SHG output increases exponentially with crystal length. The goal when picking an SHG crystal is to balance the crystal length with the limitations determined by the fundamental pulse. Decreasing the crystal thickness reduces effects related to both phase-matching bandwidth limitations and temporal walk-off, which are directly related. For example, consider a fundamental pulse propagating over a given crystal thickness. When the temporal walk-off is greater than the duration of the fundamental pulse, this pulse will also have a spectral bandwidth that greatly exceeds the spectral width of the reference pulse. A general rule for limiting the detrimental effects related to both physical effects is to choose a crystal thickness that limits the fundamental and second harmonic temporal walk-off to no more than the duration of the fundamental pulse.

It is typically necessary to focus the incident light in the crystal for SHG applications, since efficient generation of second harmonic light requires high peak intensities, such as 10⁹ W/cm². When using femtosecond pulses with moderate energies in the nanojoule range, obtaining the required peak intensities can require small focal spots. A recommended guideline for focusing femtosecond laser pulses into a nonlinear crystal is to maintain a Rayleigh range that is at least several times longer than the length of the SHG crystal. This minimizes the chance of damaging the crystal, as well as reduces the negative effects of spatial phase mismatch and spatial (Poynting vector) walk-off.

Tiberius Pulse Parameters
Wavelength		800 nm
FWHM Duration	T	140 fs
Energy	E	30 nJ
β-BBO Crystal
Thickness	d	0.6 mm
Focused Pulse Parameters
Mode Field Diametera	MFD	13.5 µm
Effective Areab		7.16 x 10-7 cm2
Peak Power		201 kW
Peak Irradiancec		281 GW/cm2

• 1/e² Full Width of a Gaussian Spatial Mode
• Effective Area of a Gaussian Spatial Mode as Defined by ISO 21254
• Assumes a pulse with a Gaussian profile.

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Figure 23: This diagram illustrates a beam's focal region in a uniaxial birefringent material. Straight lines, representing rays, are drawn through the center of the beam waist. As the beam diverges from the beam waist, the energy in the beam is distributed over a larger cross sectional area. The crystal axis is c. The chief ray is parallel to the direction used to measure the tuning angle (θ), which is optimized during phase matching. Conversion efficiency is optimized for rays parallel to the chief ray and worse for rays at larger angles (ψ) to the chief ray.

Rayleigh Length vs. Crystal Thickness
When laser light is focused to a spot, the diameter of the beam waist is the focal spot size, and the rate at which the light diverges from the beam waist is related to the Rayleigh range (z_R). The Rayleigh range quantifies the distance, measured from the beam waist, over which the beam diameter is approximately constant. The shorter the Rayleigh range, the larger the rate of divergence.

When the Rayleigh range is less than the crystal thickness, the distance over which the beam is optimally intense is less than the crystal thickness. The efficiency of the SHG process decreases substantially as the fundamental beam diverges from its focus, since the pulse energy is distributed across a larger beam cross section.

For femtosecond pulses the phase-matching/temporal walk-off limitations will set the crystal length to be relatively short (sub-mm to a few-mm). As will be discussed below, there is a focusing numerical aperture (NA) limitation to the SHG phase-matching process exactly analogous to the spectral phase matching limitation discussed in the previous section (Why is short pulse duration a concern for SHG? Phase Matching Bandwidth Limitations). This spatial focusing constraint on the SHG process will generally result in the crystal length always being less than the Rayleigh range of the input focused pulse.

Crystal Damage
There is the danger of exceeding the laser-induced damage threshold (LIDT) when ultrafast pulses are focused to small spot sizes. As an example, consider femtosecond pulses from Thorlabs' Tiberius laser (see table to the right). Selecting the crystal thickness based on the pulse duration to balance maximum length with phase-matching/temporal walk-off limitations, the 0.6 mm crystal would be recommended for the 140 fs pulse at 800 nm. If the focal spot were chosen such that the 2z_R = Crystal Length, then the focal spot size would be 13.5 μm. Using the equations provided in the table, the 30 nJ input Gaussian pulse is estimated to provide the notably high peak intensity (irradiance) of 281 GW/cm² on the crystal's input face.

Spatial Phase Matching
The Type-I phase matching process relies on matching the ordinary and extraordinary indices of refraction for the fundamental and second-harmonic frequencies, which is achieved through precise orientation of the crystal optic axis and input propagation vector. A focused laser pulse, or beam, has angular content, formally called spatial frequencies, to produce the small focal spot. We use Numerical Aperture (NA) to describe the extent of this angular content. Ideally, the fundamental laser light would be collimated throughout the crystal, since collimated light resembles a group of rays aligned to the same direction with an NA of zero.

From a ray-model perspective, the propagation direction defines the orientation of the central (chief) ray in Figure 23. The SHG process is affected by the range of angles over which incident light propagates, because light that is not parallel to the chief ray has reduced phase-matching relative to the chief ray. Instead of the reference pulse power spectrum comprising an apodization filter, in the spatial domain the filter would be in NA. The angular domain apodization filter however, is only in one-dimension defined by the plane k and .

The angular content of a focused beam can be visualized by tracing rays from the lens to the focal spot. Each ray drawn from the lens to a tightly focused spot (Figure 22) makes a different angle (ψ) with the chief ray, and the range of angles increases with the NA. As the range of angles increases beyond the reference NA filter (or angular domain apodization filter width), the phase-matching of the rays outside of the filter width diminishes therefore reducing the SHG conversion efficiency of those rays. The angular content of the resultant SHG pulse, or beam, in the dimension of the angular filter is reduced from an ideal case. In the perpendicular dimension, the SHG beam preserves the angular content of the input beam, since there’s no angular filter in that dimension. A diverging beam with asymmetric NA’s, i.e. in the horizontal and vertical dimensions, leads to an elliptical beam.

Spatial (Poynting Vector) Walk-Off
As the second harmonic light propagates, its energy flows in a different direction than that of the fundamental light. The spatial walk-off angle depends on the size of the beam and the crystal length. While the angle can be significant, it is usually small, typically on the order of 10 mrad, and can be neglected.

In the case of Type-I phase matching and β-BBO crystals, the fundamental light propagates as an extraordinary wave in the crystal, while the second harmonic light propagates as an ordinary wave. The energy in both waves propagates in directions determined by their Poynting vectors. Snell's law does not describe the direction of energy flow, but it can be used to calculate the direction of the k-vector in the crystal.