Mongodb - Find items where a property in a nested array does not contain a string

Give the following json document (below) stored in mongo, how can I find elements which meet all the below:

SearchTerm equal to a specified string
DisplayURL in the nested Ads collection which does not contain a specified string
Position in the nested Ads collection which does not contain a specified string

The query to do this is at the bottom of the page. First a bit of background.

My mongodb instance contained 3 million of these records. When I ran the query the first time, it took around 7-8 seconds to complete. I made the index:

ensureIndex({ SearchTerm:1 }, { "Ads.DisplayURL":1 })

It then took less than 0.1 seconds to run.

Here is the json document I am using:

{
  "LocalDay": "2013-11-05T00:00:00",
  "PageIndex": 1,
  "PageId": 961425400,
  "Ads": [
    {
      "Rating": "",
      "DisplayDomain": "",
      "Title": "(ACE) Top 25 at home exercises - American Council On Exercise",
      "ClickURL": "http:\/\/www.acefitness.org\/acefit\/fitness-programs-article\/2863\/Top-25-At-Home-Exercises\/",
      "Rank": 1,
      "DisplayURL": "www.test.com",
      "Description2": "",
      "UrlDomain": "",
      "Description1": "",
      "Position": "N"
    },
    {
      "Rating": "",
      "DisplayDomain": "",
      "Title": "The Ultimate Home Workout - Shape",
      "ClickURL": "http:\/\/www.shape.com\/fitness\/workouts\/ultimate-home-workout",
      "Rank": 3,
      "DisplayURL": "www.test.com",
      "Description2": "",
      "UrlDomain": "",
      "Description1": "",
      "Position": "N"
    }
  ],
  "PagesDeep": 1,
  "DBStoreTime": "2013-11-05T09:56:12.627",
  "HTML": "",
  "LocationId": 2,
  "VariantId": 2,
  "NumNatural": 10,
  "SearchTerm": "exercise at home",
  "BrowseTime": 1516,
  "ServerId": 2454,
  "Blocked": "False",
  "Page1TopAds": "",
  "EngineId": 6,
  "utcstoretime": 1383645873,
  "SearchTermId": 1643017
}

The working query:

db.extractedads.find({
"SearchTerm" : "some search"
}, {
"Ads": {
$elemMatch: {
Position: { "$regex" : "^((?!somestring).)*$" },
DisplayURL: { "$regex" : "^((?!somestring).)*$" }
}
}
})

Or in one line so that you can run in the mongodb console:

db.extractedads.find({"SearchTerm" : "somestring" }, {"Ads": { $elemMatch: { Position: { "$regex" : "^((?!somestring).)*$" }, DisplayURL: { "$regex" : "^((?!somestring).)*$" } } } })