相関光子対光源

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Figure 7.1  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 7.2  This nonlinear optical medium supports second harmonic generation and the input light is the same as in Figure 7.1. In addition to providing fundamental-frequency electric fields, as in Figure 7.1, 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 field of 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 7.1, 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 7.2, in which the light is incident on a nonlinear optical material. While the incident light is the same as in Figure 7.1, 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 7.3  Conservation of momentum in non-collinear spontaneous parametric down conversion requires an angular deviation of the signal (k_s) and idler (k_i) photons to compensate for the larger magnitude of the sum of their momenta. 

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Figure 7.4  The signal and idler photons emitted during spontaneous parametric down conversion form either a cone (type-I SPDC) or a pair of cones (type-II SPDC). The two intersection lines between the signal and idler emission cones in type-II SPDC (labeled k_s and k_i, respectively) are the wavevectors that correspond to polarization entangled photons. The optical axis (OA) of the crystal can be changed by rotating the crystal, altering the phase matching conditions. This diagram shows noncollinear SPDC emission, but collinear emission with a gaussian profile is also possible.

Spontaneous Parametric Down Conversion
Spontaneous parametric down conversion (SPDC) is the reverse process to sum frequency generation (SFG). One common example of degenerate SFG, called second harmonic generation (SHG), is used to generate 532 nm wavelength photons from a 1064 nm pump laser. Unlike SFG, however, SPDC cannot be explained by classical fields. The SPDC process is called spontaneous (as opposed to stimulated) because the photon pair is created without the presence of an external electromagnetic field at the signal wavelength. Classical theories only allow for the generation of an idler and signal beam if a weak signal beam already exists, a process called parametric amplification. Therefore, the existence of the SPDC process can only be explained with quantum mechanics.

Conservation of Energy and Momentum in SPDC
Since energy and momentum are conserved in the SPDC process, the energy and momentum of the photon being down converted must be equal to the energy and momentum of the pair of output photons. The implication of this conservation is that experimental parameters must be set up to allow for conservation of energy and momentum during SPDC or else the process cannot proceed. The following example demonstrates how this conservation determines experimental parameters such as the photon output angles, polarization states of the output photons, and the nonlinear crystal cut angle. 

In general, the output idler and signal photons will have propagation directions that fall on the surface a cone (type-I SPDC) or a pair of cones (type-II SPDC), as shown in Figure 7.4. These conical emission patterns can be observed as annuluses when viewed on a plane. Conservation of momentum requires that the sum of the wave vectors for these resulting output photons be equal to that of the input pump photon.

To simplify the calculations, let's look at the collinear case of SPDC, where the pump, signal, and idler photons are all propagating in the same direction. In this case, both the energy and momentum conservation relations simplify to scalar equations:

and

Here, ω represents the angular frequency of the photon and k the magnitude of the wave vector. The indices p, s, and i stand for pump, signal, and idler, respectively. The frequencies and wavevectors of each of these three photons are not independent but linked via the expression

where n(ω_p) is the frequency-dependent index of refraction of the crystal at the pump frequency ω_p, and c is the speed of light in vacuum. The signal and idler photons have analogous relations which together allow for the conservation of momentum expression to be rewritten as:

Degenerate SPDC
For the further simplifying case of degenerate SPDC, we can assume the wavelength (and thus frequency ω) of the signal and idler photons are the same and thus the previous expression simplifies to:

where (ω_s,i) is the frequency of the degenerate signal and idler photons. Since ω_p = 2 (ω_s,i) for energy conservation, this demonstrates that we need the same index of refraction for both the short-wavelength pump and long-wavelength signal/idler photons, i.e. n(ω_p) = n(ω_s,i). This can be accomplished through the use of birefringent crystals, which have different indices of refraction for different polarizations of light. The crystal orientation can be rotated with respect to the input polarization of the pump beam to create the appropriate phase matching conditions to conserve energy and momentum and allow for spontaneous parametric down conversion.

Categories of SPDC
Stepping back from the simple case above, SPDC can be broken down into three categories all dealing with how the process is phase matched: collinear vs non-collinear wavevector orientations, type-0 versus type-I versus type-II polarization orientations, and degenerate versus non-degenerate signal and idler wavelengths. Collinear phase matching simply means the wavevectors of the pump, signal, and idler are parallel. In type-0 SPDC, the pump is linearly polarized and parallel to both the co-polarized signal and idler photons. In type-I SPDC, the pump is linearly polarized and orthogonal to both the co-polarized signal and idler photons. In type-II SPDC, the signal and idler have polarizations perpendicular to each other, with one being parallel to the pump polarization, as shown in the lower panel of Figure 7.4. In degenerate SPDC, the signal and idler have an identical wavelength (and energy) at twice the wavelength (and energy) of the pump. However, in non-degenerate SPDC, the signal and idler can be of different wavelengths while having energies that sum to the energy of the pump. The difference in wavelength of the signal and idler also changes the particulars of the phase matching as the refractive index depends on wavelength.

