Chapter 4 Multi-reentrant ring laser
4.4 Polarization characteristics
4.4.1 Polarization measurement of the linear cavity
To measure the gain medium’s thermal birefringence and characterize the thermal effect of the gain medium at different pump powers, a linear cavity was first established to measure the polarization profile so that the polarization change due to the non-planar cavity configuration can be excluded. The optical configuration for the linear cavity is arranged to be the same as the ring cavity as shown in Fig. 4.2 except that the gain medium is located at the cavity center. A polarizer and a power meter were used to measure the polarization ellipticity as shown in Fig. 4.18.
Figure 4.18. The system set up of the polarization ellipticity measurement in the linear cavity.
As shown in Fig. 4.19, the ratio between the lengths of the long axis and the short axis is pump power dependent. The long axis of the polarization will also rotate as the pump power changes. Considering the gain medium as a thermally induced anisotropic medium, the corresponding index of refraction can be derived from the output polarization.
z
x
Figure 4.19. The output polarization of the laser at pump powers of (a) 300 mW and (b) 2000 mW.
0 5 10 15
0 30 60 90 120 150 180
1064-nm optical power (a.u.)
Polarization angle (deg.) a
b a/b=3.5
(a)
Polarization angle (deg.) 0
50 100 150 200 250
0 30 60 90 120 150 180
1064-nm optical power (a.u.)
a/b=1.9
a b
(b)
The measurement result of the polarization status can be drawn as Fig. 4.20, where Z axis direction is defined as the thermal induced optic axis at threshold pumped power. The internal part is the polarization at low pump power, which shows linear polarization. As the pump powers arise, the polarization is more elliptical, and the major axis of the ellipse rotates.
Figure 4.20. The diagram of the output polarization characteristics at varies pump powers. The inner most linear and the outer most ring were obtained at the pump power of 0.34 W and 2 W, respectively.
Depict Fig. 4.20 as Fig. 4.21, the horizontal axis represents pumping power, the vertical axis in the left shows the ellipticity of the polarization ellipse, and the right vertical axis shows the relative retardation in the linear cavity.
Y Z
Figure 4.21. Characterization of the polarization ellipse of ellipticity at different laser diode temperatures.
At low pump power, the polarization is linear as seen from the high ellipticity, which might be due to the polarization dependent gain of the Nd:YAG. However, the polarization becomes elliptical and finally near circular as the pump power was raised.
The relationship between the ellipticity, Ep, and the phase retardation, Γ, can be expressed as the following equations,
(
4.4.1)
Ellipticity of polarization ellipse Retardation (λ)
0
300 500 700 900 1100 1300 1500 1700 1900 0
where λ0 is the free space laser wavelength, lc is the crystal length, ne and no are the indices of e-ray and o-ray for thermal birefringence, respectively.
At low pump power, the thermal index quite linearly depends on the pump power, from equation (4.4.2), the effective thermally induced index change can be obtained. Figure 4.21 also shows the equivalent retardation resulted from the thermal birefringence of the gain medium. At high pump powers, the gain medium serves as a quarter-wave plate, and becomes saturated above a pump power of 1.35W. We can also see that the gain medium had larger absorption at a pump wavelength of 807.56 nm (LD temperature = 21oC) when compared to the pump wavelength of 806.85 nm (LD temperature = 18oC). Larger absorption results the earlier retardation rise.
As shown in Fig. 4.22, the polarization also rotates as the pump power increases.
For both LD temperatures, there existed a sudden jump in rotation angle at pump powers around 1.2 and 1.6 W. They are mainly due to multi-mode generations at those power levels.
Figure 4.22. Characterization of the polarization rotation angle at different laser diode temperatures.
300 500 700 900 1100 1300 1500 1700 1900 T=18℃
T=21℃
Relative polarization rotation angle (deg.)
Pumping power (mW)
Figure 4.23. A sketch of the relationship between optic axis and beam propagation.
