CmdPal: Transparent window (#45159)

## Summary

This PR adds:
- Backdrop material customization
  - Alongside acrylic, the following options are now available:
  - Transparent background
  - Mica background
- Background material opacity
  - Lets you control how transparent the background is

## Pictures? Pictures!

<img width="1491" height="928" alt="image"
src="https://github.com/user-attachments/assets/ff4e9e06-fcf1-4f05-bc0a-fb70dc4f39be"
/>



https://github.com/user-attachments/assets/84e83279-afab-481e-b904-f054318c5d2f

<img width="977" height="628" alt="image"
src="https://github.com/user-attachments/assets/241a228d-af3f-448a-94a6-0a282218bd8c"
/>


## PR Checklist

- [x] Closes: #44197
<!-- - [ ] Closes: #yyy (add separate lines for additional resolved
issues) -->
- [ ] **Communication:** I've discussed this with core contributors
already. If the work hasn't been agreed, this work might be rejected
- [ ] **Tests:** Added/updated and all pass
- [ ] **Localization:** All end-user-facing strings can be localized
- [ ] **Dev docs:** Added/updated
- [ ] **New binaries:** Added on the required places
- [ ] [JSON for
signing](https://github.com/microsoft/PowerToys/blob/main/.pipelines/ESRPSigning_core.json)
for new binaries
- [ ] [WXS for
installer](https://github.com/microsoft/PowerToys/blob/main/installer/PowerToysSetup/Product.wxs)
for new binaries and localization folder
- [ ] [YML for CI
pipeline](https://github.com/microsoft/PowerToys/blob/main/.pipelines/ci/templates/build-powertoys-steps.yml)
for new test projects
- [ ] [YML for signed
pipeline](https://github.com/microsoft/PowerToys/blob/main/.pipelines/release.yml)
- [ ] **Documentation updated:** If checked, please file a pull request
on [our docs
repo](https://github.com/MicrosoftDocs/windows-uwp/tree/docs/hub/powertoys)
and link it here: #xxx

<!-- Provide a more detailed description of the PR, other things fixed,
or any additional comments/features here -->
## Detailed Description of the Pull Request / Additional comments

<!-- Describe how you validated the behavior. Add automated tests
wherever possible, but list manual validation steps taken as well -->
## Validation Steps Performed

src/modules/cmdpal/Microsoft.CmdPal.UI.ViewModels/MainWindowViewModel.cs

@@ -22,6 +22,7 @@ public partial class MainWindowViewModel : ObservableObject, IDisposable
     public partial Stretch BackgroundImageStretch { get; private set; } = Stretch.Fill;
 
     [ObservableProperty]
+    [NotifyPropertyChangedFor(nameof(EffectiveImageOpacity))]
     public partial double BackgroundImageOpacity { get; private set; }
 
     [ObservableProperty]
@@ -39,6 +40,30 @@ public partial class MainWindowViewModel : ObservableObject, IDisposable
     [ObservableProperty]
     public partial bool ShowBackgroundImage { get; private set; }
 
+    [ObservableProperty]
+    [NotifyPropertyChangedFor(nameof(EffectiveBackdropStyle))]
+    [NotifyPropertyChangedFor(nameof(EffectiveImageOpacity))]
+    public partial BackdropStyle BackdropStyle { get; private set; }
+
+    [ObservableProperty]
+    [NotifyPropertyChangedFor(nameof(EffectiveBackdropStyle))]
+    [NotifyPropertyChangedFor(nameof(EffectiveImageOpacity))]
+    public partial float BackdropOpacity { get; private set; } = 1.0f;
+
+    // Returns null when no transparency needed (BlurImageControl uses this to decide source type)
+    public BackdropStyle? EffectiveBackdropStyle =>
+        BackdropStyle == BackdropStyle.Clear ||
+        BackdropStyle == BackdropStyle.Mica ||
+        BackdropOpacity < 1.0f
+            ? BackdropStyle
+            : null;
+
+    // When transparency is enabled, use square root curve so image stays visible longer as backdrop fades
+    public double EffectiveImageOpacity =>
+        EffectiveBackdropStyle is not null
+            ? BackgroundImageOpacity * Math.Sqrt(BackdropOpacity)
+            : BackgroundImageOpacity;
+
     public MainWindowViewModel(IThemeService themeService)
     {
         _themeService = themeService;
@@ -58,6 +83,9 @@ public partial class MainWindowViewModel : ObservableObject, IDisposable
             BackgroundImageTintIntensity = _themeService.Current.TintIntensity;
             BackgroundImageBlurAmount = _themeService.Current.BlurAmount;
 
+            BackdropStyle = _themeService.Current.BackdropParameters.Style;
+            BackdropOpacity = _themeService.Current.BackdropOpacity;
+
             ShowBackgroundImage = BackgroundImageSource != null;
         });
     }