Additional discussion on the SPDC categories and tuning between them is found below in the How is the SPDC process tuned in homogeneous nonlinear crystals? section.

Only optical materials which have a non-zero χ⁽²⁾ parameter support SPDC, and not all crystals fulfill this requirement. An example of a crystal that can be used for SPDC 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 SPDC.

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Figure 7.6  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 7.5  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 7.8  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 7.7  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 7.5. 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 7.6 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 7.6, 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 SPDC, 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 7.6 includes only even powers of x.

Non-Centrosymmetric Crystal
An example of a non-centrosymmetric material is shown in Figure 7.7. 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 7.8). 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 SPDC.

Uniaxial Crystal
A secondary requirement for efficient production of light through nonlinear processes, which will be discussed in a section below, is the so-called phase matching condition. At a high level, phase matching is the conservation of momentum between the input and output photons of the nonlinear process. 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 7.9  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 7.9 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 7.9 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 7.10  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 7.11  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 7.12  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 7.10, 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 7.10 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 7.11, 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 7.12. 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 7.13  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 7.15  How θ and γ are defined in a laboratory setup.

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Figure 7.14  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)

This is a case study using type-I second harmonic generation (SHG) as an example of an application of birefringent crystals in nonlinear optics. SHG is the reverse case of degenerate SPDC, so this treatment will be instructive.

In uniaxial crystals, the field polarizations are arranged as in Figure 7.13 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 7.13. 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 7.14 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 7.15 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 7.15  The signal and idler photons emitted during noncollinear spontaneous parametric down conversion form either a cone (type-I SPDC) or a pair of cones (type-II SPDC). The two intersection lines between the signal and idler emission cones in type-II SPDC (labeled k_s and k_i, respectively) are the wavevectors that can correspond to polarization entangled photons. This type-II SPDC diagram shows noncollinear SPDC emission, but collinear emission with a gaussian profile is also possible.

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Figure 7.16  For SPDC, it is necessary for the pump and signal/idler photons to experience the same index of refraction despite their different wavelengths. Birefringent materials like β-BBO allow for this by having different indices of refraction for two crystal axes so that photons with different polarizations and different wavelengths can be phase matched. 

Tuning Type-I, Degenerate, Non-Collinear SPDC
In the following example, degenerate non-collinear spontaneous parametric down conversion (SPDC) with a 405 nm pump is used to demonstrate how tuning is performed. Using the same logic, degenerate and non-degenerate SPDC tuning can be achieved for any pump wavelength. This example, where the degenerate signal and idler have the same polarization, perpendicular to that of the pump, is called type-I SPDC, as shown in the upper panel of Figure 7.15. In contrast, for type-II SPDC, the signal and idler have polarizations perpendicular to each other, with one being parallel to the pump polarization, as shown in the lower panel of Figure 7.15.

As discussed previously, for SPDC to occur, the index of refraction within the nonlinear crystal for the pump at 405 nm must be the same as the index of refraction for both the signal and idler at 810 nm. The refractive index as a function of wavelength for β-BBO is shown in Figure 7.16. The red solid curve indicates the refractive index for the ordinary beam (n_o). The black dashed curve is the refractive index for the extraordinary beam (n_e) if the input beam is orthogonal to the crystal plane. By rotating the crystal axes relative to the wavevector of the incident pump, the refractive index curve can be moved until n_e (405 nm, pump) = n_o (810 nm, signal/idler). This is called phase matching and allows for efficient SPDC from pump photons at 405 nm to signal and idler photon pairs at about 810 nm each. In this example, the optic axis of the crystal must be at an angle of 29.2° relative to the incident 405 nm pump beam to allow for optimal phase matching. Typically, a crystal cut to a similar angle (like our NLCQ1, NLCQ2, or NLCQ3 crystals for this example) would be used to allow for near-normal incidence and maximizing of the clear aperture.