4.4.2 Numerical simulation
To explain the pump power dependent polarization variation, it was suggested that the polarization rotation is attributed to the thermally induced optic axis rotation [128]. In literature, the induced optical axes are attributed to be originated from the thermally induced stress in the host. The thermal stress is from the nonuniform heating, which is caused by the geometric shape of the pump light. The polarization of the pump light may also have some minor effect on the thermally induce stress because the absorption coefficients for many hosts are polarization dependent.
Though YAG is a cubic material, but the local symmetry of Nd ion in YAG matrix is dodecahedrally coordinated so there exists small polarization dependent absorption, which can induce anisotropic thermal stress.
Considering the beam propagation in the gain medium is perpendicular to the optic axis as described in Fig. 4.23, it can be shown that with different pump powers, the temperature dependent dielectric tensor of the gain medium can be described as,
)
where )a(λl is the temperature coefficient of thermally induced variation of dielectric constants, λl is the laser diode wavelength, T is the gain medium g temperature. The constant a is proportional to the thermal loading, but at high pump power, the gain medium reaches thermal saturation and the optic axis introduced by the thermal effect is no longer rotated. The state of polarization is represented by a two-component vector, while each optical element is represented by a 2×2 matrix.
The Jones matrix for a anisotropic gain medium can be described by
( )
(4.4.4)where ϕ is the rotation angle of the polarization ellipse with respect to the pumping polarization.
Figure 4.24 shows the polarization rotation angle and the ellipticity of polarization ellipse as the laser diode at a pump wavelength of 808.51 nm (LD temperature = 25oC). )a(λl can be derived from Fig. 4.24, and the value is 0.00033.
Figure 4.25 is the index difference of e-ray and o-ray in the linear cavity at which the laser diode working temperature is 25oC. The index of e-ray and o-ray was also derived from equation 4.4.1 and 4.4.2.
Figure 4.24. Characterization of the polarization ellipse at laser diode working temperatures of 25oC.
0
300 600 900 1200 1500 1800 2100
1
Ellipticity of polarization ellipse
Relative polarization rotation angle (deg)
Figure 4.25. Thermal birefringence of the Nd:YAG gain medium. The left y axis is the index from simulation, the right side y axis is the index difference from experimental data.
Figure 4.26. The 3-D schematic diagram shows the rotation of the incident plane after reflection from the cavity mirror. Points 1, 2, 3, and 4 are the laser intersections with the input and output couplers. The polarization rotates 4 times in one round trip. Beam paths 12 and 34 are not crossed.
Pumping power (mW)
Index ne, no ne-no
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
0 300 600 900 1200 1500 1800 2100
0.0E+00 5.0E-06 1.0E-05 1.5E-05 2.0E-05 2.5E-05 3.0E-05
Due to the non-planar nature of the ring cavity, the polarization of the laser beam rotates when it reflected from couplers. Figure 4.26 shows a simple geometry diagram of the non-planar ring cavity; each beam path has the same length and can be expressed as, The cavity length L is,
)
where d is half of the figure “8” width and R is the radius of curvature of the input/
output coupler.
As the beam propagates a round trip, the incident plane rotates four times. In this 2-mirror ring cavity, the rotation angle θ formed by the incident-plane and the reflective-plane can be obtained by cross product of the normal vectors of the incident and the reflected planes from the following equation,
(4.4.7)
Referring to Fig. 4.26, provided that the propagation path of laser beam in one round trip is 4-1-2-3-4; the plane formed by points 4, 1, and 2 rotates by an angle θ123
from the reflected plane formed by points 1, 2, and 3. A simple legend of coordinate transfer is shown as Fig. 4.27.
Figure 4.27. The diagram of the coordinate changed as the beam reflected by the cavity mirror each time.
The transformation matrix which represents the propagation rotation is then given by
The rotation angle formed by plane 123 and plane 234 is θ234; formed by plane 234 and plane 341 is θ341; formed by plane 341 and plane 412 is θ412, respectively.
