This is part 2 in a series. To read Part one, click here.

WALKOFF SITUATIONS

Walkoff is different in different situations and with different materials.  Second harmonic generation (SHG) and third harmonic generation (THG) can be accomplished in LBO and with other materials.  For example,In Fig. 5, we could use one LBO SHG crystal and one LBO THG crystal.  If we have beams with high peak powers then we can use short crystals.  However to boost the efficiency of the THG process, we often want to make the crystals much longer.  Often the walkoff angle in LBO is 5 to 10 mr.  The crystal BBO usually offers greater fundamental efficiency; however, walkoff in BBO is typically 5 to 10 times larger than in LBO, so BBO may actually be far less efficient than LBO unless the BBO crystal is shorter or some tricky compensation techniques are available. 

NONCRITICAL PHASE MATCHING IS A CONDITION WITH NO WALKOFF

Phase matching refers to an arrangement where beams are oriented in a nonlinear crystal so as to give good nonlinear conversion.  Without phase matching, the nonlinear interaction usually becomes uselessly small.  Phase matching is only possible with precise control of the crystal orientation and (usually) the crystal temperature too.   For SHG in LBO there are conditions where this orientation may be achieved with a condition known as noncritical phase matching (NCPM) where the beams propagate along a crystal axis.  NCPM has no walkoff!  For converting 1064 nm infrared light to 532 nm green light, a crystal temperature of about 150 degrees centigrade is necessary for NCPM.  NCPM can be a great approach; however, coatings on the LBO may be more durable if they are used at closer to room temperature, and it is usually easier to mount and control an LBO crystal at closer to room temperature where the SHG walkoff angle is typically about 6 mr.  So, NCPM has trade-offs.

CRYSTAL LENGTH MATTERS

Without the issue of walkoff, longer crystals would normally provide more efficient nonlinear processes.  However, longer crystals are harder to grow and fabricate, and in reality they do have more walkoff.  Alternatively two identical shorter crystals can be used instead of one longer one.  Let’s consider that approach for the THG process.  

Walkoff can be partially compensated between crystals

Using two crystals can be essentially equivalent to using one crystal of twice the length, as shown in the top of Fig. 6.  With two crystals instead of one, there are two extra surfaces that can cause problems if the surfaces are imperfect or contaminated in any way.  If there is Poynting Vector walkoff, the amount of walkoff is the same as with a single LBO crystal that is twice as long.  However, if the second crystal is rotated 180 degrees around the X1 axis, as shown in the bottom of Fig. 6, then the walkoff in the first crystal is reversed in the second crystal.  This is the concept of walkoff compensation using two (or more) crystals.  The compensation isn’t perfect, but the walkoff is approximately cut in half, boosting the conversion efficiency and improving beam quality.  The same walkoff compensation concepts can be applied to any configuration where some of walkoff from one crystal is reversed in a second crystal.  This can often be done, even if the two crystals are not the same.  So, walkoff in an SHG crystal can partially pre-compensate walkoff from a subsequent THG crystal.  However, if we’re not careful, the walkoff can add!  

Figure 6.  A pair of identical LBO crystals can be used, one after the other, as shown in the top view.  In this case, if there is any Poynting Vector walkoff, the amount of walkoff is twice as much in two crystals as it would be in one.  However, if the second crystal is rotated 180 degrees around the x1 axis, as illustrated below, then the walkoff in the first crystal is reversed in the second crystal.  

WALKOFF CAN BE PARTLY COMPENSATED IN JUST ONE CRYSTAL

Alternatively, walkoff can be partially compensated in just one crystal.  Remember the earlier discussion concerning Fig 4 from the first part of this description of walkoff? We noticed that the effects of angular dispersion look a lot like the effects of walkoff.  Remember also that we pondered if we could determine if a beam had been subject to dispersion or to walkoff.  It’s not obvious how to do that. The two processes produce such similar effects that we can use this similarity to our benefit--dispersion will be used to partially compensate walkoff.  First, we’ll review the problem, and then we’ll describe using dispersed (non-colinear) input beams to partially compensate for walkoff.  