Thorlabs' β-BBO crystals are designed to provide near-phase-matched conditions for normally incident input light for a specified set of wavelengths. The faces of the β-BBO crystals are cut such that the angle between the normally incident light and the optic axis is approximately the phase-matching angle for the specified frequencies. See the Specs tab for detailed specifications for the crystals currently available. The phase matching angle can be tuned by rotating the crystal to alter the angle between the optical axis (OA) and the incident pump as shown in Figure 7.15. An adjustment of typically only a few degrees can either optimize the performance of the crystal in the setup or match another wavelength within the crystal's operating range.

Tuning Between Type-I and Type-II SPDC
Depending on the exact phase matching parameters, the signal and idler photon pairs either leave the crystal either collinearly with the same mode as the pump beam, or non-collinearly on a cone with a small opening angle. The angle of the cone and wavelengths of the signal/idler pair can be tuned by adjusting the angle of incidence to the crystal. 

For type-I SPDC (see the upper panel of Figure 7.15), in the specific example discussed above, the degenerate noncollinear signal and idler photons emit in a singular annular cone with a half angle of 2.88°. Rotating the crystal's optical axis to an angle of 28.64° changes the degenerate phase matching condition from non-collinear to collinear, collapsing the conical emission pattern to a Gaussian spatial mode collinear with the pump.

Due to the large difference in angle between type-I and type-II SPDC, it is often necessary to use crystals with different cut angles. For example, Thorlabs' NLCQ1 β-BBO crystal is cut at 29.2° to enable type-I phase matching with a 405 nm pump and degenerate 810 nm output at near-normal incidence. For type-II SPDC at the same wavelengths, Thorlabs' NLCQ4 BBO crystal cut at 41.8° would allow for phase matching at near normal incidence. 

For type-II SPDC (see the lower panel of Figure 7.15), the signal and idler have distinct noncollinear emission cones each with a half angle of 5°. The signal and idler emission cones are separated from each other by a half angle of 1.7°. This results in the two cones intersecting at two lines. For larger increased negative angular offset (larger optic axis angle relative to the pump propagation), the individual cone half angles increase but the separation is relatively un-changed. Due to the increased cone half angle, the intersection lines of the two emission cones move outwards (transverse). As the AOI is tuned towards the collinear case, the two noncollinear annular emission cones separate and then collapse back to the collinear Gaussian modes.

It is important to note that for a pump wavelength of 405 nm, the wavelength of the signal and idler photons will not always be exactly 810 nm. It is also allowed that photon pairs can be of differing wavelengths, such as 812 nm and 808 nm. In those cases, the photon with the longer wavelength (lower energy) makes a slightly larger angle with the pump beam and vice versa. This wavelength deviation is limited by how well the phase matching is fulfilled at the different wavelengths, as the efficiency of the SPDC process will drop sharply with non-optimal phase matching.

The broad phase matching criteria for these crystals allows for photon pairs to be emitted under a vast range of wavelengths and angles. This means that photons generated by SPDC are distributed over a large spatial region. When measuring with two detectors in coincidence, one must be careful that detectors are positioned in the correct location so that they are able to detect photons from the same signal/idler pair.

The Effect of Crystal Thickness on SPDC
While phase matching can be achieved with a wide variety of wavelengths, crystals, and geometries, optimizing the system for the desired output introduces many more variables that should be considered. One such factor is the thickness of the crystal. Thicker crystals can provide a higher photon pair generation rate for a given pump power as there is a longer path for the light to interact with the crystal. However, increasing thickness also blurs the wavelength-position correlation of the SPDC photons, as pairs created at different spots in the crystal have different points of origin for the cone on which they are emitted. In type-II SPDC, the birefringent material of β-BBO causes differing group velocities in the ordinary and extraordinary axes due to different indices of refraction, causing temporal walk-off between the perpendicularly polarized signal and idler photons. This effect acts to label the signal and idler photons, reducing the quality of entanglement. As this temporal walk-off is a propagation effect, it scales linearly with increasing crystal thickness.

Thicker crystals also tighten the pump bandwidth for a given set of phase matching conditions as seen in Figure 7.17. More consequentially, the thicker crystals also result in tighter angular bandwidth for the emitted signal and idler photons, see Figure 7.18. This tighter angular distribution reduces the wavelength-position correlation and can make it more challenging to isolate specific photon pairs, since in an actual experiment an iris is typically used to spatially filter the output signal and idler beams, thus passing a more restricted range of wavelengths.

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Figure 7.17  As crystal thickness increases, the pump bandwidth that is phase-matched gets narrower. In this series of calculations, a fixed output wavelength was chosen and the resulting pump wavelengths that can achieve phase matching are plotted. This theoretical data was calculated with the web tool SPDCalc.