Because the optical coating on the laser mirror, the oblique incidence in the ring cavity results in different reflectance between the s-polarization and p-polarization, and the Jones matrix can be expressed as,
)
where Rs is the s-polarization reflectance, and Rp is the p-polarization reflectance. In our simulation, we calculated the effective index which includes the laser mirror substrate and the dielectric optical coating on the mirror. The overall transfer matrix M(θ,R,Γ,ϕ) for the whole system is obtained by multiplying all the individual element matrices as equation (4.4.10), and the polarization state of the transmitted light can be derived from the eigenvectors of the ring cavity.
(
4.4.10)
4.4.3 Thermal birefringence of the gain medium
As we mentioned previously, to distinguish the thermal effect of the gain medium and the optical configuration, a linear cavity was first established to measure the polarization profile. To confirm the numerical simulation, starting from the linear cavity, we obtained the eigenvector by ABCD matrix. In the linear cavity, the transparent matrix of a round trip is W2
( )
Γ,ϕ as shown in Fig. 4.28.Figure 4.28. The beam passed the gain medium twice in a round trip for the linear cavity.
Figure 4.29 shows the comparison of experiment and numerical analysis of the ellipticity of polarization ellipse in the linear cavity, the trend is similar to each other.
In the linear laser cavity configuration, the output beam is basically TEM00 mode, but slight high order mode was observed, which is not taken into account in our simulation. This multi-mode component is more pronounced at low pump power.
From simulation, without gain medium, the non-planar ring configuration is preferred to linear polarization. That is due to the oblique incidence on the cavity mirrors for the non-planar configuration, the reflectivity is polarization dependent, which results in a polarization selection process.
Figure 4.29. The comparison of simulation and experimental data in the linear cavity.
At low pump power, the polarization state is linear due to the pump beam profile. At high pump power, the ellipticity of polarization ellipse becomes smaller due to thermal saturation in the gain medium.
0 2 4 6 8 10 12 14 16 18
300 600 900 1200 1500 1800 2100
experiment data simulation
Pumping power (mW)
Ellipticity of polarization ellipse
Figure 4.30 shows the ring cavity with gain medium. At high pump power, because the thermal saturation on the gain medium, the variation is reduced and tends to be a constant value.
Figure 4.30. The comparison between simulation and experimental data. The ellipticity of polarization ellipse in the non-planar ring configuration is around 1.3.
0 5 10 15 20
300 600 900 1200 1500 1800 2100
experiment data simulation
ring laser threshold
Ellipticity of polarization ellipse
Pumping power (mW)
It is also found that the polarization rotation angles of ring cavity are significantly reduced when compared with that of the linear cavity as shown in Fig.
4.31.
Figure 4.31. The polarization rotation angles are significantly reduced when compared with that of the linear cavity. This is mainly attributed to the thermal saturation on the gain medium and the optic axis is stationary at high pump power.
Based on Jones matrix analysis, full characterizations of the polarization status of the laser beam within the non-planar and reentrant 2-mirror ring cavity are performed in terms of measuring the thermally induced birefringence in the gain medium, analyzing the polarization rotation due to cavity configuration, and the polarization dependent reflectance of the cavity couplers. This analysis is important to the efficient utilization of the ring cavity for many applications, such as single frequency laser generation, intracavity frequency doubling, passive Q-switching, and
0 10 20 30 40 50 60 70 80 90 100
300 600 900 1200 1500 1800 2100
linear cavity ring cavity
Pumping power (mW)
Polarization rotation (deg)
mode-locking. A stable, low-noise single-frequency green laser was demonstrated using this ring cavity.