Assume that the fundamental wavelength is still 1064 nm, the second harmonic is 532 nm, and the desired third harmonic UV is 355 nm.  Also, assume that the second harmonic was generated using NCPM so there was no walkoff between the IR and green—they overlap each other well.  If the THG crystal is LBO and is designed to phase match these two beams for sum frequency generation at room temperature, which is about 300 K, then the walkoff angle in the THG process is about 9 mr.  The green beam walks off from the IR and UV beams.  The IR and UV beams would normally not walk off or deviate from each other in collinear phase matching.  The green beam’s walkoff will cause substantial (~1/3 mm) separation of the green from the co-propagating IR and UV over the length of a long 40 mm LBO part.  This reduces the efficiency of the THG process.

But if the fundamental (1064 nm) and second harmonic (532 nm) beams are sent into the LBO at slightly different propagating directions, then this angular dispersion can compensate the walkoff between the fundamental and the second harmonic.  How can we get the 1064 and 532 beams to overlap spatially in the crystal but propagate at slightly different angles?  This is shown in Fig. 7.  The beams enter the crystal at non-normal incidence.  The incidence angle is chosen so the wavelength-induced angular dispersion deviates the green beam from the IR beam by exactly the right amount so that the two beams continue to overlap in the presence of walkoff.  One deviation compensates another.  In this configuration the Poynting vectors of these two beams align: S1 = S2.  

Figure 7.  Infrared and green beams overlap as they enter a THG crystal.  The wavefronts of the green and red beams are parallel in air going into the crystal.  The wavefronts are not quite parallel inside the crystal due to dispersion.  However, the walkoff and the dispersion can cancel each other!  UV light is generated in the crystal, as shown in Fig. 8.  Note that the IR, green, and UV beams will not be going in exactly the same directions when these beams exit the crystals and return to air.  This is a very unusual birefringent prism!

Figure 8.  Beams of IR and Green light propagate into LBO in a manner where the Green and IR are angularly dispersed before entering the LBO (not shown) so as to have non-parallel propagation vectors, but parallel Poynting vectors.  Hence, the IR and Green are shown overlapping with energy flow going directly to the right.  For simplicity, beam focusing is not shown, and the refraction as the UV beam exits the crystal is not shown.  The LBO is phase matched for this non-collinear third harmonic generation, and the generated UV light (shown as blue rays) still has walkoff (shown going down to the right) from the IR and Green beams.  Note that the scale in the vertical dimension is 50 times magnified compared to the scale in the horizontal direction.

With the IR and green beams tracking each other, we expect to generate UV more efficiently than if the IR and green walked off from each other.  We are still left with walkoff of the UV from both the IR and green, and this is by about 6 mr, not as bad as the uncompensated 9 mr green walkoff.  It will walk off ~1/4 mm over the 40 mm LBO crystal. 

CONCLUSIONS

Using a pair of walkoff compensated crystals can improve the conversion efficiency in nonlinear processes.  This approach will also help reduce the ellipticity of the nonlinearly generated beam.  This is especially the case for narrow beams with substantial walkoff.  Using the concepts discussed here there are other ways to minimize the effects of walkoff, increase efficiency of nonlinear processes, and improve beam quality.  These include using wide beams, short crystals, or deviating one beam with respect to another as they go into a nonlinear crystal so that walkoff is compensated by an angular shift in the wave vector of one beam.  Often this deviation is done by using angular dispersion.

DR. WILLIAM GROSSMAN, AUTHOR

Will Grossman is a consultant retained by GAMDAN, and his role is to help our customers be more successful with nonlinear optics. His technical expertise includes laser design, nonlinear optics, and laser reliability. Dr. Grossman’s laser designs are used around the world in commercial products. More on the author can be found here.