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Figure 7.18  The angular dispersion of the emitted signal and idler photons becomes tighter with increasing crystal thickness. This theoretical data was calculated with the web tool SPDCalc.

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Figure 7.19  Quasi-phase matching is achieved in periodically poled materials by repeatedly inverting the crystal domain orientation at the length that maximizes constructive interference between the newly generated photons and the previously generated photons from earlier in the crystal. The optimal poling period depends on the crystal properties as well as the wavelengths of the pump, signal, and idler photons. In a non-phase matched crystal, the generated photons build up and then decay, while in a quasi-phase matched crystal, the generated photons build up to a local maximum, the domain is inverted and buildup occurs again to a next maximum, and the process repeats itself, leading to a continual buildup of generated photons.

Phase Matching and Quasi-Phase Matching
Different crystals used for SPDC are designed for different methods of phase matching. Traditional phase matching is performed for homogeneous birefringent crystals, which have different indices of refraction for different polarizations of light. The crystal orientation can be rotated with respect to the input polarization of the pump beam to create the appropriate phase matching conditions to conserve energy and momentum and allow for nonlinear processes like SHG, SFG, or SPDC to be optimally efficient. In quasi-phase matching, the crystal itself is engineered through periodic poling to correct for the differing phases within the material instead of by tuning the angle of incidence.

By designing a crystal such that a photon passing through will experience a periodic reversal of the crystal’s polarity, at a particular wavelength, this will allow the generation of SPDC photons to build constructively over the path length through the crystal, as illustrated in Figure 7.19.

Tuning Type-0 and Type-II, Degenerate, Collinear SPDC
This periodically poled crystal structure allows for type-0 SPDC, which was not achievable in homogenous crystals. Figure 7.20 illustrates the difference in signal and idler outputs between our periodically poled nonlinear crystals that are designed for type-0 or type-II SPDC. The case where the degenerate signal and idler have the same polarization, parallel to that of the pump, is called type-0 SPDC, as shown in the upper panel of Figure 7.20. In contrast, for type-II SPDC, the signal and idler have polarizations perpendicular to each other, with one being parallel to the pump polarization, as shown in the lower panel of Figure 7.20.

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Figure 7.20  The signal and idler photons emitted during collinear spontaneous parametric down conversion have polarization aligned either parallel to the pump photons (type-0) or with the signal perpendicular to the pump and the idler parallel to the pump (type-II).

Thorlabs' periodically poled potassium titanyl phosphate (PPKTP) crystals are designed for quasi-phase matching with the crystal temperature around 60 °C and either a pump wavelength around 405 nm (for Item #s NLCK1 and NLCK2) in order to efficiently produce degenerate SPDC photons at around 810 nm, or a pump wavelength around 775 nm (for Item #s NLCK3 and NLCK4) for SPDC at around 1550 nm.

Tuning Degeneracy in Quasi-Phase Matched SPDC Crystals
As an example, let's look at some data for the case of degenerate collinear SPDC with a 405 nm pump to demonstrate how tuning is performed. Using the same logic, degenerate and non-degenerate SPDC tuning can be achieved for any pump wavelength. It is important to note that for a pump wavelength of 405 nm, the wavelength of the signal and idler photons will not always be exactly 810 nm. It is also allowed that photon pairs can be of differing wavelengths, such as 812 nm and 808 nm, which is known as non-degenerate SPDC. The resulting wavelengths of the signal and idler photons in quasi-phase matched crystals are determined by the pump laser wavelength and the temperature of the crystal. Figure 7.21 shows the temperature dependence of the SPDC photon wavelengths for different pump wavelengths in type-0 crystals while Figure 7.22 shows this dependence for type-II crystals.

The Effect of Crystal Thickness on SPDC in Periodically Poled Crystals
As mentioned previously, longer crystals can provide a higher photon pair generation rate for a given pump power as there is a longer path for the light to interact with the crystal. However, increasing thickness also decreases the spectral phase-matching bandwidth, affecting the range over which quantum photon correlation effects can be observed. Periodically poled crystals are designed to limit group velocity walk-off between the pump, signal, and idler photons, leading to efficient SPDC.

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Figure 7.21  In our type-0 PPKTP nonlinear crystals, temperature can be adjusted to accommodate degeneracy with a different pump wavelength or to tune the SPDC away from degeneracy and produce two SPDC photons with different wavelengths.

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Figure 7.22  In our type-II PPKTP nonlinear crystals, temperature can be adjusted to accommodate degeneracy with a different pump wavelength or to tune the SPDC away from degeneracy and produce two SPDC photons with different wavelengths.