4.5 Exact solution of multi-reentrant ring cavity
As we mentioned earlier, ring laser cavities have been extensively studied and developed for many applications such as single-frequency laser [105,115, 129-130], mode-locked laser [131], laser gyro [132-133], etc. Compared to standing-wave linear cavities, they have the ability to oscillate in either or both of the two counter-propagating directions. By adding intracavity reciprocal and non-reciprocal polarization rotators unidirectional operation of the ring cavity can be obtained [123].
In addition, feedback from the pump beam is eliminated so that the laser amplitude noise is reduced. However, usually more cavity elements are needed to construct the ring laser and the cavities become bulky. Astigmatism produced by off-axis reflection from cavity mirrors must also be taken into account in the resonator design [134]. In the previous section, we presented a reentrant figure ”8” ring laser cavity configuration. Only two identical spherical mirrors were used to construct the laser cavity. The invention of the reentrant figure “8” ring cavity makes it possible to have a compact single-frequency laser suitable for intracavity frequency doubling as described in section 4.2. Although slight astigmatism can arise from the intracavity elements, the cavity itself is astigmatism-free. The output polarization is characterized in section 4.4.
In this section, the exact reentrant condition of the multi-reentrant ring cavity is derived. The stability of the laser cavity is also analyzed [143]. Utilizing this multi-reentrant cavity configuration, the cavity length can be reduced while maintaining the length of the round trip; it means that although the laser cavity length is short, the laser beam round-trip length can be long. A factor of 13 times enlargement in round-trip length is demonstrated. An example of the three-dimensional (3-D) drawing is shown in Fig. 4.32.
Figure 4.32. Example of 3-D view of the laser beam in a multi-reentrant cavity.
The intracavity beam path of the multi-reentrant laser looks like a cat’s cradle.
This configuration is useful for a mode-locked laser to reduce its cavity length. Take a mode-locked laser with 150-MHz repetition rate as an example, the cavity length is typically 1 meter, whereas it can be less than 10 cm if using the multi-reentrant cavity.
Another possibility to use the multi-reentrant laser is absorption spectroscopy. The multi-pass cell has been widely used for the detection of weak atomic or molecular absorption lines due to its long optical path combined with compact cell volume [135-138]. Using the multi-reentrant laser cavity, the detection sensitivity can be further increased because the laser output power is much more sensitive to intracavity loss than a passive cell.
Due to the non-orthogonality of the laser cavity [139-140], a planar beam path is employed to analyze the transverse-mode stability. This approximation is justified by comparing the simulation and experiment [143].
M2
M1
4.5.1 Reentrant condition
Considering a two mirror cavity with radius of curvature, R, and mirror separation, L, as shown in Fig. 4.33, it can be shown that planar and non-planar ring path can be supported with beam paths satisfying the following equation.
) output coupler and the optical axis.
It should be noted here that (4.5.1) is an exact solution instead of a paraxial approximation. When d becomes large compared to R, the discrepancy of beam path between (4.5.1) and that of using paraxial approximation becomes large [141-142].
For the planar ring, the reentrant condition obtained from paraxial approximation is just L = R. However, the beam path for L = R is actually a triangular-shaped folded cavity [118].
Figure 4.33. Multi-reentrant ring laser set up. I/C: input coupler, O/C: output coupler.
Laser diode with fiber
I/C O/C
Gain medium
Optic axis
Focus lens d0 Output
It can be proved [Appendix A] that a series of non-planar reentrant beams that satisfy the following criterion for the configuration shown in Fig. 4.35 can be supported,
) 2 . 5 . 4 ( cos
cos 2
2− 2ϕ θ
R= L
where ϕ is defined by Mπ 2N, and M andNare integers.
The beam path is, in general, non-planar and is composed of 2N arms with equal length, which do not intersect at the cavity center. It is a linear cavity when M equals 0. Ray tracing software by Stellar Software, Beam 4, was used to generate the beam path for various N and M, as shown in Table 4.1.
Table 4.1: Beam path for various N and M
N M Side View End View
2 1
1 3
2
1 4
3
1
2
3 5
4
The 3-D view of the beam path in Fig. 4.32 is actually for N = 5 and M = 2.
