Number of Mirrors

Up to this point, all derived conditions are general and independent of the number of optical elements. For EUVL tools, the choice of the number of mirrors in the design is a compromization between many factors, most notably between the power output and imaging quality. Regarding the power output, even with a perfectly manufactured mirror and coated with over 40 layer pairs of optical coating optimized for EUV, the reflection is only at maximum 70%, for normal incidence and decreases as the angle of incidence increases. [4] Assuming a six-mirror system, at the wafer side only 11.7% of the power from the mask remains, or 4.0% of the power from the EUV source assuming a simple three mirror illumination optics. Therefore considering the throughput efficiency, the

number of mirrors should be as few as possible.

On the other hand, regarding the EUVL tool design, increasing the number is advan-tageous for the purpose of aberration correction. With ever increasing demands for higher NA, the EUVL tool design will be unable to sufficiently mitigate the increased aberration associated with higher NA, if not given the necessary number of mirrors to design with.

As such, for most designs in the past decade, the compromization between power output and imaging quality are met at either four, six, or eight mirrors depending on the intended NA.

As for the even number of mirrors, this decision originates from a simple notion re-garding high volume manufacturability. Even number of reflection ensures that the mask (object) and the wafer (image) are positioned at the two opposing sides of the tool, al-lowing ample working space for the machineries at both the mask side and the wafer side.

3.2.1 Notes on the Aperture Stop Position

For an EUVL tool, the position of the aperture stop is often bound to a particular surface, and in many cases the second mirror is often chosen as the aperture stop, as in the case of Liu’s design. [39] While in general there are no laws or guidelines dictating that this must be so, optimizations with respect to the physical obstruction of the mirrors will most undoubtedly arrive at this result. For reflective optical systems, space is a precious and scarce resource since the mirrors will cause obstruction in the image if the mirror is placed in the path of the light, and care must be taken during the design to ensure that obstruction do not occur. If the aperture stop is located on a mirror surface, there will be one less element of uncertainty to consider for.

3.2.2 Mirror Pair Concept

Therefore, an EUVL tool can be decomposed into units of mirror pairs. As such, when simulating a mirror pair using the lens module, the optical properties must specified in accordance to what is possible for a mirror pair to produce.

For a mirror pair, its the effective optical power Φ is given by

Φ = φ₁+ φ2− φ1φ₂sep= C_(1,2) (75)

where φ₁and φ₂are the power of the first and second mirror respectively, and sep is the separation between the mirrors. In terms of GGC and its Gaussian bracket expansion, its back focal length is given by

BF = A_(1,2)

C_(1,2) = 1 − sepφ1

Φ , (76)

and likewise for the front focal length

FF = −D_(1,2)

C_(1,2) = −1 − sepφ2

Φ . (77)

Substitution of Equations76and77into75then gives the required mirror separation for the mirror pair

sep= EF +BF · FF

EF . (78)

Note that, this separation must be a positive length, otherwise ray tracing using this lens module would show that light is not reflected but instead travels through the mirror sur-faces instead.

3.2.3 Multiple mirror pair expansion

At this point, the number of mirrors in each subsystems must be decided before further derivation can continue. For the sake of simplicity, the simplest system configuration of one mirror pair on either side of the aperture stop forming a four-mirror system, is chosen for further derivations. However, the subsystem can also be expanded into multiple mirrors pairs.

Here, expansion of a subsystem into two and three mirror pairs, as shown in Figure53, are demonstrated. For expansion of the subsystem into two mirror pairs, simply substitute

EF

Figure 53: A subsystem can be further expanded into multiple mirror pairs, represented by the individual lens modules. Upper: Expansion into two mirror pairs. Lower: Three mirror pair expansion.

the following properties into the derivations for the subsystem

Φ_sub2= t₁− BF₁+ FF₂

For expansion into three mirror pairs, the substitutions are

Φ_sub3=(BF₁− FF₂− t₁)(BF₂− FF₃− t₂) − EF₂²

EF₁EF₂EF₃ (82)

BF_sub3= BF₃+ EF₃²(t₁− BF₁+ FF₂)

(BF₁− FF₂− t₁)(BF₂− FF₃− t₂) − EF₂² (83)

FF_sub3= FF₁− EF₁²(t₁− BF₁+ FF₂)

(BF₁− FF₂− t₁)(BF₂− FF₃− t₂) − EF₂² . (84) 3.2.4 Mask and Wafer Side Working Distance

The mask side working distance is the clearance between the mask and the closest element of the tool to the mask, and likewise for the wafer side. This allows space for other procedures and operations, such as transporting the post-exposure wafer onward to the next stage of lithography, and bringing in new unexposed wafer for exposure.

Therefore for a four mirror tool, the mask side working distance

WD_ob= t_im+ thc_A (85)

must be kept at a reasonable distance away, and similar is true for the wafer side

WD_im = t_ob+ thc_B . (86)

3.2.5 Subsystem Magnifications

As the t_ob is bound in a relationship to WD_mand thc_A, consequently so is the magnifica-tion of the subsystem, given by

1

M_a = D_(0,2)

= −t₀C_(1,2)+ D_(1,2) , (87)

and after substitution with Equation85gives

WD_ob+ sep_A= −FF_A−EF_A

M_A . (88)

Again, similar treatment of the wafer side gives

WD_im+ sep_B= BF_B− M_B· EF_B . (89)