Figure 4.34 is a plot of the relation between d/R and L/R for M/N from 0.1 to 0.9. It can be seen that the larger the M/N ratio, the longer the cavity length. The physical meaning of N is the number of times the beam bounces back and forth between the two mirrors, i.e. the number of beam-spots at either of the two couplers; whereas the physical meaning of M is the number of times the beam circulates in one round trip when viewed from the end. This interpretation excludes the degenerate situation when there is a common dividend between N and M.
Figure 4.34. L/R versus d/R for various M/N ratio.
From Appendix A, it can easily be seen that the round trip beam propagation length, Σ, is,
For this multi-reentrant cavity, the cavity length can be short, while the round-trip length can be long. A cavity length reduction factor, Γ, can be defined as,
(
4.5.4)
In this section, stability analysis for the transverse mode is considered. For the planar cavity described in previous section [143], the effective radii of curvature in the x- and y-directions defined in Fig. 4.35 are R cosθ and Rcosθ , respectively.
The stability criterion can, be expressed as,
(
4.5.5)
where A and D are the diagonal elements of the round trip ABCD matrix [142]. As shown in Fig. 4.35, the cavity is critically stable in the x-direction and unstable in the y-direction. Consequently, the planar figure-8 configuration is not stable for the laser cavity.
Figure 4.35. Stability condition of the planar figure-8 cavity.
Where M is not equal to zero, the cavity is not orthogonal [139]. If we ignore the non-orthogonal nature of the cavity and use the ABCD matrix to analyze the stability, the round-trip ABCD matrix starting from the cavity center can be expressed as,
Figure 4.36 shows an analysis for N = 5 configurations. The cavity is more sensitive to d for smaller M, as shown in Fig. 4.36 (a), because a smaller M corresponds to a shorter cavity length. It is noteworthy that the mode sizes for M = 2 are equal to those of M = 3 as d approaches 0, and there is a similar correspondence between M = 1 and M = 4. This phenomenon can be explained if we look closely into (4.5.6) to (4.5.8). Replacing ϕ by π− will not alter the round-trip ABCD matrix ϕ when d equals to 0.
Figure 4.36. (a) Stability and (b) mode size at z = 0 of empty ring cavities for N = 5.
Figure 4.37 shows an analysis for M = 1 configurations. The cavity is more sensitive to d for larger N, as shown in Fig. 4.37 (a), because a larger N corresponds to a shorter cavity length. As shown in Fig. 4.37 (b), the spot size is inversely proportional to d only for the N = 2 case. When compared to Fig. 4.36 (b) and other simulations, a general rule can be deduced, i.e. that the spot sizes are proportional to d when M / N is less than 0.5; otherwise, they are inversely proportional to d.
Figure 4.37. (a) Stability and (b) mode size at z = 0 of empty ring cavities for M = 1.
When placing the gain medium in the cavity, the symmetry of the cavity is destroyed. To minimize the distortion, thin gain mediums with high doping of Nd3+
are used. Figure 4.38 shows the simulated result for a non-planar figure-8 cavity with various gain medium thickness. The cavity is stable only when the offset of the gain medium from the optic axis of the cavity is larger than a critical value, which is a function of N and M.
Figure 4.38. Non-planar figure-8 cavity with various effective thickness of gain medium. The effective gain medium thickness is defined by
R t n) 1 1
( − , where n and t are the index and physical thickness of the gain medium.
0.9970
0.00 0.05 0.10 0.15 0.20 0.25 0.30
teff=0.0
Considering a Nd:YAG gain medium with a thickness of 1 mm in an R = 8 cm cavity, Fig. 4.39 shows a simulation of the mode stability for various N and M. The
Considering a Nd:YAG gain medium with a thickness of 1 mm in an R = 8 cm cavity, Fig. 4.39 shows a simulation of the mode stability for various N and M. The